Geometry of Carnot--Carath{e}odory Spaces, Differentiability and Coarea Formula

Geometr y of Carnot–Cara th ´ eodor y Sp a ces, Differentiability and Co area F ormula 1 Maria Karmano v a, Sergey V o dop y ano v Abstract W e giv e a simple pro of of Gromo v’s Theorem on nilp oten tiz ation of ve ctor ﬁelds, and exhibit a new metho d for obtaining quan titat iv e estimates of c omparing geometries of t wo diﬀeren t local Carnot groups in Carnot–Carath ´ eo dory s p aces with C 1 ,α -smo oth basis ve ctor ﬁ elds, α ∈ [0 , 1]. F rom here w e obtain t he similar estimates for compar- ing geometries of a Carnot–Carath ´ eod ory space and a lo cal C arnot group. Th ese t w o theorems imply b asic results of the theory: Gromov t yp e Lo cal Appro ximation Theorems, a nd for α > 0 Rashevski ˇ ı-Cho w Theorem and Ball–Bo x T heorem, etc. W e app ly the obtained r esults for pro ving hc -diﬀeren tiabilit y of mappings of Carnot–Carath ´ eo dory spaces with con tin uous horizon tal der iv ativ es. The latter is used in pro ving the coarea formula for some classes of con tact mappings of Carnot–Carath ´ eod ory spaces. Con ten ts 1 In tro duction 2 2 Geometry of Carnot–Carath ´ eo dory Spaces 10 2.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Gromov ’s T heorem on the Nilp oten tization of V ector F ields and Estimate of the Diameter of a Box . . . . . . . . . . . . . . . 22 2.3 Comparison of Geometries of T angent Cones . . . . . . . . . . 34 2.4 Comparison of Lo cal Geometries of T angen t Cones . . . . . . 39 2.5 The Appro xim ation Theorems . . . . . . . . . . . . . . . . . . 44 2.6 Comparison of Lo cal Geometries of Tw o Lo cal Carnot Groups 47 2.7 Comparison of Lo cal Geometries of a Carnot Manifold a nd a Lo cal Carnot Group . . . . . . . . . . . . . . . . . . . . . . . 48 2.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.8.1 Rashevski ˇ ı–Cho w Theorem . . . . . . . . . . . . . . . . 50 2.8.2 Comparison of metrics, and Ball– Bo x Theorem . . . . 53 1 Mathematics S ubje ct Classiﬁc ation (2000): Primary 5 3C17, 28 A75; Secondary 58C35, 93B29 Keywor ds: Car not–Carath´ eodory space, nilp otent tangent cone, approximation theorems, diﬀerentiabilit y , coarea fo r m ula 1 3 Diﬀeren tiabilit y on a Carnot Manifold 55 3.1 Primitiv e calculus . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1.1 Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.1.2 Chain rule . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 hc -Diﬀeren tiabilit y of Curv es on the Carnot Manifo lds . . . . 58 3.2.1 Co ordinate hc -diﬀeren tiabilit y criterion . . . . . . . . . 58 3.2.2 hc -Diﬀeren tiabilit y of absolutely con tin uous curve s . . 59 3.2.3 hc -Diﬀeren tiabilit y of scalar Lipsc hitz mappings . . . . 65 3.2.4 hc -Diﬀeren tiabilit y of rectiﬁable curv es . . . . . . . . . 69 3.3 hc -Diﬀeren tiabilit y o f Smo oth Mappings of Carnot Manifolds 72 3.3.1 Con tin uit y of horizontal deriv ative s and hc -diﬀerentiabilit y of mappings . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3.2 F unctorial prop ert y o f tangen t cones . . . . . . . . . . 76 3.3.3 Rademac her The orem . . . . . . . . . . . . . . . . . . 77 3.3.4 Stepano v Theorem . . . . . . . . . . . . . . . . . . . . 77 4 Application: The Coarea F orm ula 78 4.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.2 La y-out of the Proo f . . . . . . . . . . . . . . . . . . . . . . . 79 5 App endix 85 5.1 Pro of of Lemma 2.1 .13 . . . . . . . . . . . . . . . . . . . . . . 85 1 In tro ducti on The geometry of Carnot–Carat h ´ eo dory spaces natura lly arises in the the- ory of sub elliptic eq uations, con tact geometry , optimal control theory , non- holonomic mec hanics, neu robiology and other a reas (see w orks by A. A. Agr a- c hev [1], A. A. Agrach ev and J.-P . Gauthier [3], A. A. Agrache v and A. Mari- go [4], A. A. Agrac hev and A. V. Sarych ev [5, 6, 7, 8, 9], A. Bella ¨ ıche [15], A. Bonﬁglioli, E. Lanconelli and F. Uguzzoni [18], S. Buc kley , P . Kosk ela and G. L u [19], L. Cap ogna[21, 22], G. Citti, N. G arofalo and E. Lan- conelli [31], L. Cap ogna, D. D anielli and N. Garofalo [23, 24, 25, 2 6, 27], Y a. Eliash b erg [35, 36, 37, 38], G. B. F olland [44, 4 5], G . B. F olland and E. M. Stein [46], B. F ranchi, R. Serapioni, F. Serra Cassano [55, 56, 57 , 58], N. Garofalo [6 0], N. G arofalo and D .-M. Nhieu [62, 63], R. W. G o o d- man [65], M . Gromov [68, 69], L. H¨ ormander [74], F. Jean [75], V. Jurd- jevic [82], G. P . Leonardi, S. R igot [89], W. Liu and H. J. Sus sman [91], G. Lu [92], G. A. Margulis and G. D. Mosto w [99, 100], G. Metivier [101], J. Mitc hell [102], R. Montgomery [103, 104], R. Mon ti [1 05, 106], A. Nagel, 2 F. Ricci, E. M. Stein [108, 109], A. Nagel, E. M. Stein and S. W ainger [11 0], P . P a nsu [112, 113, 1 14, 115], L. P . Rot hsc hild and E. M. Stein [119], R. S. Stric hartz [122], A. M. V ershik and V. Y a. Gershk o vic h [124], S. K. V o dop y- ano v [125, 127, 128, 12 9, 130], S. K. V o dop y ano v a nd A. V. G reshno v [131], C. J. Xu a nd C. Zuily [138] for an introduction to this theory and some its applications). A Carnot–Carath´ eo dory space (b elo w referred to as a Carno t manifo l d s ) M (see , for example, [68, 1 24]) is a connected Riemannian manifold with a distinguished horizon tal subbundle H M in the tangen t bundle T M that meets some algebraic conditions on the comm utators of v ec tor ﬁelds { X 1 , . . . , X n } constituting a lo cal basis in H M , n = dim H M . The distance d c (the in trinsic Carnot– Carath ´ eo dory metric) b etw een p oints x, y ∈ M is deﬁned as the inﬁm um of the lengths of horizontal curv e s joining x and y and is non- Riemannian if H M is a prop er subbundle (a piecewise smo oth curv e γ is called horizon tal if ˙ γ ( t ) ∈ H γ ( t ) M ). See results on prop- erties of this metric in the monograph by D . Y u. Burago, Y u. D. Burago nd S. V. Iv anov [20]. The Carnot–Carath ´ eo dory metric is applied in t he study of h yp o ellip- tic op erators, see C. F eﬀerman and D. H. Phong [43 ], L. H¨ ormander [74], D. Jerison [76], A. Nagel, E. M. Stein and S. W ainger [110], L. P . Rothsc hild and E. M. Stein [119], A. S´ anc hez-Calle [120]. Also, this metric and its prop erties are essen tially used in theory of PDE’s (see pap ers by M. Biro li and U. Mosco [16, 17], S. M. Buc kley , P . Kosk ela and G. Lu [19], L. Ca- p ogna, D. D anielli and N. Garofalo [23, 24, 25, 26, 27], V. M. Chernik o v and S. K. V o dop’y ano v [29], D. D anielli, N. Garofalo and D .-M. Nhieu [33], B. F ra nc hi [47], B. F ranc hi, S. Gallo t and R. Whee den [48], B. F ranc hi, C. E. Guti ´ errez and R. L. Wheeden [49], B. F ranc hi and E. Lanconelli [50, 51], B. F ranc hi, G. Lu and R. Wheeden [5 2, 53], B. F ranchi and R. Serapioni [54], R. Garattini [5 9], N. Ga rofalo and E. Lanconelli [61], P . Ha j lasz and P . Strz- elec ki [71], J. Jost [77, 78, 79, 80], J. Jost and C. J. Xu [81], S. Marc hi [98], K. T. Sturm [123]). The follo wing results are usually regarded as foundations of the geometry of Carnot manifolds: 1. Rashevski ˇ ı–Cho w Theorem [30, 118] on connection of t w o p oin ts b y a horizon tal path; 2. Ball–Box Theorem [110] (saying that a ball in Carnot–Carath ´ eodory metric con tains a “ b o x” and is a subset of a “b o x” with con trolled “radii”); 3 3. Mitc hell’s Theorem [102] o n con vergenc e of rescaled Carnot–Carath ´ eo- dory spaces around g ∈ M to a nilpotent tangen t cone; 4. Gromov’s Theorem [68 ] on con v e rgence of “rescaled” with resp ect to g ∈ M basis v ector ﬁelds to nil p otentize d ( at g ) v ector ﬁelds gen erating a graded nilp otent Lie algebra (the corresp onding connec ted and simply connected L ie group is called the nilp otent tangent c one at g ); here g ∈ M is an arbitrary point; 5. Gromov Approxim ation Theorem [68] on lo cal comparison of Carno t– Carath ´ eodory metrics in the initial space and in the nilpotent ta ngen t cone, and its impro v emen ts due to A. Bella ¨ ıc he [15]. The goa l of the pap er is b oth to giv e a new approac h to the geometry of Carnot manifolds and to establish some basic results of geometric measure theory on these metric structures including an appropriate diﬀeren tiabilit y and a coarea form ula. New results in the geometry of Carnot manifolds contains essen t ially new quan titativ e estimates of closeness of geometries of diﬀeren t tang en t cones lo cated o ne near another. One of the p eculiarities of the pap er is that w e solv e all problems under minimal assumption on s mo othness of the basis v ector ﬁelds (they are C 1 ,α -smo oth, 0 ≤ α ≤ 1), although all the basic results are new ev en for C ∞ -v ector ﬁelds. In some parts of this paper, the sym bol C 1 ,α means that the deriv ative s of the basis vec tor ﬁelds are H α - con tin uous with r espect to some nonnegativ e symmetric function d : U × U → R , U ⋐ M , suc h that d ≥ C ρ , 0 < C < ∞ , where C dep ends only on U , and ρ is Riemannian distance. Some a dditional prop erties of d are described b elo w when it is necessary . Note that from the v ery b eginning it is unkno wn whether Rashevski ˇ ı–Cho w Theorem is true for C 1 ,α -smo oth basis v ector ﬁelds. Therefore Carnot–Carath ´ eo dory distance can not be w ell deﬁned. W e use the quasimetric d ∞ instead of d c , whic h is deﬁ ned a s follo ws: if y = exp  N P i =1 y i X i  ( x ), then d ∞ ( x, y ) = m ax i =1 ,...,N {| y i | 1 deg X i } , and in smo oth case is equiv alen t to d c [110, 68]. One of the main results is the followin g (see below Theorem 2.4.1 for sharp statemen t). Theorem 1.0.1. Supp ose that d ∞ ( u, u ′ ) = C ε , d ∞ ( u, v ) = C ε for some C , C < ∞ , w ε = exp  N X i =1 w i ε deg X i b X u i  ( v ) and w ′ ε = exp  N X i =1 w i ε deg X i b X u ′ i  ( v ) . 4 Then, for α > 0 , we have max { d u ∞ ( w ε , w ′ ε ) , d u ′ ∞ ( w ε , w ′ ε ) } ≤ Lερ ( u, u ′ ) α M wher e L is uniformly b ounde d in u, u ′ , v ∈ U ⊂ M , and in { w i } N i =1 b elo nging to some c omp act neig h b orho o d of 0 . In the c ase of α = 0 , w e have max { d u ∞ ( w ε , w ′ ε ) , d u ′ ∞ ( w ε , w ′ ε ) } ≤ εo (1 ) wher e o (1) is uniform in u, u ′ , v ∈ U ⊂ M , and in { w i } N i =1 b elo nging to some c om p act neighb orho o d of 0 as ε → 0 . Here w e a ssume that U ⊂ M is a compact neighborho od small enough and ρ is a Riemannian metric. The sy m b ol b X u i ( b X u ′ i ) denotes nilp oten tized at u ( u ′ ) v ector ﬁelds (see item 4 ab ov e). These v ector ﬁelds constitute Lie algebra of the nilpo ten t tangent cone a t u ( u ′ ). F urther, in Theorem 2.3.1 we extend this result to the case of a ”chain“ consisting of sev eral p oin ts. The con ten t of obta ined estimates is v ery prof ound: t hey imply b oth new prop erties of Carnot manifolds and the ab ov e-mentioned ones. The in v estigation of sub-Riemannian geometry under minimal smo oth- ness o f the basis v ector ﬁelds is motiv ated b y the recen tly constructed b y G. Citti and A. Sarti, and R. K. Hladky and S. D. P auls mo del of visualiza- tion [32, 73]. More exactly , the mo del of a brain p erception of a black -and- white plain image is constructed in these pap ers. This model mak es possible the in terpretation on a computer of a h um an bra in’s work during the vi- sualization of information. In part icular, it is sho wn how the h uman brain completes t he image part of whic h is closed. The geometry of this mo del is based on a rot o-translation gr oup whic h is a three-dimensional Carnot manifold with a tangent cone b eing a Heisen berg group H 1 at eac h p oin t. Since b y now there a re no theorems on regularit y of the image created b y a h uman brain, an y reduction of smoothness of vector ﬁelds is esse n tial for the construction of sharp visualization mo dels. The main resu lt conce rning the geometry o f Carnot manifolds is pro v ed in Section 2. The metho d of proving is new , and it essen tially uses H¨ older dep endence o f solutions to ordinary diﬀeren tial equations on parameter (see Theorem 2.1.13). Probably , t his dep endenc e is not a new result. F or reader’s con v enie nce w e g iv e its indep enden t pro o f in Section 5. In Subsection 2.1, all other auxiliary result are form ulated. In Subsection 2.2 w e prov e, in par ticular, the follo wing stat emen ts 5 A: L et X j ∈ C 1 on a Carnot manifold M . On Bo x( g , εr g ) , c onsider the ve c tor ﬁelds { ε X i } = { ε deg X i X i } , i = 1 , . . . , N . Then the u niform c onver genc e X ε i =  ∆ g ε − 1  ∗ ε X i → b X g i as ε → 0 , i = 1 , . . . , N , holds at the p oints of the b ox Bo x( g , r g ) and this c on v er genc e i s uniform in g b elongin g to som e c om p act set, wher e the c ol le ction { b X g i } , i = 1 , . . . , N , of ve c tor ﬁelds ar ound g c onstitutes a b asis of a gr ade d nilp otent Lie alge b r a ; B: Ther e exists a c onstant Q = Q ( U ) , U is a c omp act domain in M , such that the ine quality d ∞ ( u, v ) ≤ Q ( d ∞ ( u, w ) + d ∞ ( w , v )) holds for every triple of p oints u , w , v ∈ U wher e Q ( U ) dep ends on U . C: Given p oints u, v ∈ M , d ∞ ( u, v ) = C ε for some C < ∞ , w ε = exp  N X i =1 w i ε deg X i X i  ( v ) and w ′ ε = exp  N X i =1 w i ε deg X i b X u i  ( v ) , we have max  d ∞ ( w ε , w ′ ε ) , d u ∞ ( w ε , w ′ ε )  ≤ εo (1) wher e o (1) is uniform in u, v b elon ging to a c omp act neig hb orho o d U ⊂ M , and in { w i } N i =1 b elo nging to some c omp act neighb orho o d of 0 as ε → 0. Statemen t A is just Gro mo v’s Theorem [68] on the nilpo ten tization of v ector ﬁelds. G romo v has form ulated it for C 1 -smo oth ﬁelds, how ev er, Ex- ample 2.2.12 by V aleri ˇ ı Be resto vski ˇ ı mak es eviden t that argumen ts of the pro of giv en in [68, pp. 128–1 33] hav e to b e corrected. In Coro llary 2.2.11 w e giv e a new pro of of this assertion based on an another idea. Statemen t B says that the quasimetric d ∞ meets the generalised tr iangle inequalit y . T he implicatio n A = ⇒ B is prov ed in Corollary 2.2.1 4. Statemen t C gives an estimate of div ergen ce of in tegral lines of the giv en v ector ﬁelds and the nilp otentize d v ector ﬁeldes. The implication B = ⇒ C is a particular case of Theorem 2.7.1. In theory dev eloped in Subsection 2.4, is based on the generalized triangle inequalit y as a starting p oint. In Subsection 2.4, w e prov e one o f the basic results of Section 2, namely , Theorem 2.4.1 whic h compares lo cal geometries of t w o diﬀerent lo cal Carnot groups. It is essen tially based on the main theorem of Subsection 2.3 whic h compares ”global“ g eometries of diﬀeren t tangen t cones (i. e., it lo oks lik e Theorem 2.4.1 with ε = 1). Subsection 2.5 is dev oted to appro ximation 6 theorems. In particular, w e compare metrics of t w o tangent cones, and the metric of a tangen t cone with the initial one. There we give their pro ofs and the pro ofs o f some auxiliary prop erties of the g eometry . F urther, in Subsection 2.6, w e prov e Theorem 2.3.1, whic h is the ”contin ua tion“ of The - orem 2.4.1. In Subsec tion 2.7, w e compare the geometry of a Carnot manifold with the one of a tangent cone. In Subsection 2.8, w e giv e applications of our results to in v estigation of the sub-Riemannian geometry . W e pro v e Gr omo v t yp e theorem on the nilp oten tization of v ector ﬁelds [68], a new statemen t implying Rashevski ˇ ı–Cho w Theorem, Ball–Bo x Theorem, Mitc hell Theorem on Hausdorﬀ dimension of Carnot manifolds and man y other corollaries. Main results of Section 2 are fo rm ulated in short comm unications [1 32, 133]. Section 3 is dev oted to diﬀeren tiabilit y of mappings in the category of Carnot manifolds. W e recall the classical deﬁnition of diﬀeren t iabilit y for a mapping f : M → N of tw o Riemannian manifolds: f is diﬀeren tiable at x ∈ M if there exists a linear mapping L : T x M → T f ( x ) N of the tangen t spaces suc h that ρ N ( f ( exp x h ) , exp f ( x ) Lh ) = o ( k h k x ) , h ∈ T x M , where exp x : T x M → M and exp f ( x ) : T f ( x ) N → N are the exponential mappings, and ρ N is the R iemannian metric in N , k h k x is the length of h ∈ T x M . It is kno wn (see [68, 104]) that the lo cal geometry of a Carnot manifold at a point g ∈ M can b e mo delled as a graded nilp oten t Lie group G g M . It means that the ta ngen t space T g M has an additional structure of a graded nilp oten t Lie group. If M and N are tw o Carnot manifolds a nd f : M → N is a mapping then a suitable concept of diﬀerentiabilit y can b e o btained from the previous concept in the follo wing wa y: f is hc -diﬀeren tiable a t x ∈ M if there exists a horizontal homomorphism L : G x M → G f ( x ) N of the nilp oten t tangen t cones suc h that ˜ d c ( f ( exp x h ) , exp f ( x ) Lh ) = o ( | h | x ) , h ∈ G x M , where ˜ d c is the Carnot–Carath´ eo dory distance in N and | · | x is an homoge- neous norm in G x M . F or us, it is conv enien t to r egard some neigh b orho o d of a p oint g as a subspace o f the metric space ( M , d c ) and as a neigh b orho o d of unit y of the lo cal Carnot group G g M with Carnot–Cara th ´ eo dory metric d g c (see D eﬁni- tion 1.2). In the sense explained below , exp − 1 : G g M → G g M is an iso- metrical mo nomorphism of the Lie structures. Then the last deﬁnition of 7 hc -diﬀeren tiabilit y can b e reformulated as f ollo ws. Give n t w o Carnot mani- folds ( M , d c ) and ( N , ˜ d c ) and a set E ⊂ M , a mapping f : E → N is called hc -diﬀer entiable at a p oin t g ∈ E ( see the pap er b y S. K. V o dop y ano v and A. V. Greshno v [1 31], and also [127, 128, 129, 130]) if there exists a horizon tal homomorphism L :  G g M , d g  →  G f ( g ) N , d f ( g c  of the lo cal Carno t groups suc h tha t d f ( g ) c ( f ( w ) , L ( w )) = o ( d g c ( g , w )) a s E ∩ G g ∋ w → g . (1.0.1) The give n deﬁnition of hc -diﬀeren tiabilit y of mappings for Carno t man- ifolds can b e treated as a straightforw ard g eneralization of the classical deﬁnition o f diﬀerentiabilit y . Clearly , if the Carnot manifolds are Carnot groups then this deﬁnition o f hc -diﬀeren tiability is equiv alen t to the deﬁni- tion of P - diﬀeren tiabilit y introduced b y P . P ansu in [11 5] for an op en set E ⊂ G . F or an arbitrary E ⊂ G , the last concept w a s in v estigated by S. K. V o dop ′ y ano v [125] and by S. K. V o dop y ano v and A. D. Ukhlo v [136] (see also the pap er b y V. Magnani [93]). In Section 3, w e in troduce the notion of hc -diﬀeren tiabilit y , which is ad- equate to the geometry of Carnot manifold, and study its prop erties. More- o v er, in this section, w e prov e the hc -diﬀeren tiabilit y of a comp osition of hc -diﬀeren tiable mappings. In the same se ction we pro v e the hc - diﬀeren tiabilit y of rectiﬁable curv es. In the case of curv es, the deﬁnition o f the hc -diﬀeren tiabilit y is in terpreted as follow s: a mapping R ⊃ E ∋ t 7→ γ ( t ) ∈ N is hc -diﬀeren tiable at a p oint s ∈ E in a Carnot manifo ld N if the relation d γ ( s ) c  γ ( s + τ ) , exp( τ a )( γ ( s ))  = o ( τ ) as τ → 0, s + τ ∈ E , (1.0.2) holds, where a ∈ H γ ( s ) N ((1.0.2) agrees with (1 .0.1) when M = R , see also [99]). On Carnot groups, relation (1.0.2) is equiv alen t to the P -diﬀeren tiability of curv es in the sense of P ansu [1 15]. Our pro of of diﬀeren tiabilit y is new ev en for Carnot groups. W e pro v e step b y step the hc - diﬀeren tiabilit y of the absolutely diﬀeren tiable curv es , the Lipsc hitz mappings of subsets of R in to M , and the rectiﬁable curv es. Here w e generalize a classical result and obtain the following assertion: the c on tinuity of horizontal derivatives of a c ontact mapping deﬁne d on an op en set implies its p ointwise hc -diﬀer entiability (The- orem 3 .3.1). As an imp ortant corollary t o these assertion, w e infer that the nilp oten t tangen t cone is deﬁned b y the horizon tal subbundle of the Carnot manifold: tangent c ones found fr om diﬀer ent c ol le ctions of b a sis ve ctor ﬁelds ar e iso- morphic as lo c al Carnot gr oups (Corollary 3.3.3). Th us, the corresp ondence 8 “lo cal basis 7→ nilp oten t ta ngen t cone” is functorial. In the case of C ∞ - v ector ﬁelds, this result w as established by A. Agrac hev a nd A. Marigo [4], and G. A. Margulis and G. D. Mosto w [100] where a co ordinate-fr ee deﬁnition of the tangen t cone w as giv e n. Main results of Section 3 are formulated in short comm unications by S. K. V o dop y ano v [127, 128] (see some details and more general results on this sub ject including Rademac her–Stepano v Theorem in [129, 130]). Section 4 is dedic ated to suc h application of results on hc -diﬀeren tiabilit y as the sub-Riemannian analog of the coarea form ula. It is w ell known that the coa rea formula Z U J k ( ϕ, x ) d x = Z R k dz Z ϕ − 1 ( z ) d H n − k ( u ) , (1.0.3) where J k ( ϕ, x ) = p det( D ϕ ( x ) D ϕ ∗ ( x )), has many applications in analysis on Euclidean spaces. Here w e assume that ϕ ∈ C 1 ( U, R k ), U ⊂ R n , n ≥ k . F or the ﬁrst time, it w as established b y A. S. Kr onro d [88] for the case of a function ϕ : R 2 → R . Next, it w as generalized b y H. F ederer ﬁrst fo r mappings of Riemannian manifolds ϕ : M n → N k , n ≥ k , in [40], and then, for mappings of rectiﬁable sets in Euclidean spaces ϕ : M n → N k , n ≥ k , in [41]. Next, in the pap er [111], M. Oh tsuk a generalized the coarea form ula (1.0.3) for mappings ϕ : R n → R m , n, m ≥ k , with H k - σ -ﬁnite image ϕ ( R n ). An inﬁnite-dimensional ana log of the coarea formu la w as prov ed b y H. Aira ult and P . Mallia vin in 1988 [1 0] for the case of Wiener spaces. This result can b e found in the monogr aph b y P . Malliavin [97]. See other pro ofs and applications of the coarea formula in the monographs by L. C. Ev ans and R. F. Gariepy [39], M. Giaquin ta, G. Mo dica and J. Sou ˇ cek [6 4], F. Lin and X. Y ang [9 0]. F ormula ( 1.0.3) can b e a pplied in the theory of exterior forms, curren ts, in minimal surfaces problems (see, for example, paper by H. F ederer and W. H. Fleming [42]). Also, Stoke s formula can b e easily obtained by us- ing the coarea form ula (see, f or instance, lecture not es b y S. K. V o dopy - ano v [126 ]). Because of the dev elopmen t of analysis on more general struc- tures, a natural question arise to extend the coarea formula on ob jects o f more general geometry in comparison with Euclidean spaces, esp ecially o n metric spaces and structures on sub-Riemannian geometry . In 1999, L. Am- brosio and B. Kirchh eim [11] pro v ed the analog of the coarea f orm ula for Lipsc hitz mappings deﬁned on H n -rectiﬁable metric space with v alues in R k , n ≥ k . In 2004, this f orm ula w as prov ed for Lipsc hitz mappings deﬁned on H n -rectiﬁable metric space with v a lues in H k -rectiﬁable metric space, n ≥ k , b y M. Karmano v a [83, 85]. Moreo v er, nece ssary and suﬃcien t conditions o n 9 the image and the preimage of a Lipsc hitz mapping deﬁned on H n -rectiﬁable metric space with v alues in an arbitr ary metric space for the v alidit y of the coarea formula w ere found. Indep enden tly of this result, the lev el sets of suc h mappings we re in v estigated, and the metric analo g of Implicit F unction Theorem w a s prov ed by M. Kar mano v a [84, 85, 86]. All the ab o v e results are conne cted with rectiﬁable metric space s. Note that, their structure is similar to the one of Riem annian manifolds. But there are also non-r e ctiﬁable metric spaces whic h geometry is not compara ble with the Riemannian one. Carnot m anifolds ar e of sp ecial in tere st. The problem of the s ub-Riemannian coarea form ula is one of w ell-kno w n in t rinsic uns olv ed problems. A Heisen b erg group and a Carnot group are w ell-kno wn particular cases of a Carnot manifold. In 198 2, P . P ansu prov ed the coarea formula for functions deﬁne d on a Heisen b erg group [1 12]. Next, in [72], J. Heinonen ex - tended this formula to smo oth functions deﬁned on a Carnot g roup. In [107], R. Mon ti and F. Serra Cassano pro v ed the analog of the coarea form ula for B V -functions deﬁned on a t w o -step Carno t–Carath ´ eo dory space. One more result conce rning the analogue of (1.0.3) b elongs to V. Magnani. In 2000, he pro v ed a c o ar e a ine quali ty for mappings of Carnot groups [95 ]. The equalit y w as p rov ed only for the case of a mapping deﬁne d on a Heisen b erg group with v alues in Euclidean space R k [96]. Until now , the question ab out the v alidit y of coarea form ula ev en for a mo del case of a mapping of Carnot groups w as op en. Main r esults of Section 4 are formulated in [134]. 2 Geometry of Carnot –Carath ´ eo d o ry Spaces 2.1 Preliminary Results Recall t he deﬁnition of a Carnot manifold. Deﬁnition 2.1.1 (compare with [68, 110]) . Fix a connected Riemannian C ∞ -manifold M of a dimension N . The manifold M is called a Carno t manifold if, in the tangen t bundle T M , there exis ts a tangent subbundle H M with a ﬁnite collection of nat ural num bers dim H 1 < . . . < dim H i < . . . < dim H M = N , 1 < i < M , and each p oin t p ∈ M p ossesses a neigh borho od U ⊂ M with a collection of C 1 -smo oth v ector ﬁelds X 1 , . . . , X N on U enjoying the fo llo wing prop erties. F or each v ∈ U we ha v e (1) X 1 ( v ) , . . . , X N ( v ) constitutes a basis of T v M ; 10 (2) H i ( v ) = span { X 1 ( v ) , . . . , X dim H i ( v ) } is a subsp ace of T v M of a di- mension dim H i , i = 1 , . . . , M , where H 1 ( v ) = H v M ; (3) [ X i , X j ]( v ) = X deg X k ≤ deg X i +deg X j c ij k ( v ) X k ( v ) (2.1.1) where the de gr e e deg X k equals min { m | X k ∈ H m } ; (4) a quotien t mapping [ · , · ] 0 : H 1 × H j /H j − 1 7→ H j +1 /H j induced by Lie brac k ets, is an epimorphism for all 1 ≤ j < M . Remark 2.1.2. Note [110] that Condition 4 is necessary only fo r obtaining results of Subsections 2.8 and 3 .3 and its coro llaries, and Section 4. The p oin t is that in statemen ts of these subse ctions, w e use the fact that an y tw o p oin ts of a lo cal Carnot group (see the deﬁnition b elo w) can b e joined b y a horizontal (with resp ect to the lo cal Carnot group) curv e that consists of at most L segmen ts of in tegral lines of horizon tal ( with respect to the lo cal Carnot group) v ector ﬁelds. The latter is imp ossible without Condition 4. Remark 2.1.3. C onsider a C 2 -smo oth lo cal diﬀeomorphism η : U → R N , U ⊂ M . Then η ∗ X i = D η h X i i are also C 1 -v ector ﬁelds, i = 1 , . . . , N . W e ha v e the follo wing relations instead of (2.1 .1): η ∗ [ X i , X j ]( w ) = [ η ∗ X i , η ∗ X j ]( w ) = X deg X k ≤ deg X i +deg X j c ij k ( η − 1 ( w )) η ∗ X k ( w ) . Denote by X ( w ) the matrix, the i th column of which consists of the co or- dinates of η ∗ X i ( w ) in the standard basis { ∂ j } N j =1 . Then the en tries o f X ( w ) are C 1 -functions. Note that η ∗ [ X i , X j ]( w ) = X ( w )( c ij 1 ( η − 1 ( w )) , . . . , c ij N ( η − 1 ( w ))) T . Consequen tly , ( c ij 1 ( η − 1 ( w )) , . . . , c ij N ( η − 1 ( w ))) T = ( X ( w )) − 1 · η ∗ [ X i , X j ]( w ) . F rom here it fo llo ws that c ij k ◦ η − 1 are con tin uous, k = 1 , . . . , N . Since η is con tin uous, then w e ha v e that eac h c ij k = ( c ij k ◦ η − 1 ) ◦ η is also contin uous , k = 1 , . . . , N . Example (A Carnot Manifold with C 1 -Smo oth V ector Fields) . Consider the C M v ector ﬁelds X 1 , . . . , X n ∈ H . Cho ose a basis in H 2 = span { H , [ H , H ] } by the f ollo wing w a y: X k ( v ) = X i,j a k ij ( v )[ X i , X j ]( v ) + X l b k l ( v ) X l , 11 where a k ij ( v ) , b k l ( v ) ∈ C 1 , i, j, l = 1 , . . . , N , k = n + 1 , . . . , dim H 2 . Similarly , w e c ho ose the follo wing basis in H m +1 = span { H m , [ H , H m ] } , m = 2 , . . . , M − 1: X k ( v ) = X i,j a k ij ( v )[ X i , X j ]( v ) + X l b k l ( v ) X l , where a k ij ( v ) , b k l ( v ) ∈ C 1 , i = 1 , . . . , N , j, l = dim H m − 1 + 1 , . . . , dim H m , k = dim H m + 1 , . . . , dim H m +1 . Assumption 2.1.4. Throughout the pap er, w e assume that all the basis v ec- tor ﬁelds X 1 , . . . , X N are C 1 ,α -smo oth, and, conse quen tly , their comm utators are H α -con tin uous, α ∈ [0 , 1 ]. In some parts o f this pap er, w e consider cases when the deriv a tiv es o f the basis v ector ﬁelds are H α -con tin uous with resp ect to some nonnegativ e symmetric function d : U × U → R , U ⋐ M , suc h that d ≥ C ρ , 0 < C < ∞ , where C dep ends only on U , and ρ is Riemannian distance. Some additional prop erties of d a re describ ed b elo w when it is neces sary . Notation 2.1.5. In the pap er: 1. The sym b ol X ∈ C 1 , 0 means that X ∈ C 1 , and the sym bo l X ∈ C 0 means X ∈ C . 2. 0-H¨ older contin uit y me ans the or dinary contin uity . W e denote a mod- ulus of con tin uit y of a mapping f b y ω f ( δ ). 3. The Riemannian distance is denoted b y the sym bo l ρ . Theorem 2.1.6. The c o eﬃcients ¯ c ij k = c ij k ( u ) of (2.1.1) with deg X i + deg X j = deg X k deﬁne a gr ade d nilp otent Lie algebr a. Pr o of. Fix an arbitrary p oin t u ∈ M and sho w that t he collection { c ij k ( u ) } with deg X k = deg X i + deg X j , enjoy the Jacobi identit y and, th us, deﬁne the structure of a Lie algebra. The prop erty ¯ c ij k = − ¯ c j ik is ev iden t. Pro v e that the collection { ¯ c ij k } under consideration enjo ys Jacobi iden tit y . 1 st Step. W e ma y assume without loss of generality that X 1 , . . . , X N are the ve ctor ﬁelds on an op en set of R N (otherwise, consider the lo cal C 2 -diﬀeomorphism η similarly to Remark 2.1.3). F or a vec tor ﬁeld X i ( x ) = N P j =1 η ij ( x ) ∂ j , consider the molliﬁcatio n ( X i ) h ( x ) = N P j =1 ( η ij ∗ ω h )( x ) ∂ j , i = 1 , . . . , N , where the function ω ∈ C ∞ 0 ( B (0 , 1)) is such 12 that R B (0 , 1) ω ( x ) d x = 1, and ω h ( x ) = 1 h N ω  x h  . By t he prop erties of mol- liﬁcation η ij ∗ ω h , i, j = 1 , . . . , N , w e ha ve ( X i ) h C 1 − → h → 0 X i lo cally in s ome neigh b orho o d of u . Note that the v ector ﬁelds ( X i ) h ( v ), i = 1 , . . . , N , meet the Jacobi iden tit y , and are a basis o f T v M for v b elonging to some neigh- b orho o d o f u , if the parameter h is sm all enough. Consequ en tly , setting [( X i ) h , ( X j ) h ] = N P k =1 ( c ij k ) h ( X k ) h , w e ha v e X k X l ( c ij k ) h ( c k ml ) h ( X l ) h + X k X l ( c mik ) h ( c k j l ) h ( X l ) h + X k X l ( c j mk ) h ( c k il ) h ( X l ) h − X l [( X m ) h ( c ij l ) h ]( X l ) h − X l [( X j ) h ( c mil ) h ]( X l ) h − X l [( X i ) h ( c j ml ) h ]( X l ) h = 0 . Note tha t, since ( X i ) h C 1 − → h → 0 X i lo cally , and the v ector ﬁelds { ( X i ) h } N i =1 are linearly indep enden t for all h ≥ 0 small enough, we ha v e ( c ij k ) h → c ij k as h → 0. No w, ﬁx 1 ≤ l ≤ N . Since the v ector ﬁelds { ( X i ) h } N i =1 are linearly indep enden t for h > 0 small enough, w e ha v e X k ( c ij k ) h ( c k ml ) h + X k ( c mik ) h ( c k j l ) h + X k ( c j mk ) h ( c k il ) h − [( X m ) h ( c ij l ) h ] − [( X j ) h ( c mil ) h ] − [( X i ) h ( c j ml ) h ] = 0 (2 .1.2) for each ﬁxed l in some neigh bourho o d of u . Fix i, j, m a nd l suc h that deg X l = deg X i + deg X j + deg X m , and cons ider a test function ϕ ∈ C ∞ 0 ( U ) on some small compact neigh bo rho o d U ∋ u , U ⋐ M . W e m ultiply b oth sides of (2.1.2) on ϕ and in tegrate the result o v er U . F or h > 0 small enough, w e ha v e 0 = Z U h X k ( c ij k ) h ( v )( c k ml ) h ( v ) + X k ( c mik ) h ( v )( c k j l ) h ( v ) + X k ( c j mk ) h ( v )( c k il ) h ( v ) i · ϕ ( v ) dv − Z U [( X m ) h ( c ij l ) h ]( v ) · ϕ ( v ) dv − Z U [( X j ) h ( c mil ) h ]( v ) · ϕ ( v ) dv − Z U [( X i ) h ( c j ml ) h ]( v ) · ϕ ( v ) dv . 13 Sho w that, among the last three in tegrals, the ﬁrst one tends to zero as h → 0. In deed, Z U [( X m ) h ( c ij l ) h ]( v ) · ϕ ( v ) dv = − Z U [( X m ) ∗ h ϕ ]( v ) · ( c ij l ) h ( v ) d v , where ( X i ) ∗ h is an adjoint op erator to ( X i ) h . The righ t-hand par t in tegral tends to zero as h → 0 since the v alue [( X m ) ∗ h ϕ ]( v ) is uniformly b o unded in U as h → 0, and ( c ij l ) h ( v ) → 0 as h → 0 in view of the c hoice of l . The similar conclusion is true rega rding the last tw o in tegrals. Consequen tly , taking into account the fa cts that ( c ij k ) h → c ij k lo cally , and c ij k = 0 for deg X k > deg X i + de g X j , and using du Bois–R eymond Lemma for h → 0 w e infer X k : deg X k ≤ deg X i +deg X j c ij k ( v ) c k ml ( v ) + X k : deg X k ≤ deg X m +deg X i c mik ( v ) c k j l ( v ) + X k : deg X k ≤ deg X j +deg X m c j mk ( v ) c k il ( v ) = 0 (2 .1.3) for all v ∈ M close enough to u . 2 nd Step. F o r ﬁxed l , su c h t hat deg X l = deg X i + deg X j + deg X m , in v estigate the prop erties of the inde x k . Consider the ﬁrst su m. Since deg X l ≤ deg X k + de g X m , we ha v e deg X k ≥ deg X l − deg X m = deg X i + deg X j . By (2.1.1), deg X k ≤ deg X i + deg X j , and, consequen tly , deg X k = deg X i + deg X j . The other t w o cases are considered similarly . Th us, the sum (2.1.3) with deg X l = deg X i + deg X j + deg X m and v = u is X deg X k =deg X i +deg X j c ij k ( u ) c k ml ( u ) + X deg X k =deg X m +deg X i c mik ( u ) c k j l ( u ) + X deg X k =deg X j +deg X m c j mk ( u ) c k il ( u ) = 0 . The co eﬃcien ts { ¯ c ij k = c ij k ( u ) } deg X k =deg X i +deg X j enjo y the Jacobi iden tity , and, th us , they deﬁne the Lie algebra. The theorem follows . W e construc t the Lie algebra g u from Th eorem 2.1.6 as a graded nilp oten t Lie a lgebra of v ector ﬁelds { ( b X u i ) ′ } N i =1 on R N [117]. Th us the relation [( b X u i ) ′ , ( b X u j ) ′ ] = X deg X k =deg X i +deg X j c ij k ( u )( b X u k ) ′ holds for the v ec tor ﬁelds { ( b X u i ) ′ } N i =1 ev eryw here on R N . 14 Notation 2.1.7. W e use t he following standard notatio ns: for eac h N - dimensional multi-index µ = ( µ 1 , . . . , µ N ), its homo gene o us norm equals | µ | h = N P i =1 µ i deg X i . Deﬁnition 2.1.8. The Carnot group G u M c orresp onding to the Lie algebra g u , is called the nilp otent tang ent c one o f M at u ∈ M . W e construct G u M in R N as a groupalgebra [117]. By Campb ell–Hausdorﬀ formula, the group op eration is deﬁned for the basis vec tor ﬁelds ( b X u i ) ′ on R N , i = 1 , . . . , N , to b e left- in v arian t [117]: if x = exp  N X i =1 x i ( b X u i ) ′  , y = exp  N X i =1 y i ( b X u i ) ′  then x · y = z = exp  N X i =1 z i ( b X u i ) ′  , where z i = x i + y i , deg X i = 1 , z i = x i + y i + X | e l + e j | h =2 , l 0 , β > 0 F i µ,β ( u ) x µ · y β = x i + y i + X | µ + e l + β + e j | h = k , l 0 F j µ,e i ( u ) x µ if j > dim H deg X i . (2.1.5) Deﬁnition 2.1.10. Supp ose that u ∈ M and ( v 1 , . . . , v N ) ∈ B E (0 , r ) where B E (0 , r ) is an Euclidean ball in R N . Deﬁne a mapping θ u ( v 1 , . . . , v N ) : B E (0 , r ) → M as follows: θ u ( v 1 , . . . , v N ) = exp  N X i =1 v i X i  ( u ) . 15 It is kno wn, that θ u is a C 1 -diﬀeomorphism if 0 < r ≤ r u for some r u > 0. The collection { v i } N i =1 is called the normal c o or dinates or the c o or dinates of the 1 st kind ( with r esp e ct to u ∈ M ) of the p oint v = θ u ( v 1 , . . . , v N ). Assumption 2.1.11. The compactly em bedded neigh b orho o d U ⊂ M under consideration is suc h that θ u ( B E (0 , r u )) ⊃ U for all u ∈ U . By means o f the exp onen tial map w e can push-forw ard the v ector ﬁelds ( b X u i ) ′ on to U for obtaining the v ector ﬁelds b X u i = ( θ u ) ∗ ( b X u i ) ′ where ( θ u ) ∗ h Y i ( θ u ( x )) = D θ u ( x ) h Y i , Y ∈ T x R N . Note that b X u i ( u ) = X i ( u ). Indeed, o n the one hand, by the deﬁnition, w e ha v e ( θ u ) − 1 ∗ X i (0) = e i . On the other hand, Theorem 2.1.9 implies ( b X u i ) ′ (0) = e i . Th us b X u i ( u ) = X i ( u ). Theorem 2.1.12. The v e ctor ﬁelds b X u i , i = 1 , . . . , N , ar e lo c al ly H α -c o n tinu- ous on u . The pro ofs of t his theorem and of many other assertions concerning smo othness use often the following lemma (see its pro of in Section 5). Theorem 2.1.13. Con sider the ODE ( dy dt = f ( y , v , u ) , y (0) = 0 (2.1.6) wher e t ∈ [0 , 1] , y , v , u ∈ W ⊂ R N and Lip y ( f ) = L < 1 . 1. If the mapping f ( y , v , u ) = f ( y , u ) ∈ C 1 ( y ) ∩ C α ( u ) then the sol ution y ( t, u ) ∈ C α ( u ) lo c al ly. 2. If f ( y , v , u ) ∈ C 1 ,α ( y , u ) ∩ C 1 ( v ) and ∂ f ∂ v ∈ C 1 ,α ( y , u ) then dy ( t,v ,u ) dv ∈ C α ( u ) lo c al ly. Remark 2.1.14. The follow ing statemen ts a re pro v ed similarly to Theorem 2.1.13. 1. If the mapping f ( y , v , u ) from (2.1 .6) do es not dep end o n v , and it is C 1 -smo oth in y and it is lo cally α -H¨ older with respect to nonnegativ e symmetric function d deﬁned on U × U , U ⋐ M , suc h that d ≥ C ρ , 0 < C < ∞ , where C dep ends only on U , then the solution y ( t, u ) is also locally α -H¨ older with respect to d . 16 2. If f ( y , v , u ) ∈ C 1 ( y , u ) ∩ C 1 ( v ), its deriv ativ es in y and in u are lo cally α -H¨ older with resp ect to d , ∂ f ∂ v ∈ C 1 ( y , u ), and the deriv ative s of ∂ f ∂ v in y and in u are lo cally α -H¨ older with resp ect to d then dy ( t,v ,u ) dv is locally α -H¨ older with resp ect to d . Remark 2.1.15. One of particular cases of d is d ∞ . Notation 2.1.16. Hereinafter, w e denote a nonnegative symmetric function deﬁned on U × U , U ⋐ M , p ossessing prop erties from item 1 of Remark 2.1.14, b y d . Pr o of of Pr op osition 2.1.12. 1 st Step. T aking in to account Assumption 2.1.4, w e hav e the table [( b X u i ) ′ , ( b X u j ) ′ ]( v ) = X deg X k =deg X i +deg X j c ij k ( u )( b X u k ) ′ ( v ) . By means of Assumption 2.1.4 and Deﬁnition 2.1.1, the functions c ij k ( u ) from (2.1.1) are H α -con tin uous. If X = N P i =1 x i ( b X u i ) ′ and Y = N P i =1 y i ( b X u i ) ′ then b y Campb ell–Hausdorﬀ form ula w e ha v e ex p tY ◦ exp tX ( g ) = exp Z ( t )( g ) where Z ( t ) = tZ 1 + t 2 Z 2 + . . . + t M Z M and Z 1 , Z 2 , . . . are some v ector ﬁelds indep enden t of t . Dynkin form ula (see , for instance, [117]) for calculating Z l ( t ), 1 ≤ l ≤ M , giv es Z l = 1 n l X k =1 ( − 1) k − 1 k X ( p )( q ) (ad Y ) q k (ad X ) p k . . . (ad Y ) q 1 (ad X ) p 1 − 1 X p 1 ! q 1 ! . . . p k ! q k ! = X ( p )( q ) C ( p )( q ) (ad Y ) q k (ad X ) p k . . . (ad Y ) q 1 (ad X ) p 1 − 1 X , where C ( p )( q ) = cons t, ( p ) = ( p 1 , . . . , p k ), ( q ) = ( q 1 , . . . , q k ). W e sum o v er all natural p 1 , q 1 , . . . p k , q k , suc h t hat p i + q i > 0, p 1 + q 1 + · · · + p k + q k = l , and (ad A ) B = [ A, B ], (ad A ) 0 B = B . The eac h summand can b e represen ted a s a sum Z l ( v ) = N X i =1 d j,l ( u, x, y )( b X u j ) ′ ( v ) , where d j,l ( u, x, y ) are p olynomial functions of x = ( x 1 , . . . , x N ), y = ( y 1 , . . . , y N ) co eﬃcien ts of wh ic h are p olynomial functions of { c lmk ( u ) } and, consequen t ly , are H¨ o lder in u . More exactly , M X l =2 Z l = M X l =2 N X j =1 d j,l ( u, x, y )( b X u j ) ′ = N X j =1  M X l =2 X | µ + β | h = l, µ> 0 ,β > 0 F j µ,β ( u ) x µ · y β  ( b X u j ) ′ . 17 Consequen tly , d j,l ( u, x, y ) = M X l =2 X | µ + β | h = l, µ> 0 ,β > 0 F j µ,β ( u ) x µ · y β . Hence, F j µ,β ( u ) ar e H α -con tin uous in u , and ( b X u i ) ′ are also H α -con tin uous on u (see (2.1.5)). 2 nd Step. Consider the follow ing Cauc hy problem:    d Φ( t,u,ξ ) dt = N P i =1 ξ i X i (Φ) , Φ(0 , u, ξ ) = u, (2.1.7) where ξ = ( ξ 1 , . . . , ξ N ). Note that Φ( t, u, ξ ) = exp  N P i =1 tξ i X i  ( u ). W e can assume without lo ss of generalit y , tha t M = R N . If Assumption 2.1.4 holds then the mapping f ( ξ , Φ) = N P i =1 ξ i X i (Φ) is C 1 ,α -smo oth in ξ and Φ. F rom the deﬁnition, it follows, that θ u ( ξ ) = Φ(1 , u, ξ ). By theorem 2.1.13 on smo oth dep endence of ODE solution on parameters (see Section 5 for details), it is easy to see, that the diﬀeren tial D θ u ( y ) is H α -con tin uous in u . Since b X u i ( x ) = D θ u ( y )( b X u i ) ′ ( y ), x = θ u ( y ), the prop osition follo ws from results of the 1 st and 2 nd steps. Remark 2.1.17. If the deriv at iv es of X i , i = 1 , . . . , N , are lo cally H¨ older with respect to d , then b X u i , i = 1 , . . . , N , are lo cally H¨ older on u with respect to d . Deﬁnition 2.1.18. The lo cal Lie g roup corresp onding to the Lie algebra { b X u i } N i =1 , is called the lo c al Carnot gr oup G u M at u ∈ M . Deﬁne it in suc h a wa y that the mapping θ u is a gr oup isomorphism of some neigh bor ho o d of the unit y of the group G u M and G u M . The canonical Riemannian structure is deﬁned b y scalar pro duct at the unit of G u M coinciding with those in T u M . Remark 2.1.19. Recall that the v e ctor ﬁelds b X u i , i = 1 , . . . , N , are locally H α -con tin uous on M , α ∈ [0 , 1]. The exp onen tial ma pping exp  N P i =1 a i b X u i  ( g ) is not deﬁned correctly f or suc h ﬁelds . Therefore, in view o f smo othness of ( θ − 1 u ) ∗ ( b X u i ), i = 1 , . . . , N , we deﬁne the p oint a = ex p  N X i =1 a i b X u i  ( g ) 18 according to Deﬁnition 2.1.18: ﬁrst, we obtain a p oint a u = exp  N X i =1 a i · ( θ − 1 u ) ∗ ( b X u i )  ( θ − 1 u ( g )) , and then w e deﬁne a = θ u ( a u ). Moreo v er, w e similarly deﬁne the w hole curv e corresp onding to this exponen tial mapping. Supp ose that    ˙ γ u ( t ) = N P i =1 a i · ( θ − 1 u ) ∗ ( b X u i )( γ u ( t )) γ u (0) = θ − 1 u ( g ) . Then, for the curv e γ ( t ) = θ u ( γ u ( t )), we hav e    ˙ γ ( t ) = N P i =1 a i b X u i ( γ ( t )) γ (0) = g . In par ticular, we ha v e: 1. The exponential mapping b θ u ( v 1 , . . . , v n ) = exp  N P i =1 v i b X u i  ( u ) is deﬁned as θ u  exp  N X i =1 v i ( b X u i ) ′  (0)  ; and the mapping b θ w u ( v 1 , . . . , v n ) = exp  N P i =1 v i b X u i  ( w ) is deﬁned a s θ u  exp  N X i =1 v i ( b X u i ) ′  ( θ − 1 u ( w ))  . 2. The in v erse mapping exp − 1 is also deﬁned by the unique w a y for v ector ﬁelds { b X u i } N i =1 since it is deﬁne d b y the unique w a y for { ( b X u i ) ′ } N i =1 . 3. The gro up op eration is deﬁned by the follow ing wa y: if x = exp  N P i =1 x i b X u i  , y = exp  N P i =1 y i b X u i  then x · y = exp  N P i =1 y i b X u i  ◦ exp  N P i =1 x i b X u i  = exp  N P i =1 z i b X u i  where z i are taken from Deﬁnition 2.1 .8. 19 4. Using the normal co ordinates b θ − 1 u , deﬁne the action of the dilation gr oup δ u ε in the lo cal Carnot group G u M : to an elemen t x = exp  N P i =1 x i b X u i  ( u ), assign δ u ε x = exp  N P i =1 x i ε deg X i b X u i  ( u ) in the cases wh ere t he right-hand side mak es sense. Prop erty 2.1.20. F or each v ector ﬁeld b X u i , i = 1 , . . . , N , w e hav e ( δ u ε ) ∗ b X u i ( g ) = ε deg X i b X u i ( δ u ε g ). This prop erty comes from tho se on the “ canonical” Carnot gro up T u M [46]. Lemma 2.1.21 ([127]) . Supp ose that u ∈ U . The e quality j X i =1 X | µ + e i | h =deg X j , | µ + e i | = l, µ> 0 x i F j µ,e i ( u ) x µ = 0 , x = ( x 1 , . . . , x N ) ∈ R N , holds for al l deg X j ≥ 2 , l = 2 , . . . , deg X j . Pr o of. Consider a v e ctor ﬁeld X = N P i =1 x i ( b X i u ) ′ . It is kno wn that exp r sX ◦ exp r tX ( g ) = exp r ( s + t ) X ( g ). Therefore, b y (2.1.4), we ha v e X | µ + β | h =deg X j , µ> 0 , β > 0 r | µ + β | F j µ,β ( g ) s | µ | x µ · t | β | x β = 0 for all ﬁxed s and t , deg X j ≥ 2. It follo ws that the co eﬃcien ts at all p o w ers of r v anish. In particular, if | µ + β | = l ≥ 2 then X | µ + β | h =deg X j , µ> 0 , β > 0 , | µ + β | = l F j µ,β ( g ) s | µ | x µ · t | β | x β = 0 . Consequen tly , if | β | = 1 then w e infer P ( s ) = deg X j X l =2 s l − 1 j X i =1 X | µ + e i | h =deg X j , | µ + e i | = l, µ> 0 x i F j µ,e i ( g ) x µ ≡ 0 , where s is an arbitrarily small parameter. Therefore, all co eﬃcien ts of the p olynomial P ( s ) at the p o w ers of s v anish. The lemma fo llo ws. 20 Lemma 2.1.22 ([127]) . L et u ∈ U b e an arbitr ary p oint. T h en a = exp  N X i =1 a i X i  ( u ) = exp  N X i =1 a i b X u i  ( u ) for al l | a i | < r u , i = 1 , . . . , N . Pr o of. Lemma 2 .1.21 implies that the line R ∋ t 7→ t ( a 1 , . . . , a N ) is t he in te- gral line of the v ector ﬁeld N P i =1 a i ( b X u i ) ′ starting at 0 as t = 0. By the de ﬁnition of the exp onen tial map, w e infer R N ∋ ( a 1 , . . . , a N ) = N P i =1 a i ( b X u i ) ′ ( a 1 , . . . , a N ) = exp  N P i =1 a i ( b X u i ) ′  , i. e. the exp onen tial map equals the iden tit y . F rom this, it follo ws immediately that a = θ u ( a 1 , . . . , a N ) = θ u  N X i =1 a i ( b X u i ) ′  = θ u  exp  N X i =1 a i ( b X u i ) ′  = exp  N X i =1 a i b X u i  according to Remark 2.1.19. Deﬁnition 2.1.23. Supp ose that M is a Carnot manifold, and u ∈ M . F or a, p ∈ G u M , whe re a = ex p  N X i =1 a i b X u i  ( p ) , w e deﬁne the quasime tric d u ∞ ( a, p ) = max i {| a i | 1 deg X i } on G u M . The follo wing prop erties comes fr om those on the “canonical” Carnot group T u M [46]. Prop erty 2.1.24. It is easy to see that d u ∞ ( x, y ) is a quasimetric on G u M meeting the following prop erties: 1. d u ∞ ( x, y ) ≥ 0, d u ∞ ( x, y ) = 0 if and only if x = y ; 2. d u ∞ ( u, v ) = d u ∞ ( v , u ); 3. the quasime tric d u ∞ ( x, y ) is con tin uous with resp ect to eac h o f its v ari- ables; 21 4. there exists a constan t Q △ = C △ ( U ) suc h that the inequalit y d u ∞ ( x, y ) ≤ Q △ ( d u ∞ ( x, z ) + d u ∞ ( z , y )) holds fo r ev ery triple of p oin ts x , y , z ∈ U . Prop erty 2.1.25. Let w ε = exp  N X i =1 ε deg X i w i b X u i  ( v ) and g ε = exp  N X i =1 ε deg X i g i b X u i  ( v ) . Then d u ∞ ( w ε , g ε ) = εd u ∞ ( w 1 , g 1 ). By Box u ( x, r ) w e denote a set { y ∈ M : d u ∞ ( x, y ) < r } . Prop erty 2.1.26. W e ha v e δ u ε (Bo x u ( u, r )) = Box u ( u, εr ) . 2.2 Gromo v’s Theorem on the Nilp oten t ization of V ec- tor Fields a nd Esti mate of the Diameter of a Bo x Deﬁnition 2.2.1. Supp ose that M is a Carnot manifold, and let U ⊂ M b e as in Assumption 2.1.11. Giv en v = exp  N X i =1 v i X i  ( u ) u, v ∈ U , deﬁne the quasimetric d ∞ ( u, v ) = max i {| v i | 1 deg X i } . By Bo x( x, r ) w e denote a set { y ∈ M : d ∞ ( x, y ) < r } , r ≤ r x . Deﬁnition 2.2.2. Using the nor mal co ordinates θ − 1 u , deﬁne the action of the dila tion gr oup ∆ u ε in a neighborho o d of a p oint u ∈ M : to a n elemen t x = exp  N P i =1 x i X i  ( u ), assign ∆ u ε x = exp  N P i =1 x i ε deg X i X i  ( u ) in the cases where the righ t-hand side mak es sense. Prop erty 2.2.3. By Lemma 2.1.22 w e ha v e ∆ u ε x = δ u ε x . Prop erty 2.2.4. By Lemma 2.1.22 w e ha v e Box u ( u, r ) = Box( u, r ). Prop erty 2.2.5. W e ha v e ∆ u ε (Bo x( u, r )) = Box( u, εr ), r ∈ (0 , r u ]. Prop erty 2.2.6. The quasimetric d ∞ has the follow ing prop erties: 1. d ∞ ( u, v ) ≥ 0 , d ∞ ( u, v ) = 0 if and o nly if u = v ; 2. d ∞ ( u, v ) = d ∞ ( v , u ); 22 3. the quasime tric d ∞ ( u, v ) is con tin uous with resp ect to eac h of its v ari- ables; 4. there exists a constan t Q = Q ( U ) suc h that the inequalit y d ∞ ( u, v ) ≤ Q ( d ∞ ( u, w ) + d ∞ ( w , v )) holds fo r ev ery triple of p oin ts u , w , v ∈ U . Pr o of. The pro of of pro perties 1–3 is based on known pro p erties of solutions to ODE’s. W e prov e the generalized t riangle ineq ualit y at the end of curren t subsection (see Corollary 2.2.14). Theorem 2.2.7. L et X j ∈ C 1 . Fix u ∈ M . If d ∞ ( u, w ) = C ε , then b X u j ( w ) = X k : deg X k ≤ deg X j [ δ k j + O ( ε ) ] X k + X k : deg X k > deg X j o ( ε deg X k − deg X j ) X k ( w ) , j = 1 , . . . , N . Al l o ( · ) ar e unifo rm in u b elonging to some c omp act subset of U . Pr o of. 1 st Step. Apply ing the mapping θ − 1 u to eac h v ector ﬁeld b X u j , j = 1 , . . . , N , w e deduce D θ − 1 u b X u j ( s ) = N X k =1 z k j ( s ) e k , where { e k } N k =1 is the collection of the ve ctors of the standard basis in R N , and b y ( 2.1.5) z k j ( s ) = δ k j + X | µ | h =deg X k − deg X j > 0 F k µ,e j ( u ) s µ . Note that, here | s µ | = O ( ε deg X k − deg X j ), since d ∞ (0 , s ) = d ∞ ( θ − 1 u ( u ) , s ) = d u ∞ ( u, θ u ( s )) = O ( ε ) . Then b X u j ( θ u ( s )) = N X k =1 z k j ( s ) D θ u ( s ) e k = N X k =1 z k j ( s )  X k ( θ u ( s ))+ 1 2  X k , N X l =1 s l X l  ( θ u ( s ))  , since D θ u ( s ) e k = X k ( θ u ( s )) + 1 2 h X k , N P l =1 s l X l i ( θ u ( s )), where s = ( s 1 , . . . , s N ). 23 T o understand the latter, it is enough to consider the follo wing equalities θ u ( s + r e k ) = ex p  N X l =1 s l X l + r X k  ( u ) = exp  N X l =1 s l X l + r X k  ◦ exp  − N X l =1 s l X l  ◦ exp  N X l =1 s l X l  ( u ) = exp  r X k + r 2  X k , N X l =1 s l X l  + o ( r )  ( θ u ( s )) , and note (see j ustiﬁcation of this calculatio n for C 1 -v ector ﬁelds in [2]) that D θ u ( s ) e k = ∂ ∂ r θ u ( s + r e k )    r =0 = X k ( θ u ( s )) + 1 2  X k , N X l =1 s l X l  ( θ u ( s )) . In view of the pro p erties of the p oint s , w e get | s l | = O ( ε deg X l ), l = 1 , . . . , N . Moreov er, taking into accoun t the deﬁnition of a Carnot manifold, w e hav e  X k , N X l =1 s l X l  ( θ u ( s )) = N X l =1 X m :deg X m ≤ deg X k +deg X l c k lm ( θ u ( s )) X m ( θ u ( s )) . Consequen tly , b X u j ( θ u ( s )) = N X k =1 z k j ( s ) X k ( θ u ( s )) + 1 2 N X k =1 N X l =1 X deg X m ≤ deg X k +deg X l z k j ( s ) s l c k lm ( θ u ( s )) X m ( θ u ( s )) = N X k =1 h z k j ( s ) + 1 2 X m,l :deg X k ≤ deg X m +deg X l z m j ( s ) s l c mlk ( θ u ( s )) i X k ( θ u ( s )) . where   z m j ( s )   = O ( ε deg X m − deg X j ) and   z m j ( s ) s l   = O ( ε deg X m +deg X l − deg X j ) . (2.2.1) 24 Represen t the last sum as X k : deg X k < deg X j h z k j ( s ) + 1 2 X m,l :deg X k ≤ deg X m +deg X l z m j ( s ) s l c mlk ( θ u ( s )) i X k ( θ u ( s )) + X k : deg X k =deg X j h z k j ( s ) + 1 2 X m,l :deg X j ≤ deg X m +deg X l z m j ( s ) s l c mlj ( θ u ( s )) i X j ( θ u ( s )) + X k : deg X k > deg X j h z k j ( s )+ 1 2 X m,l :deg X k ≤ deg X m +deg X l z m j ( s ) s l c mlk ( θ u ( s )) i X k ( θ u ( s )) . (2.2.2) Note that, we ha v e z k j ( s ) = 0 if k < j . Next, if k < j and deg X k = deg X m + deg X l , w e hav e m < j and z m j ( s ) = 0. Th us, for t he ﬁrst sum equals X k : deg X k < deg X j h 1 2 X m,l :deg X k < deg X m +deg X l z m j ( s ) s l c mlk ( θ u ( s )) i X k ( θ u ( s )) . Similarly , f or the second sum w e ha v e z k j ( s ) = δ k j , and if deg X j = deg X m + deg X l then z m j ( s ) = 0 since this relation implie s m < j . Th us, we obtain X k : deg X k =deg X j h δ k j + 1 2 X m,l :deg X j < deg X m +deg X l z m j ( s ) s l c mlj ( θ u ( s )) i X j ( θ u ( s )) . In the third sum, the functions z k j ( s ) and z m j ( s ) can tak e an y p o ssible v alues . 2 nd Step. No w, w e calculate more exact estimates of (2.2.1). • Let deg X k > deg X j and deg X k = deg X m + deg X l . F ro m t he ab o v e estimate w e infer   z m j ( s ) s l   = O ( ε deg X k − deg X j ) . Next, suppo se that deg X k > deg X j and deg X k < deg X m + deg X l . Then all the situations deg X m > deg X j , deg X m = deg X j and deg X m < deg X j are p ossible. Here w e hav e   z m j ( s ) s l   =      εO ( ε deg X l ) ≤ ε O ( ε deg X k − deg X j ) if deg X m > deg X j , O ( ε deg X l ) ≤ ε O ( ε deg X k − deg X j ) if deg X m = deg X j , 0 if deg X m < deg X j . 25 • Let now deg X k = deg X j and deg X k < deg X m + deg X l . W e again ha v e to consider the situations deg X m > deg X j , deg X m = deg X j and deg X m < deg X j . It follows   z m j ( s ) s l   =      εO ( ε deg X l ) ≤ ε O (1) if deg X m > deg X j , O ( ε deg X l ) ≤ ε O (1) if deg X m = deg X j , 0 if deg X m < deg X j . • Finally , let deg X k < deg X j and deg X k < deg X m + deg X l . In three situations deg X m > deg X j , deg X m = deg X j and deg X m < deg X j , w e o btain the same result as in the previous case:   z m j ( s ) s l   =      εO ( ε deg X l ) ≤ ε O (1) if deg X m > deg X j , O ( ε deg X l ) ≤ ε O (1) if deg X m = deg X j , 0 if deg X m < deg X j . Th us, in the ﬁrst sum of (2.2.2 ), the coeﬃcien ts at X k equal O ( ε ), and in t he second sum the co eﬃcien t at X j equals 1 + O ( ε ), a nd the co eﬃcien ts at X k for k 6 = j equal O ( ε ) . 3 rd Step. Consider the last sum (where deg X k > deg X j ). Note that, c mlk ( θ u ( s )) = c mlk ( u ) + o (1 ) . (2.2.3) Then, taking in to accoun t (2.2.1 ) and the results of the 2 nd step, we deduce X m,l :deg X k ≤ deg X m +deg X l z m j ( s ) s l c mlk ( θ u ( s )) = X m,l :deg X k =deg X m +deg X l z m j ( s ) s l c mlk ( θ u ( s )) + X m,l :deg X k < deg X m +deg X l z m j ( s ) s l c mlk ( θ u ( s )) = X m,l :deg X k =deg X m +deg X l z m j ( s ) s l c mlk ( u ) + o (1 ) · ε deg X k − deg X j + ε · O ( ε deg X k − deg X j ) = X m,l :deg X k =deg X m +deg X l z m j ( s ) s l c mlk ( u ) + o (1) · ε deg X k − deg X j . (2.2.4) 26 Consequen tly , b X u j ( θ u ( s )) = X k : deg X k ≤ deg X j [ δ k j + O ( ε ) ] X k + X k : deg X k > deg X j h z k j ( s ) + 1 2 X m,l z m j ( s ) s l c mlk ( u ) + o ( ε deg X k − deg X j ) i X k ( θ u ( s )) , where m, l in the last sum are suc h that deg X k = deg X m + deg X l . 4 th Step. It o nly remains to show that z k j ( s ) + 1 2 X m,l :deg X k =deg X m +deg X l z m j ( s ) s l c mlk ( u ) = δ k j . (2.2.5) F or obtaining this, consider the mapping b θ u ( x ) = exp  N P i =1 x i b X u i  ( u ) = θ u ( x ), and apply the argumen ts of the 1 st step with the follo wing diﬀerence: it is kno wn, t hat the ve ctor ﬁelds b X u i , i = 1 , . . . N , are contin uous, but t hey ma y not b e diﬀeren tiable, and formally , we cannot consider comm utators of suc h v ector ﬁelds. Therefore w e mo dify previous argumen ts. F or doing this, we consider the follow ing represen tat ion of the iden tical mapping: b θ 0 ( s ) = exp  N X i =1 s i D b θ − 1 u ( b X u i )  (0) = s, and represe n t e k = D b θ 0 ( s )( e k ) as before w e repres en ted D θ u ( s )( e k ). It is po s- sible, since the v ector ﬁelds D b θ − 1 u ( b X u i ), i = 1 , . . . , N , are smo oth. Similarly to the 1 st step, w e infer D b θ 0 ( s )( e k ) = D b θ − 1 u ( b X u k )( b θ 0 ( s )) + 1 2  D b θ − 1 u ( b X u k ) , N X l =1 s l D b θ − 1 u ( b X u l )  ( b θ 0 ( s )) . Since b θ 0 ( s ) = s and in view of pro p erties of the v ector ﬁelds D b θ − 1 u ( b X u i ), i = 1 , . . . , N , w e deduce e k = D b θ − 1 u ( b X u k )( s ) + 1 2  D b θ − 1 u ( b X u k ) , N X l =1 s l D b θ − 1 u ( b X u l )  ( s ) = D b θ − 1 u ( b X u k )( s ) + 1 2 N X l =1 s l X deg X m =deg X k +deg X l c k lm ( u ) D b θ − 1 u ( b X u m )( s ) . 27 It follo ws D b θ u ( s ) e k = b X u k ( θ u ( s )) + 1 2 N X l =1 s l X deg X m =deg X k +deg X l c k lm ( u ) b X u m ( θ u ( s )) . Applying further the argumen ts o f the 1 st step, w e hav e b X u j ( θ u ( s )) = N X k =1 h z k j ( s ) + 1 2 X m,l :deg X k =deg X m +deg X l z m j ( s ) s l c mlk ( u ) i b X u k ( θ u ( s )) , and th us (2.2.5) is prov ed. T aking in to account the result of the 3 rd step, we obtain b X u j ( w ) = X k : deg X k ≤ deg X j [ δ k j + O ( ε ) ] X k + X k : deg X k > deg X j o ( ε deg X k − deg X j ) X k ( w ) , j = 1 , . . . , N . The theorem follows. Remark 2.2.8. 1. If the v ector ﬁelds X i , i = 1 , . . . , N , b elong to the class C 1 ,α , α ∈ (0 , 1], then in (2.2.3) and, consequen tly , in (2.2.4), w e obtain o (1) = O ( ρ ( u, θ u ( s )) α ). In this case, we ha v e b X u j ( w ) = X k : deg X k ≤ deg X j [ δ k j + O ( ε ) ] X k + X k : deg X k > deg X j ρ ( u, θ u ( s )) α · o ( ε deg X k − deg X j ) X k ( w ) . 2. If the deriv ativ es of the basis v ec tor ﬁelds a re H¨ o lder with respect to d ∞ , w e obtain o (1) = O ( d ∞ ( u, θ u ( s )) α ) = O ( ε α ), and b X u j ( w ) = X k : deg X k ≤ deg X j [ δ k j + O ( ε )] X k + X k : deg X k > deg X j O ( ε deg X k − deg X j + α ) X k ( w ) . 3. If the deriv ativ es of the basis v ec tor ﬁelds a re H¨ o lder with respect to d , w e ha v e b X u j ( w ) = X k : deg X k ≤ deg X j [ δ k j + O ( ε ) ] X k + X k : deg X k > deg X j d ( u, θ u ( s )) α · o ( ε deg X k − deg X j ) X k ( w ) . 28 Corollary 2.2.9. F or x ∈ Box( u, ε ) , the c o eﬃcients { a j,k ( x ) } N j,k =1 fr om the e quality X j ( x ) = N X k =1 a j,k ( x ) b X u k ( x ) (2.2.6) enjoy the fol lowing pr op erty : a j,k ( x ) =      O ( ε ) if deg X j > deg X k , δ k j + O ( ε ) if deg X j = deg X k , o ( ε deg X k − deg X j ) if deg X j < deg X k , (2.2.7) j = 1 , . . . , N . A l l “o” ar e uniform in u b elonging to so m e c omp act subset of U . Pr o of. According to Theorem 2.2.7, the coeﬃcien ts b j,k ( x ) from the relation b X u j ( x ) = N X k =1 b j,k ( x ) X k ( x ) , j = 1 , . . . , N , ha v e the same pro perties. Put A ( x ) = ( a j,k ( x )) N j,k =1 and B ( x ) = ( b j,k ( x )) N j,k =1 . Then A ( x ) = B ( x ) − 1 . W e use the w ell-kno wn fo rm ula of calculation of the en tries of the in- v erse matrix to estimate all a j,k ( x ), j, k = 1 . . . , N . W e estimate the v alue | a j,k ( x ) | = | det B j,k ( x ) | | det B ( x ) | , where the ( N − 1) × ( N − 1) -matrix B j,k is constructe d from the matrix B ( x ) b y deleting its j th column and k th line. It is easy to see that | det B ( x ) | = 1 + O ( ε ), where O ( ε ) is uniform f or x b elonging to some compact neigh b orho o d U ⊂ M . Next, we estimate | det B j,k ( x ) | . Obviously , | det B j,j ( x ) | = 1 + O ( ε ), where O ( ε ) is uniform fo r x b elonging to some compact neigh b orho o d U ⊂ M , j = 1 , . . . , N . Let now k > j . By construction, the diagonal elemen ts with n um bers ( i, i ), j ≤ i < k , equal o ( ε deg X i +1 − deg X i ), and the eleme n ts unde r thes e ones equal 1 + O ( ε ). Note that, det B j,k ( x ) up to a multiple (1 + O ( ε )) equals the pro duct of determinan ts of the following three matrices : the ﬁrst P ( x ) = p i,l ( x ) is a ( j − 1) × ( j − 1)-matrix with p i,l ( x ) = b i,l ( x ), the sec ond Q ( x ) = q i,l ( x ) is a ( k − j ) × ( k − j )-matrix with q i,l ( x ) = b i + j − 1 ,l + j ( x ), and the third R ( x ) = r i,l ( x ) with r i,l ( x ) = b i + k − 1 ,l + k − 1 ( x ). F or the matrices P ( x ) and R ( x ) w e hav e | det P ( x ) | = 1 + O ( ε ) and | det R ( x ) | = 1 + O ( ε ). By construction, q i,i ( x ) = o ( ε deg X i +1 − deg X i ) and q i +1 ,i ( x ) = 1 + O ( ε ). W e hav e that the pro duct of the diagonal elemen ts of 29 Q ( x ) equals k − 1 Y i = j o ( ε deg X i +1 − deg X i ) = o ( ε deg X k − deg X j ) . It is easy t o se e that, for all other summands constituting det Q ( x ), we ha v e the same estimate. Similarly , w e sho w that for k < j w e ha v e | det B j k ( x ) | = O ( ε ). Here O ( ε ) is uniform for x b elonging to s ome compact neigh b orho o d U ⊂ M . The lemma follo ws. Remark 2.2.10. Similarly to Remark 2.2.8: • if X i ∈ C 1 ,α then a j,k ( x ) =      O ( ε ) if deg X j > deg X k , δ k j + O ( ε ) if deg X j = deg X k , ρ ( u, x ) α · o ( ε deg X k − deg X j ) if deg X j < deg X k , • if the deriv ativ es of the basis ve ctor ﬁelds are H¨ older with resp ect to d ∞ then a j,k ( x ) =      O ( ε ) if deg X j > deg X k , δ k j + O ( ε ) if deg X j = deg X k , O ( ε deg X k − deg X j + α ) if deg X j < deg X k , • if the deriv ativ es of the basis v e ctor ﬁelds are H¨ older with resp ect to d then a j,k ( x ) =      O ( ε ) if deg X j > deg X k , δ k j + O ( ε ) if deg X j = deg X k , d ( u, x ) α · O ( ε deg X k − deg X j ) if deg X j < deg X k , j = 1 , . . . , N . Corollary 2.2.9 imply instan tly G romo v’s Theorem on the nilp oten tization of ve ctor ﬁelds [68]. Corollary 2.2.11 (G romo v’s Theorem [68]) . L et X j ∈ C 1 . On Bo x( g , εr g ) , c on sider the ve ctor ﬁelds { ε X i } = { ε deg X i X i } , i = 1 , . . . , N . Then the uniform c on ver genc e X ε i =  ∆ g ε − 1  ∗ ε X i → b X g i as ε → 0 , i = 1 , . . . , N , holds a t the p oints of the b ox Bo x( g , r g ) and this c onver genc e is uniform in g b elo nging to some c omp act set. 30 Pr o of. Really , by (2.2.6), (2.2.7) and in view of Corollary 2.2.9 and Prop- ert y 2.1.20, w e inf er X ε i ( x ) =  ∆ g ε − 1  ∗ ε X i  ( x ) = ε deg X i N X k =1 a i,k  ∆ g ε ( x )  ∆ g ε − 1  ∗ b X g k  ( x ) = N X k =1 ε deg X i − deg X k a i,k  ∆ g ε ( x )  b X g k ( x ) = X k : deg X k ≤ deg X i ε deg X i − deg X k ( δ ik + O ( ε ) ) b X g k ( x ) + X k : deg X k > deg X i o (1) b X g k ( x ) as ε → 0. It follow s the uniform con v ergence X ε i =  ∆ g ε − 1  ∗ ε X i → b X g i as ε → 0 , i = 1 , . . . , N , at the p oin ts of the box Box( g , r g ) and this con v ergence is unifor m in g b elonging to some compact set. Remark 2.2.12. F or C ∞ -v ector ﬁelds, the ab ov e corollary is formulated in [101, 119] in another wa y: b X g i is an ho mogeneous part o f X i , 1 = 1 , . . . , N . This statemen t implies Corollary 2.2.11. It is sho wn in [67] that, applying similar argumen ts, the smo othness of vec tor ﬁelds can b e reduced to b e 2 M + 1 . Estimates (2.2.7) w ere written in the pro o f of [13 0, Thereom 3 .1] as a consequenc e of the Gromov’s Theorem whic h can b e pro v ed by metho d of [119] under an additional smo othness of vec tor ﬁelds: X j ∈ C 2 M − deg X j . Corollary 2.2.11 sho ws that estimates (2 .2.7) a re no t only nec essary but also suﬃcien t for the v alidit y of the Gromov ’s Theorem. In our pap er es timates (2.2.7) are obtained under minimal assumption on the smoo thness of v ec tor ﬁelds: X j ∈ C 1 , j = 1 , . . . , N . Th us, taking in to account t he fo otnote in [130, p. 253], all results of pap ers [127, 128, 129, 1 30, 132] are v alid under the same assumptions on the smo othness of basis v ector ﬁelds. Recall that Gromov [68 , p. 130] has formulated the theorem under as- sumption X j ∈ C 1 . V a leri ˇ ı Beresto vs ki ˇ ı sen t us the follow ing example con- ﬁrming that arguments of Gromo v’s pro of hav e to b e corrected. Example. Let X = ∂ ∂ x , Y = xy ∂ ∂ x + ∂ ∂ y + x ∂ ∂ z . Then Z := [ X, Y ] = y ∂ ∂ x + ∂ ∂ z , [ X , Z ] = 0, [ Y , Z ] = ∂ ∂ x − y  y ∂ ∂ x + ∂ ∂ z  = (1 − y 2 ) ∂ ∂ x − y ∂ ∂ z . One can easily see that X, Y , Z constitutes a global frame of smo oth v ec tor ﬁelds o v er the ring of smo oth functions in R 3 . Also for corresponding one-parameter sub groups X ( x ), Y ( y ), Z ( z ), w e ha v e ( X ( x ) ◦ Y ( y ) ◦ Z ( z ))(0 , 0 , 0) = ( x, y , z ). Under this X = ∂ ∂ x on R 3 , Y = ∂ ∂ y on x = 0, Z = ∂ ∂ z on z -line (ev e n on y = 0). On the other hand, ∂ ∂ y Z = X 6 = [ Y , Z ] (see a bov e) on x = 0. This con tradicts to the Gromo v’s statemen t that (A) of [68, p. 131] implies (B) of [6 8, p. 13 2] in general case . 31 Corollary 2.2.13 (Estimate o f the Diameter of a Bo x ) . In a c omp act n e igh- b orh o o d U ⊂ M , for e ach p oint u ∈ U and e ach ε > 0 smal l enough, we have diam(Box( u, ε ) ) ≤ Lε , wher e L dep ends only on U . Pr o of. Assume the con trary: t here exist sequences { ε k } k ∈ N , { u k } k ∈ N , { v k } k ∈ N and { w k } k ∈ N suc h that ε k → 0 a s k → ∞ , d ∞ ( u k , v k ) = ε k and d ∞ ( u k , w k ) ≤ ε k but d ∞ ( v k , w k ) > k ε k . Sinc e U ⊂ M is compact, w e may assume without loss of generalit y that u k → u 0 as k → ∞ . Then v k → u 0 and w k → u 0 as k → ∞ . Assume without loss of generalit y that ε deg X i D ∆ u k ε − 1 X i ( x ) → b X u k i ( x ) as ε → 0 for x ∈ Bo x( u 0 , K r 0 ) uniformly in u k , i = 1 , . . . , N , where K = max { 5 , 5 c 4 } , c is suc h tha t d u k ∞ ( v , w ) ≤ c ( d u k ∞ ( u, v ) + d u k ∞ ( u, w )) for all k ∈ N big e nough, and k ∈ N is big enough (see C orollary 2.2.11). No te that, c < ∞ since c ( u k ) con tin uous ly dep ends on v alues of { F j µ,β ( u k ) } j,µ,β , consequen tly , it dep ends con tin uously on u k . Moreo v er, the c hoice of K implies the follo wing: 1. F o r k big enough, w e hav e that an integral curv e of a v ector ﬁeld with constan t co eﬃcien ts connecting ∆ u k r 0 ε − 1 k ( w k ) and ∆ u k r 0 ε − 1 k ( v k ) in the lo cal Carnot group G u k M lies in Box( u k , K r 0 ); 2. W e ma y c ho ose k b y the follo wing w a y: d ∞ ( u 0 , u k ) < r 0 and the Rie- mannian distance b et w een the in tegral curve s corresp onding to the col- lections { b X u k i } N i =1 and { ( r 0 − 1 ε k ) deg X i D ∆ u k r 0 ε − 1 k X i } N i =1 (with constan t co- eﬃcien ts) that connect p oints ∆ u k r 0 ε − 1 k ( w k ) and ∆ u k r 0 ε − 1 k ( v k ), is less than r 0 . Fix k ∈ N . Then v k = exp  N P i =1 ξ i ε deg X i k X i  ( u k ), w k = exp  N P i =1 η i ε deg X i k X i  ( u k ), and w k = ex p  N P i =1 ζ i ( ε k ) ε deg X i k X i  ( v k ). Apply the mapping ∆ u k r 0 ε − 1 k to v k and w k . W e hav e ∆ u k r 0 ε − 1 k ( w k ) = exp  N X i =1 ζ i ( ε ) ε deg X i k D ∆ u k r 0 ε − 1 k X i   ∆ u k r 0 ε − 1 k ( v k )  . Note that, d ∞  u k , ∆ u k r 0 ε − 1 k ( v k )  = r 0 and d ∞  u k , ∆ u k r 0 ε − 1 k ( w k )  ≤ r 0 . In view of Corollary 2.2 .11, the v ector ﬁelds ( r 0 − 1 ε k ) deg X i D ∆ u k r 0 ε − 1 k X i ( x ) = b X u k i ( x ) + o (1), i = 1 , . . . , N , where o (1) is uniform in x and in u k . Conseq uen tly , since dim span { b X u k i ( x ) } N i =1 = N at each x ∈ Box( u 0 , r 0 ), the Riemannian distance b et w ee n ∆ u k r 0 ε − 1 k ( w k ) and ∆ u k r 0 ε − 1 k ( v k ) is bounded f rom ab ov e f or all k ∈ N big enough. Therefore, the co eﬃcien ts ζ i ( ε k ), i = 1 , . . . , N , are b ounded f rom 32 ab o v e fo r all k ∈ N big enough. The a ssum ption d ∞ ( v k , w k ) > k ε k con tradicts this conclusion. Th us there exists a constan t L = L ( U ) suc h that diam(Bo x( u, ε )) ≤ Lε for u ∈ U . The statemen t follows. F rom the previous statemen t we come immediately to the following Corollary 2.2.14 (T ria ngle inequalit y) . The quasimetric d ∞ ( x, y ) m e ets lo- c al ly the gen e r alize d triangle ine quality ( se e Pr op erty 2.2.6 ) . Corollary 2.2.15 (Decomp osition of the basis v ector ﬁelds) . Fix a p oint θ u ( s ) ∈ Bo x( u, O ( ε )) . R emarks 2.2.8 and 2.2.1 0 imply the fol lowing de c om- p os i tion of Dθ − 1 u X i , i = 1 , . . . , N : [ D θ − 1 u X i ( s )] j = [( b X u i ) ′ ( s )] j + N X k =1 ( a i,k ( θ u ( s )) − δ ik )[( b X u k ) ′ ( s )] j . If d ∞ ( u, θ u ( s )) = O ( ε ) , we have [ D θ − 1 u X i ( s )] j = z j i ( u, s ) + X k : deg X k ≤ deg X i O ( ε ) z j k ( u, s ) + X k : deg X k > deg X i a i,k ( θ u ( s )) z j k ( u, s ) . If deg X j ≤ deg X i then [ D θ − 1 u X i ( s )] j = δ ij + O ( ε ) . F or deg X j > deg X i we have: • If the b asis ve ctor ﬁelds ar e C 1 -smo oth then we de duc e [ D θ − 1 u X i ( s )] j = z j i ( u, s ) + O ( ε deg X j − deg X i +1 ) + o (1) · ε deg X j − deg X i , and ther efor e [ D θ − 1 u X i ( s )] j = z j i ( u, s ) + o ( ε deg X j − deg X i ) . • If the d e rivatives of the b asis ve ctor ﬁelds ar e H α -c o n tinuous with r e- sp e ct to d , then if deg X j > deg X i we have [ D θ − 1 u X i ( s )] j = z j i ( u, s ) + d ( u, θ u ( s )) α · O ( ε deg X j − deg X i ) . In p articular, for α = 1 and d = d ∞ or d = d z ∞ , wher e d ∞ ( u, z ) = O ( ε ) , we have [ D θ − 1 u X i ( s )] j = z j i ( u, s ) + O ( ε deg X j − deg X i +1 ) . 33 2.3 Comparison of Geometrie s of T angen t Cones The goal of Subsec tions 2.3, 2.4 and 2.6 is to compare the geometries of t w o lo cal Carnot groups. The main result of Section 2 is the follo wing Theorem 2.3.1. L et u , u ′ ∈ U b e s uch that d ∞ ( u, u ′ ) = C ε . F or a ﬁxe d Q ∈ N , c onsider p oints w 0 , d ∞ ( u, w 0 ) = C ε , and w ε j = exp  N X i =1 w i,j ε deg X i b X u i  ( w ε j − 1 ) , w ε j ′ = exp  N X i =1 w i,j ε deg X i b X u ′ i  ( w ε ′ j − 1 ) , w ε ′ 0 = w ε 0 = w ′ 0 = w 0 , j = 1 , . . . , Q . ( Her e Q ∈ N is such that al l these p oin ts b elo ng to the neighb orho o d U ⊂ M , for al l ε > 0 . ) Then for α > 0 , max { d u ∞ ( w ε Q , w ε ′ Q ) , d u ′ ∞ ( w ε Q , w ε ′ Q ) } = ε · [Θ( C , C , Q, { F j α,β } j,α,β )] ρ ( u, u ′ ) α M . (2.3.1) In the c ase of α = 0 , we have max { d u ∞ ( w ε Q , w ε ′ Q ) , d u ′ ∞ ( w ε Q , w ε ′ Q ) } = ε · [Θ( C , C , Q, { F j α,β } j,α,β )][ ω ( ρ ( u, u ′ ))] 1 M wher e ω → 0 is a m o dulus of c on tinuity. ( Her e Θ is uniform in u, u ′ , w 0 ∈ U and { w i,j } , i = 1 , . . . , N , j = 1 , . . . , Q , b elonging to some c omp act ne i g hb or- ho o d of 0 , a nd it dep end s on Q and { F j µ,β } j,µ,β . ) Remark 2 .3.2. If the deriv atives of X i , i = 1 , . . . , N , are lo cally H¨ older with respect to d , then w e hav e d ( u, u ′ ) α M instead of ρ ( u, u ′ ) α M in ( 2.3.1). In the curren t subs ection w e pro v e the ” base“ o f the main result, i. e., we obtain it for Q = 1 and ε = 1. The full pro of is written in Subsection 2.6. Fix po in ts u, u ′ ∈ U , where U is suc h that Assumption 2.1.4 holds. Recall that the collections of v ector ﬁelds { b X u i } N i =1 and { b X u ′ i } N i =1 are frames in G u M and in G u ′ M res p ectiv ely . Deﬁnition 2.3.3. By b X p ( q ), we denote the matrix, suc h that its i th column consists of the co ordinates of the v ector b X p i ( q ), i = 1 , . . . , N , p ∈ M , q ∈ G p M , in t he frame { b X j } N j =1 . Lemma 2.3.4. Supp ose that Assumption 2.1.4 h olds. L et Ξ( u, u ′ , q ) , q ∈ M , b e the matrix such that b X u ′ ( q ) = b X u ( q )Ξ( u, u ′ , q ) . (2.3.2) Then the entries o f Ξ( u , u ′ , q ) ar e ( lo c al ly ) H α -c o n tinuous i n u and u ′ . 34 Pr o of. The pro of of this statemen t follows from Theorem 2.1 .12. Indeed, it implies that the ve ctor ﬁelds { b X u i } N i =1 are lo cally H α -con tin uous in u . Since w e pro v e a lo cal prop ert y , and M is a Riemannian manifold, then, instead of M , we ma y consider without loss of generalit y some neigh borho o d U ⊂ R N con taining u and u ′ . Then it is easy to see that the en tr ies o f the matrices b X u and b X u ′ are (lo cally) H α -con tin uous on U × U . Since b oth matrices are non-degenerate in U ⊂ M , w e ha v e t hat Ξ( u, u ′ , q ) = b X u ( q ) − 1 b X u ′ ( q ) is also non-degenerate, and its en tries Ξ ij ( u, u ′ , q ) b elong lo cally to C α ( U × U ), i, j = 1 , . . . , N . Remark 2 .3.5. If the deriv atives of X i , i = 1 , . . . , N , are lo cally H¨ older with resp ect to d , then the en tries of Ξ are also lo cally H¨ o lder with resp ect to d (see Remark 2.1.17). Remark 2.3.6. Supp ose tha t Assumption 2.1.4 holds. Since Ξ( u, u ′ , q ) equals the unit matrix if u = u ′ then Ξ ij = δ ij + Θ ρ ( u, u ′ ) α where Θ = Θ( u, u ′ , q ) is a bounded me asurable function: | Θ | ≤ C , and the constan t C ≥ 0 depends only on the neigh bor ho o d U ⊂ M . Pr o of. Note that Ξ( u, u, q ) equals the unit matrix. Then the α -H¨ older con- tin uit y o f all v ector ﬁelds implies | Ξ ij ( u, u ′ , q ) − δ ij | ≤ C ( ρ ( u, u ′ ) α ), where C = sup u,u ′ ,q ∈ U | Ξ ij ( u, u ′ , q ) − δ ij | ρ ( u, u ′ ) α < ∞ dep ends only on the neigh b orho o d U ⊂ M . Remark 2 .3.7. If the deriv atives of X i , i = 1 , . . . , N , are lo cally H¨ older with respect to d , then Ξ ij ( u, u ′ , q ) = δ ij + Θ d ( u , u ′ ) α . Notation 2.3.8. Throughout the paper, b y the sym b ol Θ, we denote some b ounded function absolute v alues of whic h do not exceed some 0 ≤ C < ∞ , where C dep ends only on the neigh b orho o d where Θ is deﬁned (i. e., it do es not depend on p oints of this neigh b orho o d). Theorem 2.3.9. L et w = exp  N X i =1 w i b X u i  ( v ) , w ′ = exp  N X i =1 w i b X u ′ i  ( v ) . Then, for α > 0 , we have max { d u ∞ ( w , w ′ ) , d u ′ ∞ ( w , w ′ ) } = Θ[ ρ ( u , u ′ ) α ρ ( v , w )] 1 M , (2.3.3) wher e u, u ′ , v ∈ U , { w i } N i =1 ∈ U (0) ⊂ R N . In the c ase of α = 0 , max { d u ∞ ( w , w ′ ) , d u ′ ∞ ( w , w ′ ) } = Θ[ ω Ξ ( ρ ( u, u ′ )) ρ ( v , w )] 1 M . 35 Remark 2.3.10. Here (see Notation 2.3.8), the v alue sup | Θ( u, u ′ , v , { w i } N i =1 ) | < ∞ dep ends only on U ⊂ M and U (0) ⊂ R N . Pr o of of The or em 2.3.9. 1 st Step. Fix q ∈ M . Notice that b oth collections of vec tors { b X u i ( q ) } N i =1 and { b X u ′ i ( q ) } N i =1 are frames of T q M . Consequen tly , there exists the transition ( N × N )- matrix Ξ( u, u ′ , q ) = (Ξ( u, u ′ , q )) i,k suc h that b X u ′ i ( q ) = N X k =1 (Ξ( u, u ′ , q )) i,k b X u k ( q ) . (2.3.4) Remark 2.3.6 implies that Ξ( u, u ′ , q ) i,j = ( 1 + Θ i,j ρ ( u, u ′ ) α if i = j, Θ i,j ρ ( u, u ′ ) α if i 6 = j. (2.3.5) Th us b X u ′ i ( q ) = b X u i ( q ) + [Ξ( u, u ′ , q ) − I ] b X u i ( q ) where | [Ξ( u, u ′ , q ) − I ] | k ,j = Θ k ,j ρ ( u, u ′ ) α for all k , j = 1 , . . . , N . 2 nd Step. Consider the integral line γ ( t ) of the ve ctor ﬁeld N P i =1 w i b X u ′ i starting at v with the endpoint w ′ . Rewrite the tangen t v ector to γ ( t ) in the frame { b X u i } N 1 i =1 as ˙ γ ( t ) = N P i =1 w u i ( γ ( t )) b X u i ( γ ( t )). F rom (2.3.4) it follo ws that w u i ( q ) = N X k =1 w k (Ξ( u, u ′ , q )) k ,i . F rom (2.3.5) we can estimate the co eﬃcien t w u i at b X u i : w u i = w i + N X k =1 [ w k Θ k ,i ρ ( u, u ′ ) α ] , i = 1 , . . . , N . (2.3 .6) 3 rd Step. Next, w e estimate the Riemannian distance b et w een w a nd w ′ . By κ ( t ) denote the in tegral line of the v ector ﬁeld N P i =1 w i b X u i connecting v and w , i. e., a line suc h that κ (0) = v and ˙ κ ( t ) = N X i =1 w i b X u i ( κ ( t )) . 36 By means of the mapping θ − 1 u w e transpo rt κ ( t ) and γ ( t ) to R N . Let κ u ( t ) = θ − 1 u ( κ ( t )) and γ u ( t ) = θ − 1 u ( γ ( t )). Then ˙ κ u ( t ) = ( θ − 1 u ) ∗ ( κ ( t )) ˙ κ ( t ) = N X i =1 w i ( b X u i ) ′ ( κ u ( t )) and similarly ˙ γ u ( t ) = N X i =1 w i ( θ − 1 u ) ∗ b X u ′ i = N X i =1 w u i ( t )( b X u i ) ′ ( γ u ( t )) since ( θ − 1 u ) ∗ b X u ′ i ( q ) = N P k =1 (Ξ( u, u ′ , q )) i,k ( b X u i ) ′ ( q ) (see (2 .3.2)). Using fo rm ula (2.1.5) rewrite the tangen t ve ctors in Cartesian co ordinates: ˙ κ u ( t ) = N X i =1 w i N X j =1 z j i ( u, κ u ( t )) ∂ ∂ x j = N X j =1 W j ( u, κ u ( t )) ∂ ∂ x j where W j ( u, κ u ( t )) = N X i =1 w i z j i ( u, κ u ( t )) = w j + j − 1 X i =1 w i z j i ( u, κ u ( t )) . Similarly ˙ γ u ( t ) = N X j =1 W u j ( u, γ u ( t )) ∂ ∂ x j where W j ( u, γ u ( t )) = w u j ( t ) + j − 1 X i =1 w u i ( t ) z j i ( u, γ u ( t )) . No w w e es timate the length of the curv e λ u ( t ) = γ u ( t ) − κ u ( t ) + θ − 1 u ( w ) with endp oin ts θ − 1 u ( w ) and θ − 1 u ( w ′ ). The ta ngen t v ector to λ u ( t ) equ als ˙ λ u ( t ) = ˙ γ u ( t ) − ˙ κ u ( t ) = N X j =1 [ W u j ( u, γ u ( t )) − W j ( u, κ u ( t ))] ∂ ∂ x j = N X j =1 h ( w u j ( t ) − w j ) + X i 0 , we have max { d u ∞ ( w ε , w ′ ε ) , d u ′ ∞ ( w ε , w ′ ε ) } = ε [Θ ( C, C )] ρ ( u, u ′ ) α M . (2.4.1) In the c ase of α = 0 , we have max { d u ∞ ( w ε , w ′ ε ) , d u ′ ∞ ( w ε , w ′ ε ) } = ε [Θ( C , C )] max { ω Ξ ( ρ ( u, u ′ )) , ω ∆ u ε − 1 ,v ◦ ∆ u ′ ε,v ( ρ ( u, u ′ )) } α M , wher e ∆ u ε − 1 ,v is deﬁne d b elow in (2.4.4) and (2.4.5) . ( Her e Θ is uniform in u, u ′ , v ∈ U ⊂ M , a nd in { w i } N i =1 b elo nging to some c om p act neighb orho o d of 0 ( se e Notation 2.3.8 ) . ) Remark 2.4.2. If the deriv ativ es of X i , i = 1 , . . . , N , are lo cally α -H¨ o lder with resp ect to d ( instead of ρ ), then we ha v e d ( u, u ′ ) α M instead of ρ ( u, u ′ ) α M in ( 2.4.1) (the pro of is similar, see Remark 2.3.11). Pr o of of The or em 2.4.1. 1 st Step. Let w = w 1 and w ′ = w ′ 1 as it w as earlier. In the frame { b X u i } N i =1 w e hav e w ′ = exp  N X i =1 w ′ i b X u i  ( v ) . Consider the p oin t ω ε = exp  N X i =1 w ′ i ε deg X i b X u i  ( v ) . Note that ω 1 = w ′ . In view of the generalized triangle inequalit y , d u ∞ ( w ε , w ′ ε ) ≤ c ( d u ∞ ( w ε , ω ε ) + d u ∞ ( ω ε , w ′ ε )). By the ab ov e estimate d u ∞ ( ω ε , w ε ) = εd u ∞ ( w , w ′ ) = ε Θ( ρ ( u , u ′ ) α d u ∞ ( v , w )) 1 M . ( 2.4.2) Note that, if α = 0, then w e obtain here ω Ξ ( ρ ( u, u ′ )). 39 No w w e estimate the dis tance d u ∞ ( ω ε , w ′ ε ). Represen t w ′ ε in the frame { b X u i } N i =1 : w ′ ε = exp  N X i =1 α i ( ε ) ε deg X i b X u i  ( v ) , (2.4.3) and conside r the p oint ω ′ = exp  N X i =1 α i ( ε ) b X u i  ( v ) . Here the co eﬃcien ts α i ( ε ), i = 1 , . . . , N , depend on u and { w i } N i =1 . 2 nd Step. Next, w e sho w that the co eﬃcien ts α i ( ε ), i = 1 , . . . , N , are uniformly b ounded for all ε > 0 uniformly on u and { w i } N i =1 . By ano ther w ords, there exists S < ∞ suc h that d u ∞ ( v , w ′ ε ) ≤ S ε for all ε > 0 small enough and all u and { w i } N i =1 . Indeed, b y the generalized triangle inequalit y for Carnot groups, w e hav e d u ∞ ( v , w ′ ε ) ≤ c ( d u ∞ ( u, v ) + d u ∞ ( u, w ′ ε )) . Next, d u ∞ ( u, w ′ ε ) = d ∞ ( u, w ′ ε ). Since d ∞ ( u, v ) = C ε , it is enough to sho w that d ∞ ( u, w ′ ε ) ≤ K ε . T o do this, we es timate the v alue d ∞ ( u ′ , w ′ ε ). Since d ∞ ( u ′ , w ′ ε ) = d u ′ ∞ ( u ′ , w ′ ε ), t hen in view of the generalized triangle inequalit y for Carnot groups, w e hav e d u ′ ∞ ( u ′ , w ′ ε ) ≤ c ( d u ′ ∞ ( u ′ , v ) + d u ′ ∞ ( v , w ′ ε )) . The conditions d ∞ ( u, u ′ ) = C ε , d ∞ ( u, v ) = C ε and The orem 2.2.13 imply d u ′ ∞ ( u ′ , v ) = d ∞ ( u ′ , v ) ≤ L max { C , C } ε. Applying Theorem 2.2.13 again, we infer d ∞ ( u, w ′ ε ) ≤ K ε. F rom here a nd from the fact that d ∞ ( u, v ) = C ε , we ha v e d u ∞ ( v , w ′ ε ) ≤ S ε for all ε > 0 small enough and all u and { w i } N i =1 b elonging to some compact neigh b orho o ds. F rom here, w e hav e that all α i ( ε ), i = 1 , . . . , N , are b ounded unifo rmly in ε > 0. 40 3 rd Step. Note that d u ∞ ( ω ε , w ′ ε ) = εd u ∞ ( ω ′ , w ′ ). Consider the mapping ∆ u ε,v ( x ) = exp  N X i =1 x i ε deg X i b X u i  ( v ) . (2.4.4) More exactly , M ∋ x 7→ { x 1 , . . . , x N } by suc h a w a y that x = exp  N X i =1 x i b X u i  ( v ) ∆ u ε,v 7− → exp  N X i =1 x i ε deg X i b X u i  ( v ) . (2.4 .5) Sho w that the co ordinate functions are H α -con tin uous in u ∈ M uniformly on ε > 0. 1. The case of α > 0 . Indeed, the mapping θ v,u ( x 1 , . . . , x N ) = exp  N X i =1 x i b X u i  ( v ) , where ( x 1 , . . . , x N ) ∈ Box(0 , T ε ), is H α -con tin uous in u ∈ M as a solu- tion to an eq uation with H α -con tin uous righ t-hand part (see Section 5), and its H¨ older constan t do es not dep end on v b elonging to some compact set. This mapping is also quasi-isometric on ( x 1 ε − deg X 1 , . . . , x N ε − deg X N ) ∈ R N with resp ect to t he R iemannian metric. C onsider no w the in v ers e mapping, whic h assigns to a giv en p oint x ∈ M , d u ∞ ( v , x ) ≤ T ε , the “co ordinates” x 1 ( u, x ) ε − deg X 1 , . . . , x N ( u, x ) ε − deg X N suc h tha t x = exp  N X i =1 x i ( u, x ) b X u i  ( v ) . Note tha t the quasi-isometric co eﬃcien ts of the mapping θ v,u are inde- p enden t fr om ( x 1 , . . . , x N ), u and v belonging to some compact set (here we supp ose that d u ∞ ( v , x ) ≤ T ε ). Show that the functions x 1 ( u, x ) ε − deg X 1 , . . . , x N ( u, x ) ε − deg X N are H α -con tin uous in u ∈ U for a ﬁxed x ∈ M , and their H¨ older constan ts a re b ounded lo cally uniformly in x , v and in ε > 0. (Here, to guaran tee the uniform b oundedness of x 1 ( u, x ) ε − deg X 1 , . . . , x N ( u, x ) ε − deg X N , w e a ssume that • b oth v alues d ∞ ( u, v ) and d u ∞ ( v , x ) are comparable to ε • the p oin t u can b e c hanged only b y a p oin t u ′ , suc h that the distance d ∞ ( u, u ′ ) is also comparable to ε (see 2 nd step).) 41 The latter stateme n t follo ws from the fact, that θ u,v ( x 1 , . . . , x N ) is lo cally H¨ older in u , and its H¨ older constan t is indep enden t of v b elonging t o some compact set, a nd of ( x 1 , . . . , x N ) belonging to some compact neigh borho o d U (0) of zero. Since w e prov e a lo cal pro p ert y of a mapping then w e may as- sume that u , u ′ , x a nd v meet our ab ov e condition on d ∞ -distances and they b elong to some compact neigh borho o d U suc h that the mapping θ u,v is bi- Lipsc hitz on ( x 1 ε − deg X 1 , . . . , x N ε − deg X N ) if u ∈ U ; moreo v er, its bi-Lipschitz co eﬃcien ts are independen t of u , ( x 1 ε − deg X 1 , . . . , x N ε − deg X N ) and v b elong- ing to some compact set. Indeed, consider t he mapping θ v ( u, x 1 , . . . , x N ) = θ u,v ( x 1 , . . . , x N ) and supp ose that for any L > 0 there exist ε > 0, p oin ts v , x ∈ U , a lev el set θ − 1 v ( x ), and p oints ( u, x 1 ( u ) , . . . , x N ( u )) and ( u ′ , x 1 ( u ′ ) , . . . , x N ( u ′ )) on it suc h that    ( x 1 ( u ) ε − deg X 1 , . . . , x N ( u ) ε − deg X N ) − ( x 1 ( u ′ ) ε − deg X 1 , . . . , x N ( u ′ ) ε − deg X N )    ≥ L | u − u ′ | α (2.4.6) for some u and u ′ . The assumption (2.4.6) leads to the following con tradic- tion: 0 =    θ v ( u, x 1 ( u ) ε − deg X 1 , . . . , x N ( u ) ε − deg X N ) − θ v ( u ′ , x 1 ( u ′ ) ε − deg X 1 , . . . , x N ( u ′ ) ε − deg X N )    ≥    θ v ( u, x 1 ( u ) ε − deg X 1 , . . . , x N ( u ) ε − deg X N ) − θ v ( u, x 1 ( u ′ ) ε − deg X 1 , . . . , x N ( u ′ ) ε − deg X N )    −    θ v ( u, x 1 ( u ′ ) ε − deg X 1 , . . . , x N ( u ′ ) ε − deg X N ) − θ v ( u ′ , x 1 ( u ′ ) ε − deg X 1 , . . . , x N ( u ′ ) ε − deg X N )    ≥ C x    ( x 1 ( u ) ε − deg X 1 , . . . , x N ( u ) ε − deg X N ) − ( x 1 ( u ′ ) ε − deg X 1 , . . . , x N ( u ′ ) ε − deg X N )    − C u | u − u ′ | α ≥ ( LC x − C u ) | u − u ′ | α > 0 ( 2.4.7) if L > C u C x . Note t hat ω ′ = ∆ u ε − 1 ,v (∆ u ′ ε,v ( w ′ )), and w ′ = ∆ u ′ ε − 1 ,v (∆ u ′ ε,v ( w ′ )). Here, f or the p oin t w ′ ε = ∆ u ′ ε,v ( w ′ ), w e ha v e x i ( u, w ′ ε ) = α i ( ε ) · ε deg X i on the one hand, and w e ha v e x i ( u ′ , w ′ ε ) = w i · ε deg X i on the other hand, i = 1 , . . . , N . Since the p oin ts u, u ′ , v a nd w ′ ε meet our assumption on p oin ts, w e ha v e that the H¨ older 42 constan ts of x i ( u, x ) ε − deg X i are b ounded uniformly in { w j } N j =1 b elonging to some neigh borho o d of zero. Hence, ρ ( ω ′ , w ′ ) = Θ ρ ( u, u ′ ) α , and d u ∞ ( ω ′ , w ′ ) = Θ ρ ( u, u ′ ) α M . (2.4.8) 2. The case of α = 0 is prov ed similarly to the previous case. W e pro v e that the functions x 1 ( u, x ) ε − deg X 1 , . . . , x N ( u, x ) ε − deg X N are uniformly con tin uous in u ∈ U for a ﬁxed x ∈ M , a nd this con tin uit y is uniform in x, v and ε > 0. The points under conside ration meet the ab ov e condition. T o prov e our result, w e a ssume the con trary that there exists σ > 0 suc h that fo r an y δ > 0 there exist ε > 0, p oints v , x ∈ U , a level set θ − 1 v ( x ), and p oints ( u, x 1 ( u ) , . . . , x N ( u )) a nd ( u ′ , x 1 ( u ′ ) , . . . , x N ( u ′ )) on it suc h that | u − u ′ | < δ , and in the righ t-hand part of (2.4.6) instead of L | u − u ′ | α , we obtain σ . Rep eating further the sche me of the pro of almost v erbatim and replacing ( LC x − C u ) | u − u ′ | α b y σ C x − ω θ v ( u ) in the righ t-hand pa rt of (2.4.7), w e deduce ρ ( ω ′ , w ′ ) = ω ∆ u ε − 1 ,v ◦ ∆ u ′ ε,v ( ρ ( u, u ′ )) . (2.4.9) W e ma y assume without loss of generality , that ω ∆ u ε − 1 ,v ◦ ∆ u ′ ε,v do es not dep end on x and v (see (2 .4.6) and (2.4.7)). 4 th Step. T aking (2.4.2), (2.4.8) and (2.4.9) in t o account w e obtain d u ∞ ( w ε , w ′ ε ) = ε [Θ( C, C )] ρ ( u, u ′ ) α M for α > 0. Similarly , w e obtain the theorem for α = 0. The theorem follo ws. Corollary 2.4.3. 1. Note that d ∞ ( u, u ′ ) = C ε implies ρ ( u, u ′ ) < C ε . Then, for α > 0 , we have d u ∞ ( w ε , w ′ ε ) = O ( ε 1+ α M ) as ε → 0 wher e O is uniform in u, u ′ , v ∈ U ⊂ M , and i n { w i } N i =1 b elo nging to some c om p act neighb orho o d of 0 , and dep ends on C and C . 2. If α = 0 then d u ∞ ( w ε , w ′ ε ) = o ( ε ) as ε → 0 wher e o is uniform in u, u ′ , v ∈ U ⊂ M , and in { w i } N i =1 b elo nging to some c om p act neighb orho o d of 0 , and dep ends on C and C . 43 Remark 2.4.4. The estimate O ( ε 1+ α M ) is also true for the case of vec tor ﬁelds X i , i = 1 , . . . , N , whic h are H¨ o lder with respect to suc h d that d ∞ ( u, u ′ ) = C ε implies d ( u, u ′ ) = K ε , where K is bo unded fo r u, u ′ ∈ U . A particular case is d = d z ∞ , where d ∞ ( z , u ) ≤ Qε (see Lo cal Appro xima- tion Theorem 2.5.4, case α = 0, b elow). Remark 2.4.5. The estimate O ( ε 1+ α M ) is also true for the case of vec tor ﬁelds X i , i = 1 , . . . , N , whic h are H¨ o lder with respect to suc h d that d ∞ ( u, u ′ ) = C ε implies d ( u, u ′ ) = K ε , where K is bo unded fo r u, u ′ ∈ U . A particular case is d = d z ∞ , where d ∞ ( z , u ) ≤ Qε (see Lo cal Appro xima- tion Theorem 2.5.4, case α = 0, b elow). 2.5 The Appro ximation T heorems In this subsection, we prov e tw o Appro ximation Theorems. Their pro ofs use the follow ing geometric prop erty . Prop osition 2.5.1. F or a neighb orho o d U , ther e exist p ositive c onstants C > 0 and r 0 > 0 dep ending on U , M , and N , such that for any p oints u and v fr om a neighb orho o d U the fol lowin g incl usio n is valid: [ x ∈ Box u ( v,r ) Bo x u ( x, ξ ) ⊆ Box u ( v , r + C ξ ) , 0 < ξ , r ≤ r 0 . Pr o of. Let x = ex p  N P i =1 x i b X u i  ( v ), d u ∞ ( v , x ) ≤ r , and z = exp  N P i =1 z i b X u i  ( x ), d u ∞ ( x, z ) ≤ ξ . W e estimate the distance d u ∞ ( v , z ) applying (2.1.4) to points x and z . Let z = exp  N P i =1 ζ i b X u i  ( v ). Case of deg X i = 1. Then | ζ i | ≤ | x i | + | z i | ≤ ( r + ξ ) deg X i . Case of deg X i = 2. Then | ζ i | ≤ | x i | + | z i | + X | e l + e j | h =2 , l 2. Then w e obta in analogo usly to the previous case | ζ i | ≤ | x i | + | z i | + X | µ + β | h = k ,µ> 0 ,β > 0 | F i µ,β ( u ) | x µ · z β ≤ r k + ξ k + X | µ + β | h = k c µβ i ( u ) r | µ | h ξ | β | h ≤ ( r + C i ( u ) ξ ) deg X i . Here we assume t hat C i ( u ) , c i ( u ) ≥ 1. Denote b y C ( u ) = max i C i ( u ). F rom ab o v e es timates w e obta in d u ∞ ( v , x ) = max i {| ζ i | deg X i } ≤ ma x i { ( r + C i ( u ) ξ ) deg X i deg X i } ≤ r + C ( u ) ξ . Since all the C i ( u )’s a re contin uous on u then w e ma y c ho ose C < ∞ such that C ( u ) ≤ C for all u b elonging to a compact neigh bo rho o d. The lemma follo ws. Theorem 2.5.2 (Appro ximation Theorem) . L et u, u ′ , v , w ∈ U . Th en the fol lowing estimate is valid: | d u ∞ ( v , w ) − d u ′ ∞ ( v , w ) | = Θ[ ρ ( u, u ′ ) α ρ ( v , w )] 1 M . (2.5.1) Pr o of. Let p = exp  N P i =1 p i b X u i  ( v ) and p ′ = ex p  N P i =1 p i b X u ′ i  ( v ). Notice that if z ∈ Bo x u ( v , d u ∞ ( v , w )) then z ′ ∈ Box u ′ ( v , d u ∞ ( v , w )) and z ∈ Bo x u ′ ( z ′ , R ( u , u ′ )), where R ( u, u ′ ) = sup p ′ ∈ Bo x u ′ ( v,d u ∞ ( v,w )) d u ′ ∞ ( p, p ′ ) . Using Prop osition 2.5.1 w e hav e that Bo x u ( v , d u ∞ ( v , w )) ⊂ [ x ∈ Box u ′ ( v,d u ∞ ( v,w )) Bo x u ′ ( x, R ( u, u ′ )) ⊂ Box u ′ ( v , d u ∞ ( v , w ) + C R ( u, u ′ )) for some C > 0 . Consequen tly , in view o f Theorem 2.3.9 w e can write Bo x u ( v , d u ∞ ( v , w )) ⊂ Bo x u ′ ( v , d u ∞ ( v , w ) + C R ( u, u ′ )) ⊂ Bo x u ′ ( v , d u ∞ ( v , w ) + Θ[ ρ ( u, u ′ ) α ρ ( v , w )] 1 M ) . If d u ∞ ( v , w ) ≤ Θ[ ρ ( u , u ′ ) α ρ ( v , w )] 1 M then the theorem fo llo ws: | d u ∞ ( v , w ) − d u ′ ∞ ( v , w ) | ≤ d u ∞ ( v , w ) + d u ′ ∞ ( v , w ) = Θ[ ρ ( u, u ′ ) α ρ ( v , w )] 1 M . 45 If d u ∞ ( v , w ) > Θ[ ρ ( u, u ′ ) α ρ ( v , w )] 1 M then applying again Prop osition 2.5.1 w e o btain Bo x u ′ ( v , d u ∞ ( v , w ) − Θ[ ρ ( u , u ′ ) α ρ ( v , w )] 1 M ) ⊂ Box u ( v , d u ∞ ( v , w )) . F rom the latter relation it follow s t hat d u ∞ ( v , w ) − Θ[ ρ ( u , u ′ ) α ρ ( v , w )] 1 M ≤ d u ′ ∞ ( v , w ) ≤ d u ∞ ( v , w ) + Θ[ ρ ( u, u ′ ) α ρ ( v , w )] 1 M , and the theorem follo ws. Remark 2 .5.3. If the deriv atives of X i , i = 1 , . . . , N , are lo cally H¨ older with res p ect to d , then w e ha v e d ( u , u ′ ) α instead of ρ ( u, u ′ ) α in (2.5 .1) (the pro of is similar). Appro ximation Theorem and lo cal estimates (see Theorem 2.4.1) imply Lo cal Appro ximation Theorem. Theorem 2.5.4 (Lo cal Appro ximation Theorem) . Assume that d ∞ ( u, u ′ ) = C ε , d ∞ ( u, v ) = C ε and d ∞ ( u, w ) = C ε for some C, C , C < ∞ . 1. If α > 0 , then | d u ∞ ( v , w ) − d u ′ ∞ ( v , w ) | = ε Θ[ ρ ( u, u ′ )] α M Θ( d u ∞ ( v , w ) + o (1)) . (2.5.2) Mor e over, if u ′ = v and α > 0 , then | d u ∞ ( v , w ) − d ∞ ( v , w ) | = ε Θ[ ρ ( u, v )] α M Θ( d u ∞ ( v , w ) + o (1)) . 2. If α = 0 , then | d u ∞ ( v , w ) − d u ′ ∞ ( v , w ) | = ε o (1) = o ( ε ) wher e o i s uniform in u , u ′ , v , w ∈ U ⊂ M . Mor e over, if u ′ = v and α = 0 , then | d u ∞ ( v , w ) − d ∞ ( v , w ) | = o ( ε ) wher e o is uniform in u, v , w ∈ U ⊂ M . Pr o of follows the sc heme as the pro of o f Appro ximation Theorem 2.5.2 with R ( u, u ′ ) = ε [Θ( C, C , C ) ] ρ ( u, u ′ ) α M . The latter equality is v alid b y the unifor- mit y asse rtion of The orem 2.4.1. Remark 2 .5.5. If the deriv atives of X i , i = 1 , . . . , N , are lo cally H¨ older with respect to d , then we ha v e d ( u, u ′ ) α M instead of ρ ( u, u ′ ) α M in (2.5.2) (the pro of is similar). 46 2.6 Comparison of Lo cal Geomet ries of Tw o Lo cal Ca rnot Groups Pr o of of The or em 2.3.1. 1 st Step. Conside r the case of α > 0. The case of Q = 1 is prov ed in Theorem 2 .4.1. 2 nd Step. Consider the case of Q = 2 . First, for the p oints w 2 = w 1 2 and w ′ 2 = w 1 2 ′ , w e ha v e w 2 = exp  N X i =1 ω i, 2 b X u i  ( w 0 ) (2.6.1) and w ′ 2 = exp  N X i =1 ω ′ i, 2 b X u ′ i  ( w 0 ) . (2.6.2) By the form ulas of group op eration, ω i, 2 diﬀers from ω ′ i, 2 in the v alues of { F j µ,β ( u ) } j,µ,β . By Assumption 2.1.4, F j µ,β ( u ′ ) = F j µ,β ( u ) + Θ ρ ( u , u ′ ) α . Consider the auxiliary points w ′′ 2 = exp  N X i =1 ω i, 2 b X u ′ i  ( w 0 ) and w ′′ 2 ε = exp  N X i =1 ω i, 2 ε deg X i b X u ′ i  ( w 0 ) and estimate t he v alue d u ′ ∞ ( w ′′ 2 , w ′ 2 ). F or doing this, w e use the gro up op eration in the lo cal Carnot gro up G u ′ M and Approximation Theorem 2.5.2. Note that, | ω i, 2 − ω ′ i, 2 | = Θ ρ ( u, u ′ ) α . Next, note that while applying the group op eration, all summands lo o k lik e ω i, 2 − ω ′ i, 2 or ω i, 2 − ω ′ i, 2 + P Θ( ω k , 2 ω ′ j, 2 − ω j, 2 ω ′ k , 2 ). By (2.1.4), we deduce ω k , 2 ω ′ j, 2 − ω j, 2 ω ′ k , 2 = ω k , 2 ( ω j, 2 + Θ ρ ( u, u ′ ) α ) − ω j, 2 ( ω k , 2 + Θ ρ ( u, u ′ ) α ) = Θ ρ ( u, u ′ ) α , d u ′ ∞ ( w ′′ 2 , w ′ 2 ) = Θ( ρ ( u, u ′ ) α M ). Here Θ dep ends on C , C , Q = 2 and { F j µ,β ( u ′ ) } j,µ,β . It follows from the formulas of g roup op erat ion in G u M and G u ′ M , that w ε 2 = exp  N X i =1 ω i, 2 ε deg X i b X u i  ( w 0 ) and w ε 2 ′ = exp  N X i =1 ω ′ i, 2 ε deg X i b X u ′ i  ( w 0 ) . 47 By Theorem 2.4.1, w e hav e d u ′ ∞ ( w ′′ 2 ε , w ε 2 ) = ε Θ ρ ( u , u ′ ) α M . By the homogeneit y of the distance d u ′ ∞ w e hav e d u ′ ∞ ( w ′′ 2 ε , w ε ′ 2 ) = εd u ′ ∞ ( w ′′ 2 , w ′ 2 ) = ε Θ ρ ( u , u ′ ) α M , and from the generalize d triangle inequalit y w e deduce d u ′ ∞ ( w ε 2 , w ε 2 ′ ) = ε Θ ρ ( u , u ′ ) α M . In view of Lo cal Appro ximation Theorem 2.5 .4, w e deriv e d u ∞ ( w ε 2 , w ε 2 ′ ) = ε Θ ρ ( u , u ′ ) α M . 3 rd Step. In the case o f Q = 3, it is easy to see from the previous case and the group operatio n, that if w 3 = exp  N X i =1 ω i, 3 b X u i  ( w 0 ) and w ′ 3 = exp  N X i =1 ω ′ i, 3 b X u ′ i  ( w 0 ) , then again | ω i, 3 − ω ′ i, 3 | = Θ ρ ( u, u ′ ) α . Here Θ dep ends on C , C , Q = 3 and { F j µ,β } j,µ,β . (It suﬃces to apply the group op eration in local Carnot groups G u M and G u ′ M to expressions (2.6.1) and (2.6.2) and to p oints w 3 and w ′ 3 , resp ectiv ely .) F rom now on, for obtaining estimate (2.3.1) a t Q = 3, w e rep eat t he argumen ts of the 2 nd Step. 4 th Step. It is easy to see ana logously to the 3 rd Step, that the group op eration and the induction hy p othesis | ω i,l − 1 − ω ′ i,l − 1 | = Θ ρ ( u, u ′ ) α , 3 < l < Q , imply | ω i,l − ω ′ i,l | = Θ ρ ( u, u ′ ) α . Indeed, it suﬃces to put ω i,l and ω ′ i,l instead of ω i, 3 and ω ′ i, 3 , a nd ω i,l − 1 and ω ′ i,l − 1 instead of ω i, 2 and ω ′ i, 2 in the 3 rd Step, and apply argumen ts from the 2 nd Step. The case of α = 0 can b e prov e d b y applying the similar argumen ts. The theorem follow s. 2.7 Comparison of Lo cal Geometries of a Carnot Man- ifold and a Lo cal Carnot Group In this subsection, w e compare the lo cal geometry of a Carnot manifold with the one o f a lo cal Carnot group. 48 Theorem 2.7.1. Fix Q ∈ N . Consider p oints w 0 , u such that d ∞ ( u, w 0 ) = C ε for some C < ∞ , an d b w ε j = exp  N X i =1 w i,j ε deg X i b X u i  ( b w ε j − 1 ) , w ε j = exp  N X i =1 w i,j ε deg X i X i  ( w ε j − 1 ) , w ε 0 = b w ε 0 = b w 0 = w 0 , j = 1 , . . . , Q . ( Her e Q ∈ N is such that al l these p oints b elo ng to a neighb orho o d U ⊂ M smal l enough for al l ε > 0 . ) Then for α > 0 max { d u ∞ ( b w ε Q , w ε Q ) , d ∞ ( b w ε Q , w ε Q ) } = Q X k =1 Θ( C , k , { F j µ,β } j,µ,β ) · ε 1+ α M . (2.7 .1) In the c ase of α = 0 we have { d u ∞ ( b w ε Q , w ε Q ) , d ∞ ( b w ε Q , w ε Q ) } = ε · Θ( C , Q, { F j µ,β } j,µ,β )[ ω ( ε )] 1 M wher e ω ( ε ) → 0 as ε → 0 . Her e | w i,j | ar e b ounde d, and Θ is uniformly b ound e d for u , w 0 ∈ U and { w i,j } , i = 1 , . . . , N , j = 1 , . . . , Q , b elonging to some c omp act neighb orho o d o f 0 , and it dep en d s on Q and { F j µ,β } j,µ,β . Pr o of. F or simplifying the nota tion w e denote the points b w 1 i b y b w i , and w e denote w 1 i b y w i for ε = 1. First, consider the p oin ts b w Q and w Q . Now w e construct a follow ing sequ ence of po in ts. Let ω k ,j = exp  N X i =1 w i,j b X w k i  ( ω k ,j − 1 ) , k = 0 , . . . , Q − 1 , j = 1 , . . . , Q − k , ω k , 0 = w k . Hence, ω Q − 1 , 1 = w Q and d u ∞ ( w Q , w ′ Q ) = O  d u ∞ ( w Q , ω 0 ,Q ) + Q − 1 X k =1 d u ∞ ( ω k ,Q − k , ω k − 1 ,Q − k +1 )  . If α > 0 then, b y Theorem 2.3 .1, d u ∞ ( b w Q , ω 0 ,Q ) = Θ( C , Q, { F j α,β } j,α,β ) ρ ( u, w 0 ) α M , and eac h of the summands d u ∞ ( ω k ,Q − k , ω k − 1 ,Q − k +1 ) = Θ( C , Q − k , { F j α,β } j,α,β ) ρ ( w k , w k − 1 ) α M . 49 By the same t heorem, if w e r eplace w i,j b y w i,j ε deg X i then it is easy to see using induction b y k that ﬁrs tly d u ∞ ( w ε k , w ε k − 1 ) = O ( ε ), secondly d u ∞ ( u, w ε k ) ∼ ε and d u ∞ ( u, ω ε k ,Q − k ) ∼ ε for all k , a nd thirdly d u ∞ ( b w ε Q , ω ε 0 ,Q ) = ε Θ( C , Q, { F j α,β } j,α,β ) ρ ( u, w 0 ) α M and d u ∞ ( ω ε k ,Q − k , ω ε k − 1 ,Q − k +1 ) = ε Θ( C , Q − k , { F j α,β } j,α,β ) ρ ( w ε k , w ε k − 1 ) α M . Th us we obtain d u ∞ ( b w ε Q , w ε Q ) = Q P k =1 Θ( C , k , { F j α,β } j,α,β ) · ε 1+ α M . Since d u ∞ ( b w ε Q , w ε Q ) = O ( ε ) and d u ∞ ( u, b w ε Q ) = O ( ε ) then, b y Lo cal Appro x- imation Theorem 2.5.4, w e hav e d ∞ ( b w ε Q , w ε Q ) = Q X k =1 Θ( C , k , { F j α,β } j,α,β ) · ε 1+ α M . If α = 0, then w e repeat the ab o v e a rgumen ts replacing ρ ( · , · ) α M b y o (1). The theorem follow s. Remark 2 .7.2. If the deriv atives of X i , i = 1 , . . . , N , are lo cally H¨ older with r espect to d , suc h that d ∞ ( x, y ) ≤ ε implies d ( x, y ) ≤ K ε , where K is b ounded on U , then the same estimate as in (2.7 .1) is true (the pro of is similar). A particular case of suc h d is d z ∞ , d ∞ ( u, z ) ≤ Qε (see Lo cal Appro ximation Theorem 2.5.4). 2.8 Applications 2.8.1 Rashevski ˇ ı–Cho w Theorem Deﬁnition 2.8.1. An absolutely con tin uous curv e γ : [0 , a ] → M is said to b e horizontal if ˙ γ ( t ) ∈ H γ ( t ) M for almost all t ∈ [0 , a ]. Its length l ( γ ) equals a R 0 | ˙ γ ( t ) | dt , where the v alue | ˙ γ ( t ) | is calculated using the Riemann tensor on M . Analogously , the canonical Riemann tensor on G u M deﬁnes a length b l of an absolutely contin uous curv e b γ : [0 , a ] → G u M . Deﬁnition 2.8.2. The Carnot–Car ath´ eo d o ry distanc e b et w een p oints x, y ∈ M is deﬁned as d c ( x, y ) = inf γ l ( γ ) where the inﬁm um is tak en o v er all hori- zon tal curves with endp oin ts x and y . 50 Corollary 2.8.3 (of Theorem 2.7.1) . Supp ose that Assumption 2.1.4 holds for α ∈ (0 , 1] . L et g ∈ M . L et also ε b e smal l en ough, and u, v , b w ∈ Bo x( g , ε ) . The p o ints v , b w ∈ Bo x( g , ε ) c an b e joine d in the lo c al Carnot gr oup ( G u M , d u 1 ) ⊃ Bo x( g , ε ) by a horizontal curve b γ c omp ose d by a t most L se gments of inte gr al curves of horizontal ﬁelds b X u i , i = 1 , . . . , dim H 1 . T o the curve b γ it c orr esp onds a curve γ , horizontal with r esp e ct to the initial horizontal distribution H M , c ons titute d by at most L se gments of inte gr al curves of the given horizontal ﬁelds X i , i = 1 , . . . , n . Mor e over, 1. the curve γ ha s en d p oints v , w ∈ Bo x( g , O ( ε )) ; 2. | l ( γ ) − b l ( b γ ) | = o ( b l ( b γ )) ; 3. max { d u ∞ ( b w , w ) , d ∞ ( b w , w ) } ≤ C ε 1+ α M wher e C is indep endent of g , u, v , b w in some c omp act set. Pr o of. The des ired curv e comes from those on an y Carnot group [46]: give n a Carnot group G with the v ector ﬁelds b X 1 , . . . , b X N , eac h p oint x can b e join t with 0 b y a horizon tal curv e γ constituted b y at most L segmen ts γ j , j = 1 , . . . , L , of in tegral curv es of the given basic horizon tal ve ctor ﬁelds b X 1 , b X 2 , . . . , b X dim H 1 , i. e., ( ˙ γ 1 ( t ) = a 1 b X i 1 ( γ 1 ( t )) γ 1 (0) = 0 , ( ˙ γ j ( t ) = a j b X i j ( γ j ( t )) γ j (0) = γ j − 1 (1) , j = 2 , . . . , L , and from here w e hav e x = γ i L (1). By another w ords, x = exp ( a L b X i L ) ◦ · · · ◦ exp( a 1 b X i 1 ) , i j = 1 , . . . , dim H 1 , where | a j | is controlled by the distance d c (0 , x ), j = 1 , . . . , L , and L is inde- p enden t o f x . No w w e carr y o v er a construction describ ed ab o v e to the lo cal Carnot group ( G u M , d u 1 ) ⊃ Box( g , ε ): the giv e n p oints b w, v ∈ G u M can be join t b y a horizon tal curv e b γ : w = exp( a L b X i L ) ◦ · · · ◦ exp( a 1 b X i 1 )( v ) , i j = 1 , . . . , dim H 1 , (2.8.1) j = 1 , . . . , L . Then the curv e γ deﬁned as w = exp( a L X i L ) ◦ · · · ◦ exp( a 1 X i 1 )( v ) , i j = 1 , . . . , dim H 1 , (2.8.2) 51 is horizontal and its length equals b l ( b γ )(1 + o (1)). The estimate max { d u ∞ ( w , w ′ ) , d ∞ ( w , w ′ ) } ≤ C ε 1+ α M follo ws immediately from (2.7.1). Theorem 2.8.4. Supp ose that Assumption 2.1.4 holds for some α ∈ (0 , 1] . L et g ∈ M . Given two p oints w , v ∈ B ( g , ε ) wher e ε i s s mal l enough, ther e exist a curve γ , ho ri z ontal with r e sp e ct to the initial horizontal distribution H M , with endp oints w and v , and a horizontal curve b γ in the lo c al Carnot gr oup ( G g M , d g 1 ) with the same endp oints, such that 1. b l ( b γ ) is e quivalent to d g ∞ ( w , v ); 2. | l ( γ ) − b l ( b γ ) | = o ( b l ( b γ )); 3. if v = g then the length l ( γ ) is e quiva l e n t to d ∞ ( g , w ) . A l l these estima tes ar e uniform in w , v and g of some c omp act neighb orh o o d as ε → 0 . Pr o of. W e can c ho ose ε fro m the conditio n of the theorem b y requests C 2+ α M ε α 2 M 2 ≤ 1 and ε ≤ 1 2 , whe re C is the constan t from Corolla ry 2.8.3. Apply Corollar y 2.8.3 to the p oin ts u = g , v and w . It gives a horizontal curv e γ 1 ( b γ ) with resp ect to the initial horizon tal distribution H M (in the lo cal C arnot group ( G g M , d g 1 )) w ith endpo in ts v and w 1 ( v and w ) constituted b y at most L segmen ts of integral curv es of giv en ho rizon tal ﬁelds X i ( b X g i ), i = 1 , . . . , n . The curv es b γ and γ 1 ha v e lengths comparable with d g 1 ( v , w ), and max { d g ∞ ( w 1 , w ) , d ∞ ( w 1 , w ) } ≤ C ε 1+ α M . Next, w e apply again Corollary 2.8.3 to the p oin ts u = v = w 1 and w . It giv es a horizon tal curv e γ 2 with resp ect t o H M with endp oin ts w 1 and w 2 . Its length is O ( ε 1+ α M ) where O is uniform in u , w ∈ Bo x( g , ε ), and d ∞ ( w 2 , w ) ≤ C ( C ε 1+ α M ) 1+ α M ≤ ε 1+ 2 α M . Assume that we hav e points w 1 , . . . , w k and horizon tal curve s γ l , l = 2 , . . . , k , with resp ect to H M with endpoints w l − 1 and w l , suc h that γ l has a length O ( ε 1+ l − 1 M α ), and d ∞ ( w l , w ) ≤ ε 1+ lα M . W e con tin ue, b y the induc tion, applying Corollary 2.8.3 to the p oints u = v = w k and w . It results a horizon tal curv e γ k +1 with endpoints w k and w k +1 , suc h that γ k +1 has a length O ( ε 1+ kα M ) and d ∞ ( w k +1 , w ) ≤ C ( C ε 1+ kα M ) 1+ α M ≤ ε 1+ k +1 M α . A curve Γ m = γ 1 ∪ . . . ∪ γ m is horizon tal, has endp oints v and w m , its length do es not exc eed l ( γ 1 ) + C ∞ P l =1 ε 1+ lα M ≤ l ( γ 1 ) + C ε 1+ α M and d ∞ ( w m , w ) → 0 a s 52 m → ∞ . Therefore the sequen ce Γ m con v erges to a horizontal curv e γ as m → ∞ with prop erties 1–2 mentioned in the theorem. Under v = g w e can tak e d ∞ ( g , w ) a s ε in ab ov e estimates: it giv es an ev aluation l ( γ ) ≤ C d ∞ ( g , w ). The opp osite inequalit y can b e v eriﬁed directly b y means of the ab ov e obtained estimate: indeed, if d ∞ ( g , w ) = ε then d ∞ ( g , w ) = d g 1 ( g , w ) ≤ C b l ( b γ ) ≤ C l ( γ ) + o ( b l ( b γ )); it follow s that d ∞ ( g , w ) − o  d ∞ ( g , w )  ≤ C l ( γ ) and the estimate d ∞ ( g , w ) ≤ C 1 l ( γ ) holds with C 1 indep enden t of g from some compact neighborho o d if v is close enough to g . Th us we hav e obtained the prop erty 3. As an application of Theorem 2.8.4 we obtain a vers ion of Rashevski ˇ ı– Cho w t y p e connectivit y t heorem. Theorem 2.8.5. Supp ose that Assumption 2.1 . 4 holds for α ∈ (0 , 1] . Every two p oints v , w of a c onne cte d Ca rnot man i f o ld c an b e joine d by a r e cti- ﬁable absolutely c ontinuous horizontal curve γ c om p ose d by n ot mor e than c ountably many se gme n ts of i n te gr al lines of given horizontal ﬁelds . 2.8.2 Comparison of metrics, and Ball–Bo x Theorem Corollary 2.8.6. Supp ose that Assumption 2.1.4 holds for α ∈ (0 , 1] . I n some c omp ac t neighb orho o d the distanc e d c is e quivalen t to the quasimetric d ∞ . Pr o of. An estimate d c ( x, y ) ≤ C 1 d ∞ ( x, y ) for p o in ts x, y from a compact part M fo llo ws from Theorem 2.8.4 . Our nex t goal is to prov e the con v erse estimate. Fix a compact part K ⊂ M and assume the con trary: for a n y n ∈ N there exist points x n , y n ∈ K suc h that d ∞ ( x n , y n ) ≥ nd c ( x n , y n ). In this case w e ha v e d ∞ ( x n , y n ) → 0 as n → ∞ since otherwise w e ha v e sim ultaneously d c ( x n , y n ) → 0 as n → ∞ , and d ∞ ( x n , y n ) ≥ α > 0 for all n ∈ N what is imp ossible. W e can assume also that x n → x ∈ K as n → ∞ and x n 6 = y n . Setting d ∞ ( x n , y n ) = ε n w e ha v e d ∞  x n , ∆ x n r 0 ε − 1 n y n  = r 0 , and d n c  x n , ∆ x n r 0 ε − 1 n y n  ≤ r 0 n − 1 , where the distance d n c is measured with r espect to the frame  X ε n i  with pushed-forw ard Riemannian tensor. As far as the length of v ectors X ε n i , i = 1 , . . . , dim H 1 , is closed to the lengths o f corresp onding nilp oten tized vec tor ﬁelds b X i , i = 1 , . . . , dim H 1 , by Corollary 2.2.11, the Riemannian distance ρ  x n , ∆ x n r 0 ε − 1 n y n  → 0 as n → ∞ . It is in a con tradiction with d ∞  x n , ∆ x n r 0 ε − 1 n y n  = r 0 for all n ∈ N (see Prop osition 2.8.12 for a comparison of metrics). 53 Remark 2.8.7. Note that, for obtaining the estimate d ∞ ( x, y ) ≤ C 2 d c ( x, y ), the v alue α need not to be strictly greater than zero. Th us, the estimate d ∞ ( x, y ) ≤ C 2 d c ( x, y ) is v alid also for α = 0. Another corollary is so called ba ll-b o x theorem pro v ed for smo oth v ector ﬁelds in [110, 68]. Theorem 2.8.8 (Ball– Bo x Theorem ) . Supp ose that Assumption 2.1.4 holds for α ∈ (0 , 1] . Th e shap e of a smal l b al l B ( x, r ) in the metric d c lo o k s lik e a b ox : given c omp act set K ⊂ M ther e ar e c onstants 0 < C 1 ≤ C 2 < ∞ and r 0 indep endent fr om x ∈ K such that Bo x( x, C 1 r ) ⊂ B ( x, r ) ⊂ Box( x, C 2 r ) (2.8.3) for al l r ∈ (0 , r 0 ) . Theorem 2.8.8 implies Corollary 2.8.9. Supp ose that Assumption 2.1.4 holds for α ∈ (0 , 1] . The Hausdorﬀ dimensio n of M e quals ν = M X i =1 i (dim H i − dim H i − 1 ) wher e dim H 0 = 0 . This Corollary extends Mitchell The orem [102] to Carnot–Carath ´ eo dory spaces with minimal smoo thness of v ec tor ﬁelds. Remark 2.8.10. Let Assumption 2.1.4 holds for α ∈ (0 , 1 ]. Applying Corol- lary 2.8.8, w e obtain 1. the generalization of Theorem 2.3 .9 for p oin ts w and w ′ close enough: max { d u c ( w , w ′ ) , d c ( w , w ′ ) } = Θ[ ρ ( u , v ) ρ ( v , w )] 1 M ≤ Θ[ d c ( u, v ) d c ( v , w )] 1 M ; 2. the generalization of Theorem 2.4 .1: max { d u c ( w ε , w ′ ε ) , d c ( w ε , w ′ ε ) } = ε [Θ ( C, C )] ρ ( u, v ) α M ( d u c ( v , w ) + o (1)); 3. the generalization of Theorem 2.7 .1: max { d u c ( b w ε Q , w ε Q ) , d c ( b w ε Q , w ε Q ) } = Q X k =1 Θ( C , k , { F j µ,β } j,µ,β ) · ε 1+ α M . 54 Corollary 2.8 .6 and Theorem 11.11 [7 0] imply the follo wing statemen t con taining a r esult of [6 6] where only the ﬁrst assertion is obta ined under assumption of higher smoothness of v ec tor ﬁelds. Prop osition 2.8.11. L et X and Y b e two families of ve c tor ﬁelds on M with the same horizontal distribution H M for b oth of whi c h Assumption 2.1.4 holds w ith some α ∈ (0 , 1] . Then in some c omp ac t neighb orho o d the fol lowing assertions ar e e quivalent : 1) Ther e exis ts a c onstant C ≥ 1 such that C − 1 d X ∞ ≤ d Y ∞ ≤ C d X ∞ . 2) Ther e e xists a c onstant C ≥ 1 such that C − 1 | X H ϕ | ≤ | Y H ϕ | ≤ C | X H ϕ | for al l ϕ ∈ C ∞ ( M ) . Here d X ∞ and d Y ∞ are quasime trics constructed with resp ect to the bases X and Y , and X H ϕ and Y H ϕ are subgradien ts o f ϕ . Deﬁne the R iemannian quasimetric d riem ( u, v ) betw een a p oint u and a p oin t v = exp  N P i =1 x i X i  ( u ) as d riem ( u, v ) = max {| x i | | i = 1 , . . . , N } . The well-kno wn facts of diﬀerential geometry imply that the metric d riem is equiv alen t to the Riemannian metric ρ on ev ery compactly em b edded domain U ⋐ M , i.e., there exis ts a constan t c indep enden t of the choice o f the p oin ts u , v ∈ U and suc h that c − 1 ρ ( u, v ) ≤ d riem ( u, v ) ≤ cρ ( u, v ) for all u, v ∈ U fo r whic h the quan tities under consideration are deﬁned. Hence, w e ha v e Prop osition 2.8.12. The r elations c − 1 ρ ( u, v ) ≤ d riem ( u, v ) ≤ cd ∞ ( u, v ) ≤ cd riem ( u, v ) 1 M hold for al l x, y ∈ U . Remark 2.8.13. If the deriv at iv es of X i , i = 1 , . . . , N , are lo cally H¨ older with resp ect to d , where d meets conditions o f Remark 2.7.2, the statemen ts of Corollary 2.8.3, Theorem 2.8 .4, Theorem 2.8.5, Corollary 2.8.6, Theo- rem 2.8.8, Corollary 2.8 .9, Remark 2.8.10 and Prop osition 2.8.11 are a lso true. 3 Diﬀeren tiability on a Carn ot Manifold 3.1 Primitiv e calculus F urther, we extend the dilations δ g t to negativ e t b y setting δ g t x = δ g | t | ( x − 1 ) for t < 0. The con v enie nce of t his deﬁnition is seen from the comparison of diﬀeren t kinds of diﬀeren t iabilit y . 55 3.1.1 Deﬁnition Let M , N b e tw o C arnot manifolds. W e denote the v ector ﬁelds on N b y Y i . W e lab el the remaining ob j ects on N (the distance, the tangent cone etc.) with the same sym b ols as on M but with a tilde ˜ excluding the cases where the ob j ects under conside ration a re o b vious: for example, fo r a given mapping f : E → N , it is clear that G g M is the tangen t cone at a p oin t g ∈ M and G f ( g ) N is the tangen t cone at the p oin t f ( g ) ∈ N ; d g c is the metric in t he cone G g M , d f ( g ) c is the metric in G f ( g ) N , etc. Recall that a h o rizontal homomorphism of Carnot groups is a con tin uous homomorphism L : G → e G of Carnot groups suc h that 1) D L (0)( H G ) ⊂ H e G . The notion of a horizon tal homomorphism L :  G g M , d g c  →  G q N , ˜ d q c  , g ∈ M , q ∈ N , of local Carnot gro ups is diﬀerent from this o nly in that the inclusion L ( G g M ∩ exp H G g M ) ⊂ G q N ∩ exp H G q N holds only for v ∈ G g M ∩ exp H G g M suc h that L ( v ) ∈ G q N . Since a homomorphism of Lie groups is con tin uous, it can b e pro v ed that a horizon tal homomorphism L : G → e G also has the prop ert y 2) L ( δ t v ) = ˜ δ t L ( v ) for all v ∈ G and t > 0 (in the case of a hor izon tal ho - momorphism L :  G g M , d g c  →  G q N , ˜ d q c  of lo cal Carnot groups, the equalit y L ( δ t v ) = ˜ δ t L ( v ) is fulﬁlled only for v ∈ G g M and t > 0 suc h that δ t v ∈ G g M and ˜ δ t L ( v ) ∈ G q N ). Deﬁnition 3.1.1. G iv en tw o Carnot manifolds M and N , and a se t E ⊂ M , a mapping f : E → N is called hc -diﬀer entiable at a p oin t g ∈ E if there exists a horizon tal homomorphism L :  G g M , d g c  →  G f ( g ) N , d f ( g ) c  of the nilp oten t tangen t cones suc h that d f ( g ) c ( f ( v ) , L ( v )) = o  d g c ( g , v )  as E ∩ G g M ∋ v → g . (3.1.1) A hor izon tal homomorphism L :  G g M , d g c  →  G f ( g ) N , d f ( g ) c  satisfying condition (3.1 .1), is called a hc - diﬀer e n tial of the mapping f : E → N at g ∈ E on E and is denoted b y D f ( g ). It can b e prov ed [129] that if g is a densit y po in t then the hc -diﬀeren tial is unique. Moreo v er, it is easy to verify that the hc -diﬀeren tial comm utes with the one-parameter dilation gro up: δ f ( g ) t ◦ D f ( g ) = Df ( g ) ◦ δ g t . (3.1.2) Prop osition 3.1.2 ([129]) . De ﬁ nition 3.1.1 is e quivalent to e ach of the fol- lowing assertion: 56 1. d f ( g ) c  ∆ f ( g ) t − 1 f  δ g t ( v )  , L ( v )  = o (1) as t → 0 , wher e o ( · ) is uniform in the p oints v of any c omp act p a rt of G g M ; 2. ˜ d ∞ ( f ( v ) , L ( v )) = o  d g c ( g , v )  as E ∩ G g M ∋ v → g ; 3. ˜ d ∞ ( f ( v ) , L ( v )) = o  d ∞ ( g , v )  as E ∩ G g M ∋ v → g ; 4. ˜ d ∞  ∆ f ( g ) t − 1 f  δ g t ( v )  , L ( v )  = o (1) as t → 0 , wher e o ( · ) is uniform in the p oi n ts v of any c omp act p art of G g M . Pr o of. Consider a p oin t v of a compact part of G g M and a sequence ε i → 0 as i → 0 suc h that δ g ε i v ∈ E for all i ∈ N . F rom (3.1.1) w e hav e d f ( g ) c  f  δ g ε i v  , L  δ g ε i v  = o  d g c  g , δ g ε i v  = o ( ε i ). In view of (3.1.2), we in- fer d f ( g ) c  ∆ f ( g ) ε i  ∆ f ( g ) ε − 1 i f  δ g ε i v  , δ f ( g ) ε i L ( v )  = o ( ε i ) uniformly in v . F rom here w e o btain item 1. Obv iously , the argumen t is rev e rsible. Item 1 is equiv alen t to item 4 since αρ ( x, y ) ≤ d ∞ ( x, y ) ≤ β ρ ( x, y ) 1 M ( αρ ( x, y ) ≤ d g ∞ ( x, y ) ≤ β ρ ( x, y ) 1 M ) on a n y compact part of M ∩ G g M ( α and β dep end on the c hoice of the compact part). By comparing the metrics of Subsection 2 .8: d f ( g ) ∞ ( g , v ) = O  d f ( g ) c ( g , v )  , and b y Lo cal Approx imation Theorem 2.5.4, w e obtain the equiv alence of (3.1.1) to the item 2. The equiv alence of items 2 and 3 is obtained b y compar- ing the metrics of Subsection 2.8: d g c ( g , v ) = O  d g ∞ ( g , v )  and d g ∞ ( g , v ) = O  d ∞ ( g , v )  as v → g . 3.1.2 Chain rule In this subsubsec tion, w e pro v e the c hain rule. Theorem 3.1.3 ([12 9]) . Supp ose that M , N , X ar e Carnot ma nifolds, E is a set in M , and f : E → N is a mapping fr om E into N hc -diﬀer entiable at a p oi n t g ∈ E . Supp ose also that F is a set in N , f ( E ) ⊂ F and ϕ : F → X is a mappi n g fr om F into X hc -diﬀer entiable at p = f ( g ) ∈ N . Then the c om p osition ϕ ◦ f : E → X is hc - d iﬀer entiable at g and D ( ϕ ◦ f )( g ) = D ϕ ( p ) ◦ D f ( g ) . Pr o of. By h yp othesis, d f ( g ) c ( f ( v ) , D f ( g )( v )) = o  d g c ( g , v )  as v → g a nd also 57 d ϕ ( p ) c ( ϕ ( w ) , D ϕ ( p )( w )) = o  d p c ( p, w )  as w → p . W e now infer d ϕ ( p ) c (( ϕ ◦ f )( v ) , ( Dϕ ( p ) ◦ D f ( g ))( v )) ≤ d ϕ ( p ) c ( ϕ ( f ( v )) , D ϕ ( p )( f ( v ))) + d ϕ ( p ) c ( D ϕ ( p )( f ( v )) , D ϕ ( p )( D f ( g )( v ))) ≤ o  d p c ( p, f ( v ))  + O  d p c  f ( v ) , D f ( g )( v )   ≤ o  d g c ( g , v )  + O  o  d g c ( g , v )  = o  d g c ( g , v )  as v → g , since d p c  p, f ( v )  ≤ d p c  p, D f ( g )( v )  + d p c  f ( v ) , D f ( g )( v )  = O  d g c ( g , v )  + o  d g c ( g , v )  = O  d g c ( g , v )  as v → g . (The estimate d p c  p, D f ( g )( v )  = O  d g c ( g , v )  as v → g f ollo ws from the con tin uit y of the homomorphism D f ( g ) and (3.1.2).) 3.2 hc -Diﬀeren tiabilit y of Curv es on the Carnot Man- ifolds 3.2.1 Co ordinate hc -diﬀeren tiabilit y criterion Recall that a mapping γ : E → M , where E ⊂ R is an a rbitrary set, is called a Lip s chitz mapping if there exists a constan t L suc h that the inequality d ∞ ( γ ( y ) , γ ( x )) ≤ L | y − x | holds for all x, y ∈ E . Deﬁnition 3.2.1. A mapping γ : E → M , where E ⊂ R is an arbitra ry set, is called hc -diﬀer entiable at a limit p oint s ∈ E t o E if there exists a horizon tal v ector a = dim H 1 P i =1 α i b X γ ( s ) i ( γ ( s )) ∈ H γ ( s ) M suc h that the lo cal ho- momorphism τ 7→ exp  τ dim H 1 P i =1 α i b X γ ( s ) i  ( γ ( s )) ∈ G γ ( s ) M as the hc -diﬀeren tial of the mapping γ : E → M : d γ ( s ) c  γ ( s + τ ) , δ γ ( s ) τ a  = o ( τ ) for τ → 0, s + τ ∈ E . The point exp  dim H 1 P i =1 α i b X γ ( s ) i  ( γ ( s )) ∈ G γ ( s ) M is called the hc - derivative 2 . Some prop erties of the in tro duced notion of hc -diﬀeren tiabilit y can b e obtained from Prop osition 3.1.2. F or instance, the co eﬃcien ts α i are deﬁned uniquely: if, in the normal c o ordinates, γ ( s + τ ) = exp  N P i =1 γ i ( τ ) b X γ ( s ) i  ( γ ( s )), s + τ ∈ E , for suﬃcien tly small τ then Prop osition 3.1.2 implies: 2 If the hc - deriv a tiv e do es not exis t in G γ ( s ) M then it b elongs in G γ ( s ) M : w e cons ide r the “preimage” under θ γ ( s ) being equal exp  dim H 1 P i =1 α i ( b X γ ( s ) i ) ′  (0) in all the necessary cases. 58 Prop erty 3.2.2 ([129]) . A mapping γ : [ a, b ] → M is hc -diﬀeren tiable at a p oin t s ∈ ( a, b ) if and only if one of the following assertions holds: (1) γ i ( τ ) = α i τ + o ( τ ), i = 1 , . . . , dim H 1 , and γ i ( τ ) = o ( τ deg X i ), i > dim H 1 , as τ → 0, s + τ ∈ E ; (2) the v ector dim H 1 P i =1 α i b X γ ( s ) i ( γ ( s )) ∈ H γ ( s ) M is the Riemannian deriv ativ e of γ : [ a, b ] → M at a point s ∈ ( a, b ), and γ i ( τ ) = o ( τ deg X i ), i > dim H 1 , as τ → 0 , s + τ ∈ E . 3.2.2 hc -Diﬀeren tiabilit y of absolu tely contin u ous curv es If a curv e γ : [ a, b ] → M is absolutely con tin uous in the Riemannian sense then all co ordinat e functions γ i ( t ) are absolutely contin uo us on the closed in terv al [ a, b ] (it is clear that this pro perty is indep enden t of the c hoice of the co ordinate system). Therefore the tangen t v ector ˙ γ ( t ) is deﬁne d a lmost ev eryw here on [ a, b ]. If, moreov er, ˙ γ ( t ) ∈ H γ ( t ) M at the p oin ts t ∈ [ a, b ] of Riemannian d iﬀeren tiabilit y then the curv e γ : [ a, b ] → M is called horizontal . It is well kno wn that almost all p oints t of a closed inte rv a l E = [ a, b ] are Leb esgue p oin ts of the deriv at iv es of the horizon tal comp onen ts, that is, if, in the normal co ordinates γ ( t + τ ) = exp  N P j =1 γ j ( τ ) X j  ( γ ( t )) then the horizontal compo nen ts γ j ( σ ), j = 1 , . . . , dim H 1 , ha v e the prop ert y Z { σ ∈ ( α,β ) | t + σ ∈ E } | ˙ γ j ( σ ) − ˙ γ j (0) | d σ = o ( β − α ) as β − α → 0 (3.2.1) on in terv als ( α , β ) ∋ 0. Note tha t prop erty 3.2.1 is indep enden t of the c hoice of the co ordinate sys tem in a neigh b orho o d of γ ( t ). Theorem 3.2.3 ([129]) . L et a curve γ : [ a, b ] → M on a Carnot m anifold b e ab solutely c ontinuous in the Riemannian sense a nd horizontal. Th e n γ : [ a, b ] → M is hc -diﬀer entiable almost everywher e : any p oint t ∈ [ a, b ] which is a L eb esgue p oint of the derivatives of its ho rizontal c omp onents is also a p oi n t at w h ich γ is hc -diﬀer entiable. If γ ( t + τ ) = exp  N P j =1 γ j ( τ ) X j  ( γ ( t )) then hc -derivative ˙ γ ( t ) e quals exp  dim H 1 X j =1 ˙ γ j (0) b X γ ( t ) j  ( γ ( t )) = exp  dim H 1 X j =1 ˙ γ j (0) X j  ( γ ( t )) . 59 Pr o of. Fix a Leb esgue p oin t t 0 ∈ ( a, b ) of the deriv ativ es of the horizon tal comp onen ts of the mapping γ ( t 0 + τ ) = exp  N P j =1 γ j ( τ ) X j  ( g ), g = γ ( t 0 ). In this pro of, w e also ﬁx a normal co ordinate sy stem θ g at g . T o simplify the notation, we write the v ector ﬁelds e X g i = ( θ − 1 g ) ∗ X i and b X ′ i g = ( θ − 1 g ) ∗ b X g i de- ﬁned in a neigh borho o d of 0 ∈ R N without the superscript g : e X i = ( θ − 1 g ) ∗ X i and b X ′ i = ( θ − 1 g ) ∗ b X g i resp ectiv ely . F or proving the hc -diﬀeren tiabilit y of the mapping γ at t 0 , w e need to establish the estimate γ j ( τ ) = o ( τ deg X j ) as τ → 0 for all j > dim H 1 , t 0 + τ ∈ [ a, b ] (see Prop ert y 3.2.2). Partition the pro of of the desired estimate in to sev eral steps . 1 st Step. Here w e sho w that the h y p othesis implies the Riemannian diﬀeren tiabilit y of the mapping γ at t 0 and ˙ γ ( t 0 ) ∈ H g M . Put Γ( t 0 + τ ) = θ − 1 g ( γ ( t )) = ( γ 1 ( τ ) , . . . , γ N ( τ )). The curv e Γ( τ ) is absolutely con tin uous, and its tangent v ector ˙ Γ( τ ) is horizon tal in a neigh borho o d of 0 ∈ T g M with resp ect to the ve ctor ﬁelds { e X i } : ˙ Γ( τ ) ∈ ( θ − 1 g ) ∗ H γ ( t 0 + τ ) M for almost all τ . F rom here, fo r almost all τ suﬃcien tly closed to 0, we infer ˙ Γ( τ ) = N X j =1 ˙ γ j ( τ ) ∂ ∂ x j = dim H 1 X i =1 a i ( τ ) e X i (Γ( τ )) . (3.2.2) The Riemann tensor pulled bac k from the manifold M onto a neighbor- ho o d of 0 ∈ T g M is contin uous a t the zero. Therefore, using this con tin uit y , w e see that, for an y τ , t 0 + τ ∈ [ a, b ], (3.2.1) implies d c ( γ ( t 0 ) , γ ( t 0 + τ )) ≤ c 1 Z (0 ,τ ) | ˙ Γ( σ ) | r dσ ≤ c 2 dim H 1 X j =1 Z (0 ,τ ) ( | ˙ γ j ( σ ) − ˙ γ j (0) | + | ˙ γ j (0) | ) d σ = O ( τ ) as τ → 0, where | ˙ Γ( σ ) | r stands for the length of t he tangen t v ector in the pulled-bac k Riemannian metric. By Prop osition 2.8.6 and Remark 2.8.7, w e ha v e d ∞ ( γ ( t 0 ) , γ ( t 0 + τ )) = O ( d c ( γ ( t 0 ) , γ ( t 0 + τ )) as τ → 0. Therefore the co ordinate components γ j ( τ ) of t he mapping γ satisfy γ j ( τ ) = O ( τ deg X j ) as τ → 0 for all j ≥ 1 . (3.2.3) It follo ws that the curv e Γ( τ ) is diﬀeren tiable at 0 and ˙ Γ(0) = ( ˙ γ 1 (0) , . . . , ˙ γ dim H 1 (0) , 0 , . . . , 0) . 60 Hence, the curv e γ is diﬀeren tiable in the Riemannian sense at t 0 and ˙ γ ( t 0 ) ∈ H g M . F ro m (3.2.3) w e also obtain γ ( τ ) ∈ B ( g , O ( τ ) ). 2 nd Step. Corollary 2.2.9 and the fact that γ ( τ ) ∈ B ( g , O ( τ )) imply that, in a neigh b orho o d of 0, the v ector ﬁelds e X i can b e exp ressed via b X ′ k so that e X i (Γ( τ )) = N X k =1 α ik ( τ ) b X ′ k (Γ( τ )) , where α ik ( τ ) =      o ( τ deg X k − deg X i ) if deg X k > deg X i , δ ik + o (1) otherwise as τ → 0. Now, us ing ex pansion (2.1.5) of the v ector ﬁelds b X ′ i in the standard Euclidean basis, for all p oin ts τ suﬃc ien tly close to 0, from (3.2.2) we now obtain N X j =1 ˙ γ j ( τ ) ∂ ∂ x j = dim H 1 X i =1 a i ( τ ) e X i (Γ( τ )) = N X k =1 dim H 1 X i =1 a i ( τ ) α ik ( τ ) b X ′ k (Γ( τ )) = N X j =1 j X k =1 dim H 1 X i =1 a i ( τ ) α ik ( τ ) ˆ z j k (Γ( τ )) ∂ ∂ x j . (3.2.4) 3 rd Step. F or 1 ≤ j ≤ dim H 1 , we ha v e deg X j = 1. Then from (3.2.3) and (2.1.5) we conclude that ˆ z j k (Γ( τ )) = δ j k + O ( τ ). The refore, from (3.2.4) w e infer ˙ γ j ( τ ) = dim H 1 P i =1 a i ( τ ) ˜ α ij ( τ ), where, as b efore, ˜ α ij ( τ ) = δ ij + o (1) . Hence, a i ( τ ) = dim H 1 P n =1 ˙ γ n ( τ ) β ni ( τ ), where { β ni ( τ ) } , n, i = 1 , . . . , dim H 1 , is a matrix inv erse to { ˜ α ij ( τ ) } , has t he elemen ts β ni ( τ ) = δ ni + o (1). Conse- quen tly , a i ( τ ) = dim H 1 X i =1 ˙ γ n ( τ ) β ni ( τ ) = dim H 1 X n =1 ˙ r n ( τ ) β ni ( τ ) + dim H 1 X n =1 ˙ γ n (0) β ni ( τ ) , where r n ( τ ) = τ Z 0 ( ˙ γ n ( σ ) − ˙ γ n (0)) dσ . ( 3.2.5) 4 th Step. Fix dim H l − 1 < j ≤ dim H l , 1 < l ≤ M . F or estimating ˙ γ j ( τ ), 61 from (3.2.4) w e ha v e ˙ γ j ( τ ) = dim H 1 X k ,i,n =1 ˙ γ n ( τ ) β ni ( τ ) α ik ( τ ) ˆ z j k (Γ( τ )) + j X k =dim H 1 +1 dim H 1 X i,n =1 ˙ γ n ( τ ) β ni ( τ ) α ik ( τ ) ˆ z j k (Γ( τ )) = I j + I I j . (3.2.6) Since in this case α ik ( τ ) = o ( τ deg X k − deg X i ), and ˆ z j k (Γ( τ )) = O ( τ deg X j − deg X k ) b y (3.2.3) then all comp onen ts in the double sum in (3.2.6) hav e a fa ctor o ( τ l − 1 ). Therefore I I j = dim H 1 X n =1 ˙ γ n ( τ ) o ( τ l − 1 ) . (3.2.7) F rom another side I j = dim H 1 X n =1 ˙ γ n ( τ ) ˆ z j n (Γ( τ )) + dim H 1 X k ,n =1 ˙ γ n ( τ ) o ( 1) ˆ z j k (Γ( τ )) = dim H 1 X n =1 ˙ γ n (0) ˆ z j n (Γ( τ )) + dim H 1 X n =1 ˙ r n ( τ ) ˆ z j n (Γ( τ )) + dim H 1 X k ,n =1 ˙ γ n ( τ ) o ( 1) ˆ z j k (Γ( τ )) = dim H 1 X n =1 ˙ γ n (0) X | α + e n | h =deg X j , α> 0 F j α,e n ( g )Γ( τ ) α + dim H 1 X n =1 ˙ r n ( τ ) O ( τ l − 1 ) + dim H 1 X n =1 ˙ γ n ( τ ) o ( τ l − 1 ) . (3.2.8) In the estimation of the increm en t of γ j ( τ ) on [0 , τ ] b y the Newton–Leibnitz form ula, the comp onents of (3.2.7) a nd the last tw o summands in (3.2.8) ha v e order o ( τ l ). Indeed, f or all 1 ≤ i ≤ dim H 1 and s > 0, from (3.2 .1) and (3.2.5) w e ha v e | ˙ γ n ( τ ) | ≤ | ˙ γ n (0) | + | ˙ r i ( τ ) | from (3.2.5), | r i ( τ ) | ≤ τ R 0 | ˙ γ i ( σ ) − ˙ γ i (0) | d σ = o ( τ ) and     τ Z 0 ˙ r i ( σ ) O ( σ s ) dσ     ≤ | O ( τ s ) | τ Z 0 | ˙ γ i ( σ ) − ˙ γ i (0) | d σ = o ( τ s +1 ) . 5 th Step. In the remaining double sum in (3.2 .8), the summands with index α for whic h | α + e n | < deg X j con tain the f actor Γ( τ ) α = o ( τ l − 1 ), 62 since, in this case, the pro duct Γ( τ ) α necessarily con t ains the f actor γ j ( τ ) = ˙ γ j (0) τ + o ( τ ) = o ( τ ), j > dim H 1 . Therefore, expression (3.2.8) fo r ˙ γ j ( τ ) is reduced t o the follo wing: ˙ γ j ( τ ) = dim H 1 X i =1 ˙ γ i (0) X | α + e n | h =deg X j , | α + e n | =deg X j F j α,e n ( g )Γ( τ ) α + o ( τ l − 1 ) . (3.2.9) Since also Γ ( τ ) = ˙ Γ(0) τ + o ( τ ) , w e see that eac h summand in (3.2.9 ) is equal t o ˙ γ i (0) F j α,e n ( g )Γ( τ ) α = τ l − 1 ˙ γ i (0) F j α,e n ( g ) ˙ Γ(0) α + o ( τ l − 1 ). Consequen tly , lea v ing only the summands of order τ l − 1 in ( 3.2.9), w e ha v e ˙ γ j ( τ ) = dim H 1 X i =1 τ l − 1 X | α | = | α | h = l − 1 ˙ γ i (0) F j α,e n ( g ) ˙ Γ(0) α + o ( τ l − 1 ) . (3.2.10) Similarly , the second summand in the estimation of the incremen t γ j ( τ ) is equal to o ( τ l ). Consequ en tly , for the v alidit y of the theorem, it is necessary and suﬃcien t that the double sum in (3.2.10) b e zero. This w as established in L emma 2.1.21. Th us, we ha v e pro v ed that γ j ( τ ) = o ( τ deg X j ) for all j > dim H 1 . Since the horizon tal components of γ are diﬀeren tiable at t 0 , b y Prop erty 3.1, the estimate γ j ( τ ) = o ( τ deg X j ) for all j > dim H 1 yields the hc - diﬀeren tiabilit y of γ a t t 0 . The metho d o f proving Theorem 3 .2.3 is applicable to a wider class of mappings a nd mak es it p ossible to make additional conclusions ab out the nature of hc -diﬀeren tiabilit y . Corollary 3.2.4. Supp ose that a curve γ : [ a, b ] → M on a Carnot manifold is Lipschitz with r esp e ct to the Riemannian metric and horizontal, i.e., ˙ γ ( s ) ∈ H γ ( s ) M for almost every s ∈ [ a, b ] . Th en the curve γ : [ a, b ] → M is hc - diﬀer entiable almost everywher e 3 . Pr o of. Ev ery Lipschitz curv e with resp ect to the Riemannian metric is also absolutely con tin uous in the Riemannian sense. Th us all conditions of The- orem 3 .2.3 hold. Corollary 3.2.5. Supp ose that w e have a family of curves γ : [ a, b ] × F → M on a Carnot manifold M that is b ounde d and c ontinuous in the totality of its va ria bles, wher e F is a lo c al ly c omp act metric sp ac e. Supp ose that, 3 In pap ers [129, 130 ], a wr ong Coro llary 3 .1 is formulates instead of this. 63 for e ach ﬁxe d u ∈ F , the curve γ ( · , u ) is diﬀer entiable i n the R iemannian sense at al l p oints of [ a, b ] and h orizontal, i. e ., d ds γ ( s, u ) ∈ H γ ( s,u ) M for al l s ∈ [ a, b ] . I f the R iemannian derivative d ds γ ( s, u ) is b ounde d and c ontinuous in the totality of its variables s and u then its hc -derivative is also b ounde d and c ontinuous on [ a, b ] × F . F urthermor e, the c on v e r genc e ∆ γ ( s ) τ − 1 γ ( s + τ , u ) to ˙ γ ( s, u ) ∈ G γ ( s,u ) M is lo c al ly uniform in the totality of s ∈ [ a, b ] and u ∈ F . Pr o of. It suﬃces to pro v e in a ll items of the pro of of Theorem 3.2.3 that the smallness of all quan tities con v erging to zero is lo cally uniform o n [ a, b ] × F (see Proposition 2.8.6 for the estimate C 0 d ∞ ( g , v ) ≤ d c ( g , v )) Corollary 3.2.6. Supp ose that a curve γ : [ a, b ] → M on a Carnot manifold b elo ngs to C 1 and its Riemannian tangen t ve ctor ˙ γ i ( t ) is horizontal for al l t ∈ [ a, b ] . Then the curve γ : [ a, b ] → M is hc -diﬀer entiable at al l t ∈ [ a, b ] . F urthermor e, the c o nver genc e of ∆ γ ( s ) τ − 1 γ ( s + τ ) to ˙ γ ( s ) ∈ G γ ( s ) M is uniform in s ∈ [ a, b ] . Pr o of. F or an y x, y ∈ [ a, b ], the length L ( γ | [ x,y ] ) of the curv e γ : [ x, y ] → M is deﬁned; moreo v er, d ∞ ( γ ( y ) , γ ( x )) ≤ c 1 L ( γ | [ x,y ] ) ≤ c 1 C | y − x | , where C = max t ∈ [ a,b ] | ˙ γ ( t ) | . Th us, t he curv e γ : [ a, b ] → M meets the conditions of The- orem 3 .2.3 at all p oints of [ a, b ] a nd, therefore, is uniformly hc -diﬀeren tiable b y Corollary 3.2.5. The last assertion of this corollary follo ws Lemma 3.2.7. Every Lipschitz mapping γ : E → M is diﬀer e ntiable a l m ost everywher e in the Riemannian sense, and ˙ γ ( t ) ∈ H γ ( t ) M at the p oints of the Riemannian di ﬀ er entiability of γ Pr o of. In the normal co ordinates at a p oint g = γ ( t ), w e ha v e γ ( t + τ ) = exp  N X j =1 γ j ( τ ) X j  ( g ) , t + τ ∈ E . The Lipsc hitzity of the mapping γ : E → M and the prop erties of d ∞ imply the estimate γ j ( τ ) = O ( τ deg X j ) for all j ≥ 1, t + τ ∈ E . Since deg X j ≥ 2 for j > dim H 1 , the deriv ativ e ˙ γ j (0) exists and is zero for all j > dim H 1 . Consequen tly , the Riemannian diﬀeren tiabilit y of γ at t is equiv alent to the diﬀeren tiabilit y of the horizon tal comp onents γ j , j = 1 , . . . , n , of γ at 0. No w, the Lipsc hitz mapping γ : E → M is a lso Lipschitz with resp ect to the Riemannian metric (see Prop osition 2.8 .12). Th us, b y Rademac her’s classical theorem, the Riemannian deriv ativ e ˙ γ ( t ) ∈ T γ ( t ) M ex ists for almost ev ery t ∈ [ a, b ]. The ab ov e implies that, at ev ery suc h p oin t, ˙ γ ( t ) ∈ H γ ( t ) M 64 Since a Lipsc hitz mapping γ : [ a, b ] → M is absolutely contin uo us in the Riemannian se nse (see the comparison of the metrics in Proposition 2.8.12), from Lemma 3.2.7 and Theorem 3.2.3 w e infer Corollary 3.2.8. Every Lipschitz ma p ping γ : [ a, b ] → M is hc -diﬀe r entiable almost everywher e on [ a, b ] : if t ∈ [ a, b ] is a L eb esgue p oint of the derivatives of its horizo n tal c omp onents then this p oint is its hc -diﬀer entiability p oint. 3.2.3 hc -Diﬀeren tiabilit y of scalar Lipsc hi tz mappings In this s ubsubsection, we e stablish the hc -diﬀeren tiabilit y of the Lipsc hitz mappings γ : E → M where E ⊂ R is an arbitrary se t. Recall that x ∈ A , where A ⊂ R is a measurable set, is t he densit y p oint of A if | A ∩ ( α , β ) | 1 = β − α + o ( β − α ) for β − α → 0, x ∈ ( α, β ) (here | · | 1 stands for the one-dimensional Leb esgue me asure). It is kno wn that almost all p oin ts of a measurable set A are its densit y po in ts (for example , see [4 1]). It is explic itly see n from the abov e pro of of Lemma 3.2.7 tha t the q uestion of hc -diﬀeren tiabilit y for a Lipsc hitz mapping dep ends o n the diﬀeren tia bil- it y of its horizon tal comp onen ts. If a Lipschitz mapping γ : E → M (w e ma y assume that E ⊂ R is closed) is written in the normal c o ordinates: γ ( t + τ ) = exp  N P j =1 γ j ( τ ) X j  ( γ ( t )), t ∈ E is a ﬁxed num ber, t + τ ∈ E , then, b y Lemma 3 .2.7, its comp onen ts γ j ( τ ), j = 1 , . . . , N , are diﬀeren tiable almost ev e rywhere on E . It is kno wn that almost all densit y p oin ts o f E are Leb esgue p o in ts of the deriv ativ e of the horizon tal comp onen ts, i.e., fo r in terv als ( α, β ) ∋ τ , t + τ ∈ E , w e infer Z { σ ∈ ( α,β ) | t + σ ∈ E } | ˙ γ j ( σ ) − ˙ γ j ( τ ) | dσ = o ( β − α ) for β − α → 0 (3.2.11) for all j = 1 , . . . , dim H 1 . No te that prop erty ( 3.2.11) do es not dep end on the c hoice of the co ordinate system in a neigh borho o d of the point g = γ ( t ). Theorem 3.2.9 ([129]) . Every Lipschitz map ping γ : E → M , E ⊂ R is close d, is hc -di ﬀ er entiable almost everywher e on E : the mapping γ : E → M is hc -diﬀer entiable at every p oint t ∈ E such that 1. t is the density p oint of E ; 65 2. ther e exist de riv atives ˙ γ j (0) , j = 1 , . . . , dim H 1 , of the horizo ntal c om - p on ents of γ , wher e γ ( t + τ ) = exp  N P j =1 γ j ( τ ) X j  ( γ ( t )) , t + τ ∈ E ; 3. c ondition (3.2.11) is fulﬁl le d at the p oint τ = 0 . The hc -derivative ˙ γ ( t ) e q uals exp  dim H 1 X j =1 ˙ γ j (0) b X γ ( t ) j  ( γ ( t )) = exp  dim H 1 X j =1 ˙ γ j (0) X j  ( γ ( t )) . Pr o of. 1 st Step. Supp ose that t ∈ E is a p oint at whic h conditions 1–3 of the theorem hold and g = γ ( t ). Since the result is lo cal, w e may also assume that E is included in an in terv al [ a, b ] ⊂ R , t ∈ [ a, b ], a, b ∈ E , whose image is included in G g M (w e ma y assume b y diminishing the in terv al [ a, b ] if nece ssary that γ ([ a, b ] ∩ E ) ⊂ G γ ( η ) M for ev ery η ∈ [ a, b ] ∩ E ). The open b ounded set Z = ( a, b ) \ E is represe n table as the union of an at most coun table collection of disjoin t in terv als: Z = S j ( α j , β j ), where, for con v enie nce of the subseque n t e stimates, w e put α j < β j if t ≤ α j and β j < α j if α j < t . It is kno wn (for example, see [46]), that, in G γ ( α j ) M , there exists a horizontal curv e ˜ σ j : [0 , b j ] → G γ ( α j ) M joining the p oints ˜ σ j (0) = γ ( α j ) and ˜ σ j ( b j ) = γ ( β j ) and par ameterized b y the arc length; moreo v er, b j = d γ ( α j ) c ( γ ( α j ) , γ ( β j )) ≤ cd γ ( α j ) ∞ ( γ ( α j ) , γ ( β j )) = cd ∞ ( γ ( α j ) , γ ( β j )) ≤ cL | β j − α j | , where c is indep enden t of j (see the relation betw een the metrics in Subsection 2.8). Consequen tly , the mapping σ j : [ α j , β j ] → M deﬁned by the rule [ α j , β j ] ∋ η 7→ σ j ( η ) = ˜ σ j  b j | β j − α j | | η − α j |  ∈ G γ ( α j ) M is Lipsc hitz in the metric d γ ( α j ) c with the Lipsc hitz constan t cL for all j ∈ N . Deﬁne now the extension f : [ a, b ] → M as follo ws: f ( η ) = ( γ ( η ) , if η ∈ E , σ j ( η ) , if η ∈ ( α j , β j ) . 2 nd Step. The ma pping f : [ a, b ] → M has the follow ing prop erties: (1) f : [ α, β ] → M is a Lipsc hitz mapping with respect to the Riemannian metric; (2) the Riemannian deriv a tiv e of f e xists for almost eve ry η ∈ [ a, b ] and is b ounded; (3) the v ector ˙ f ( η ) b elongs to the horizontal space H γ ( η ) M for almost ev ery η ∈ E ; 66 (4) the mapping f : [ a, b ] → M has a Riemannian deriv a tiv e at t equal to ˙ γ ( t ); if f ( t + τ ) = exp  N P j =1 f j ( τ ) X j  ( g ), t + τ ∈ [ a, b ], then (5) f j ( τ ) = O ( τ deg X j ) as τ → 0 for all j ≥ 1; (6) 0 is a Leb esgue p oint for the deriv atives ˙ f j ( τ ), j = 1 , . . . , dim H 1 . Indeed, if t ≤ α j < η 1 < β j < α k < η 2 < β k ≤ b then, taking the relations b et w ee n the metrics in to accoun t, w e obtain the estimates ρ ( f ( η 1 ) , f ( η 2 )) ≤ ρ ( f ( η 1 ) , γ ( β j )) + ρ ( γ ( β j ) , γ ( α k )) + ρ ( γ ( α k ) , f ( η 2 )) ≤ C (( β j − η 1 ) + ( α k − β j ) + ( η 2 − α k )) = C | η 2 − η 1 | . The other case s of m utual disp osition of η 1 and η 2 with resp ect to t are considered similarly . Hence w e obt ain prop erties (1) and (2). Next, if t ≤ α j < t + τ < β j then d ∞ ( f ( t + τ ) , f ( t )) ≤ C ( d ∞ ( f ( t + τ ) , γ ( α j )) + d ∞ ( γ ( α j ) , γ ( t ))) ≤ C 1  d γ ( α j ) ∞ ( f ( t + τ ) , γ ( α j )) + ( α j − t )  = C 2 (( t + τ − α j ) + ( α j − t )) = C 2 τ b y the t riangle inequality , the construction of f , and the relations betw een the metrics. F rom this w e obta in property (5) and, hence, t he diﬀeren tiabilit y of all components f j at 0, j > dim H 1 : ˙ f j (0) = 0. Since the deriv ativ es of Lipsc hitz functions are bo unded and t is the de n- sit y p o in t of E , for in terv a ls ( r, s ) ∋ 0 w e ha v e Z ( r ,s ) | ˙ f j ( σ ) − ˙ γ j (0) | d σ = Z { σ ∈ ( r,s ) | t + σ ∈ E ∩ [ a,b ] } | ˙ γ j ( σ ) − ˙ γ j (0) | d σ + Z { σ ∈ ( r,s ) | t + σ / ∈ E ∩ [ a,b ] } | ˙ f j ( σ ) − ˙ γ j (0) | d σ = o ( | s − r | ) (3.2 .12) as s − r → 0 for all j = 1 , . . . , dim H 1 . Hence, τ R 0 ( ˙ f j ( σ ) − ˙ γ j (0)) dσ = f j ( τ ) − ˙ γ j (0) τ = o ( τ ) and d f j dτ (0) = ˙ γ j (0) for all j = 1 , . . . , dim H 1 . Th us, w e hav e pro v ed pro p erties (4) and (6 ). Note that the precedin g arguments are independen t of the co or dinate system. They a re based on the following principle: if η is the densit y p oin t for E , the mapping f | E has a Riemannian deriv ativ e at η ∈ E , and η ∈ E is a Leb esgue p oin t for the horizon tal co ordinate functions of f | E then, with regard to Lemma 3.2.7 and what has b een pro v ed abov e , f has a Riemannian deriv ative a t η ; moreov er, the Riemannian tangen t vec tor b elongs to the horizon tal space H γ ( η ) M . This prov es pro perty (3). 3 rd Step. Since the Riemannian deriv ativ e ˙ f ( η ) o f the mapping f : [ a, b ] → M belongs to the horizontal sp ace H f ( η ) M only at almost ev e ry point η ∈ E , a direct application of T heorem 3 .2.3 is imp ossible. Ho w e v er, gran ted 67 the fact that the complemen t [ a, b ] \ E has densit y zero at t , the metho d of its proo f can be adapted also to this case. W e no w indicate the c hanges to the pro of of Theorem 3.2 .3 nec essary for obtaining the hc -diﬀeren tiabilit y of f at the p oin t t ﬁxed ab ov e. In tro duce the notation Γ( τ ) = ( ( γ 1 ( τ ) , . . . , γ N ( τ )) , if t + τ ∈ E , ( f 1 ( τ ) , . . . , f N ( τ )) , if t + τ / ∈ E . It has b een prov ed ab ov e that ˙ Γ(0) = ( ˙ γ 1 (0) , . . . , ˙ γ dim H 1 (0) , 0 , . . . , 0). De- duce (3.2.2) for the p oin ts τ suﬃcien tly close to 0 and suc h that t + τ ∈ E . A t the p oin ts t + τ ∈ ( α j , β j ), w e ha v e ˙ Γ( τ ) = N X j =1 ˙ f j ( τ ) ∂ ∂ x j = dim H 1 X i =1 a i ( τ ) b X ′ i f ( α j ) (Γ( τ )) . (3.2.13) By Prop osition 2.2.7, at the p oints t + τ ∈ ( α j , β j ) the relatio n f ( τ ) ∈ B ( g , O ( τ )) implies that, in a neigh borho o d of 0, the v e ctor ﬁelds b X ′ i f ( α j ) are expresse d via the v ector ﬁelds b X ′ k (here w e write b X ′ k instead o f b X ′ k g ) in the form b X ′ i f ( α j ) (Γ( τ )) = N X k =1 γ ik ( τ ) b X ′ k (Γ( τ )) , where γ ik ( τ ) =      o ( τ deg X k − deg X i ) , if deg X k > deg X i , δ ik + o (1) otherwise as τ → 0. Really , by (2.2.6), w e ha v e b X ′ i f ( α j ) (Γ( τ )) = N P l =1 β il ( τ ) e X l (Γ( τ )) at p oin ts f ( τ ) ∈ B ( g , O ( τ )), where β il ( τ ) = ( o ( τ deg X l − deg X i ) if deg X l > deg X i , δ il + o (1) otherwise (3.2.14) as τ → 0, and e X l (Γ( τ )) = N P k =1 α lk ( τ ) b X ′ k (Γ( τ )) where α lk ( τ ) = ( o ( τ deg X k − deg X l ) if deg X k > deg X l , δ ik + o (1) otherwise (3.2.15) as τ → 0. It follo ws b X ′ i f ( α j ) (Γ( τ )) = N P k =1 N P l =1 β il ( τ ) α lk ( τ ) b X ′ k (Γ( τ )) . No w taking in to accoun t (3 .2.14) a nd (3.2.15), and represen ting the last double sum a s 68 P k ≤ i N P l =1 + P k >i  P l ≤ i + P i k . Then the inequalit y of (3.2.17) holds for y ∈ E ∩ ( x − l − 1 , x + l − 1 ) 69 with l instead of k . Since A l ∩ ( x − l − 1 , x + l − 1 ) ⊃ A k ,j ∩ ( x − k − 1 , x + k − 1 ), w e hav e f ( x ) = lim y → x, y ∈ A l f ( y ) = lim y → x, y ∈ A k,j f ( y ) = ˜ f k ,j ( x ) . By Theorem 3.2.9, the mapping ˜ f k ,j : A k ,j → M is hc -diﬀeren tiable almost ev eryw here in A k ,j . W e are left with c hec king the hc -diﬀeren tiabilit y of the mapping f : E → M at the points of hc -diﬀeren tiabilit y of the mapping ˜ f k ,j : A k ,j → M ha ving densit y one with resp ect to A k ,j . F or brevit y , denote the set A k ,j b y A and denote the mapping f k ,j b y f . Extend the Lipsc hitz mapping f : A → M b y con tin uit y to a Lipsc hitz mapping ˜ f : A → M . Supp ose now that a p oin t a ∈ A is a p oint of hc - diﬀeren tiabilit y fo r ˜ f and the p oint densit y of A . Recall that, b y the deﬁnition of A , the inequalit y d ∞ ( f ( y ) , f ( z )) ≤ k | y − z | holds for all y ∈ A and all z ∈ ( y − k − 1 , y + k − 1 ) ∩ E . Note that this inequalit y is extendable to A b y con tin uit y . Conseque n tly , t he inequalit y d ∞ ( ˜ f ( y ) , f ( z )) ≤ k | y − z | holds for all y ∈ A a nd all z ∈ ( y − k − 1 , y + k − 1 ) ∩ E . If z ∈ E belongs to the neigh b orho o d ( a − k − 1 , a + k − 1 ) of a then, b y the w ell-kno wn prop erty of a densit y p oin t (see, for example, [1 21]), there exists a p oint y ∈ A suc h that | y − z | = o ( | z − a | a s z → a . Let X b e the horizon tal v ector ﬁeld of the deﬁnition of hc -diﬀeren tiabilit y for the restriction ˜ f : A → M at a p oint a . Then, in a suﬃ cien tly small neigh borho o d of a , from what w as said ab ov e w e hav e d ∞ ( f ( z ) , exp(( z − a ) X )( f ( a )) ≤ c 2 ( d ∞ ( f ( z ) , ˜ f ( y ))+ d ∞ ( ˜ f ( y ) , exp( y − a ) X ) ( f ( a ))) + d ∞ (exp( y − a ) X )( f ( a )) , exp( z − a ) X )( f ( a )) ≤ c 2 ( k | y − z | + o ( | y − a | ) + k X k| y − z | ) = o ( | z − a | ) as z → a , z ∈ E . Hence, the mapping f : E → M is hc -diﬀeren tiable at a . Supp ose now that k 1 < k 2 < k 3 . . . is a sequen ce of naturals suc h t hat the measure of the comple men t B k j = A k j \ A k j − 1 is nonzero for ev ery j ≥ 2 . Ob viously , the abov e argument applies to each of the sets B k j , j ≥ 2, whic h pro v es the theorem. No w w e can pro v e the hc -diﬀeren tiabilit y of rectiﬁable curve s. Consider a curve (con tin uous mapping) γ : [ a, b ] → M . By a partition I n = I n ([ a, b ]) of the segme n t [ a, b ] w e mean any ﬁnite sequence of p oin ts { s 1 , . . . , s n } with a = s 1 < · · · < s n = b . T o ev ery partition I n ([ a, b ]), w e assign a n um b er 70 M ( I n ) b y setting M ( I n ) = n X i =1 d ∞ ( γ ( s i ) , γ ( s i +1 )) . Put m n = max { s i +1 − s i | i = 1 , . . . , n − 1 } . Deﬁnition 3.2.11 ([20]) . A curv e γ : [ a, b ] → M is called r e ctiﬁable if L ([ a, b ]) = lim m n → 0 sup I n M n < ∞ . Making use of a standard argumen t, w e ma y pro v e: Prop erty 3.2.12. Suppose t hat a sequenc e of curv es γ n : [ a, b ] → M , n ∈ N , con v erges p oint wis e to a curv e γ : [ a, b ] → M : γ n ( s ) → γ ( s ) for eve ry s ∈ [ a, b ]. Then the lengths L n ([ a, b ]) of γ n p ossess the semicon tin uit y pro p ert y: L ([ a, b ]) ≤ lim n →∞ L n ([ a, b ]) . Prop osition 3.2.13. Every r e ctiﬁable curve γ : [ a, b ] → M me ets (3.2.1 6) . Pr o of. Consider the following set function Φ deﬁned on interv als included in [ a, b ]: the v alue Φ( α, β ) at an in terv al ( α, β ) ⊂ [ a, b ] is eq ual to L ([ α, β ]), the length of the curv e γ : [ α, β ] → M . The se t function Φ is quasiadditiv e: the inequality X i Φ( α i , β i ) ≤ Φ( α , β ) holds for ev ery ﬁnite collection of pairwise disjoin t in terv als ( α i , β i ) with ( α i , β i ) ⊂ ( α, β ), where ( α, β ) ⊂ [ a, b ] is some in terv al. It is kno wn (see, for example, [136]), that Φ has a ﬁnite deriv ativ e Φ ′ ( x ) = lim ( α,β ) ∋ x, β − α → 0 Φ( α, β ) β − α = lim ( α,β ) ∋ x, β − α → 0 L ([ α, β ]) β − α almost ev erywhere in [ a, b ]. Hence, lim y → x d ∞ ( f ( y ) , f ( x )) | y − x | ≤ lim ( α,β ) ∋ x, β − α → 0 d ∞ ( f ( α ) , f ( β )) L ([ α, β ]) · lim ( α,β ) ∋ x, β − α → 0 L ([ α, β ]) β − α ≤ Φ ′ ( x ) < ∞ for almo st all x ∈ [ a, b ]. Theorem 3.2.10 and Proposition 3.2.13 imply: Prop osition 3.2.14. Every r e ctiﬁable curve γ : [ a, b ] → M is hc -diﬀ er entiable almost everywher e. 71 Remark 3.2.15. If the Carnot manifold is a Carnot group our deﬁnition of the hc -diﬀeren tiabilit y of curv es coincides with the P - diﬀeren tiabilit y o f curv es give n by P . P ansu in [115]. He prov e d also [115, Pro po sition 4.1] the P -diﬀerentiabilit y almost ev erywhere o f rectiﬁable curv es on Carnot groups using a diﬀeren t metho d. 3.3 hc -Diﬀere n tiabilit y of Smo oth Mappings of Carnot Manifolds 3.3.1 Con tin uit y of horizon tal deriv ativ es and hc -diﬀeren tiabilit y of map pings In this subsubsection, w e generalize the classical property that the con- tin uit y of the partial deriv ativ es of a function deﬁned on a Euclidean space guaran tees its diﬀeren tiabilit y . In what follo ws, w e rep eatedly use the follo wing correspondence: to ar- bitrary elemen t a = exp  N P i =1 a i b X g i  ( g ) ∈ G g and p oin t w ∈ G g , assign the elemen t ∆ w ε a = exp  N X j =1 a j ε deg X j X j  ( w ) (3.3.1) for those ε for whic h the right-hand side of (3.3.1) exists. Not e that, by Prop ert y 2.2.3, w e ha v e ∆ g ε a = δ g ε a f or all a ∈ G g . Theorem 3.3.1. Supp o se that f : M → N is a Lipschitz m apping of Carno t manifolds such that, at e ach p oint g ∈ M , ther e exist h o rizontal deriva- tives X i f ( g ) ∈ H f ( g ) N c ontinuous on M , i = 1 , . . . , dim H 1 . Then f is hc -diﬀer entiable at every p oint of M . The Lie alg e br a homomorphism c orr e- sp o n ding to the hc -diﬀer ential is uniquely deﬁne d by the mapping H g M ∋ X i ( g ) 7→ X i f ( g ) = d dt f (exp tX i ( g )) | t =0 = dim e H 1 X j =1 b ij Y j ( f ( g )) ∈ H f ( g ) N of the b as i s horizontal ve ctors X i ( g ) , i = 1 , . . . , dim H 1 , to horizontal ve ctors in H f ( g ) N : H G g M ∋ b X g i 7→ dim e H 1 X j =1 b ij b Y f ( g ) j ∈ H G f ( g ) N . 72 Pr o of. 1 st Step. Fix a p oin t g ∈ U a nd a compact neighborho od F ⊂ G g of the lo cal Carnot group G g . F or eac h horizon tal v ector ﬁeld X i , a family of curv es γ : [ − ε, ε ] × F → N is deﬁned: for u ∈ F , put γ i ( s, u ) = f (exp ( sα i X i )( u )), where α i ∈ A , A ⊂ R is a bounded neigh borho o d of 0 ∈ R . This family of curv es mee ts the conditions of Corollary 3.2 .5 and, hence , the con v ergence ∆ f ( u ) s − 1 γ ( s, u ) → δ f ( u ) α i exp([ X i f ]( u ))( f ( u )) ∈ G f ( u ) (3.3.2) is uniform on F × A and the hc -deriv ative δ f ( u ) α i exp( X i f ( u ))( u ) is contin uous with respect to ( u, α i ) ∈ F × A . Denote b y x i the “horizon tal basis elemen t” exp( X i )( g ) = exp( b X g i )( g ) ∈ G g and, for all 1 ≤ i ≤ dim H 1 , denote b y a i the horizon tal deriv ativ e ex p( X i f ( g ))( f ( g )). It is kno wn [46] that an y e lemen t v ∈ F can b e represen ted (non uniquely) in t he form δ g α 1 x j 1 · · · · · δ g α S x j S , 1 ≤ j i ≤ dim H 1 , (3.3.3) where S is indep enden t of the choice of the p oin t and the n um bers α i are b ounded b y a common constan t. T og ether with the mapping [0 , ε ) ∋ t 7→ ˆ v i ( t ) = δ g tα 1 x j 1 · · · · · δ g tα i x j i , 1 ≤ j k ≤ dim H 1 , 1 ≤ k ≤ i ≤ S, consider the mapping (see (3.3 .1)) [0 , ε ) ∋ t 7→ v i ( t ) = ∆ v i − 1 ( t ) tα i x j i = exp( tα i X j i )( v i − 1 ( t )) , 2 ≤ i ≤ S, where v 1 ( t ) = ∆ g tα 1 x j 1 = exp( tα 1 X j 1 )( g ) . By Theorem 2.7.1, d ∞ ( v i ( t ) , ˆ v i ( t )) = o ( t ) as t → 0 uniformly in g ∈ F and α i ∈ A , i ≤ S . Since the ma pping f is Lipsc hitz on F , the limits lim t → 0 ∆ f ( g ) t − 1 f ( ˆ v S ( t )) and lim t → 0 ∆ f ( g ) t − 1 f ( v S ( t )) exist simultaneously . Conseque n tly , it suﬃc es to prov e the existe nce of t he second limit. 2 nd Step. F or pro ving this, b y (3.3.2), w e infer that w 1 ( t ) = f ( v 1 ( t )) = exp  e N X k =1 z 1 k ( t ) Y k  ( f ( g )) has hc -deriv ativ e δ f ( g ) α 1 a j 1 ∈ G f ( g ) at t = 0 . Here Y k , k = 1 , . . . , e N , is a lo cal basis on N around the p oin t f ( g ) . Assume that the mapping t 7→ w i ( t ) = f ( v i ( t )) = exp  e N X k =1 z i k ( t ) Y k  ( f ( v i − 1 ( t ))) has hc - deriv ativ e δ f ( g ) α 1 a j 1 · . . . · δ f ( g ) α i a j i ∈ G f ( g ) , at t = 0 , 2 ≤ i < S. 73 Our next goa l is to sho w that hc -deriv ative of the mapping t 7→ w i +1 ( t ) = f ( v i +1 ( t )) = exp  e N P k =1 z i +1 k ( t ) Y k  ( f ( v i ( t ))) equals δ f ( g ) α 1 a j 1 · . . . · δ f ( g ) α i a j i · δ f ( g ) α i +1 a j i +1 . T ogether with the mapping w i +1 ( t ), conside r the mapping t 7→ b w i +1 ( t ) = ex p  e N X k =1 z i +1 k ( t ) b Y g k  ( f ( v i ( t ))) . By Theorem 2.7.1 w e ha v e d f ( g ) c ( w i +1 ( t ) , b w i +1 ( t )) = o ( t ) as t → 0 . Therefore, the relatio n d f ( g ) c  w i +1 ( t ) , δ f ( g ) t  δ f ( g ) α 1 a j 1 · . . . · δ f ( g ) α i +1 a j i +1  = o ( t ) as t → 0 holds if and only if d f ( g ) c  b w i +1 ( t )) , δ f ( g ) t  δ f ( g ) α 1 a j 1 · . . . · δ f ( g ) α i +1 a j i +1  = o ( t ) as t → 0. By Prop erty 3.1.2, this is equiv alen t to the relation d f ( g ) c  δ g t − 1 b w i +1 ( t )) , δ f ( g ) α 1 a j 1 · . . . · δ f ( g ) α i +1 a j i +1  = o (1) as i → ∞ . Note that, b y the contin uity of the group op eration in G g , we alw a ys ha v e the conv ergence δ g t − 1 b w i +1 ( t )) → δ f ( g ) α 1 a j 1 · · · · · δ f ( g ) α i +1 a j i +1 as t → 0. Th us, by induction, the hc -deriv ativ e of the mapping [0 , ε ) ∋ t 7→ f ( v S ( t )) at 0 is equal to δ f ( g ) α 1 a j 1 · · · · · δ f ( g ) α S a j S ; moreo v er, the con v ergenc e is uniform in v ∈ F and α i , 1 ≤ i ≤ S . Consequen tly , grante d the equalit y v S ( t ) = δ g t v , w e inf er d f ( g ) c  f  δ g t v  , L  δ g t v  = o  d g c  g , δ g t v  = o ( t ) (3.3.4) uniformly in v ∈ F , where L stands fo r the correspondence G g ∋ v = δ g α 1 x j 1 · · · · · δ g α S x j S 7→ δ f ( g ) α 1 a j 1 · · · · · δ f ( g ) α S a j S ∈ G f ( g ) . F or ﬁnishing the pr o of, it remains to c hec k that the corresp ondence L : G g → G f ( g ) is a homomorphism of the lo cal Carnot groups. 3 rd Step. Note that L ( v ) is the hc -deriv ativ e at 0 of the mapping t 7→ f  δ g t v  for a ﬁxed v ∈ G g (see (3.3.4)), whic h is obv iously indep enden t of represen tation (3.3.3). Consequ en tly , L : G g → G f ( g ) is a mapping of the lo cal groups. Clearly , this mapping is con tin uous. Demonstrate that it is a group homomorphism. Consider a second ele- men t v = δ g β 1 x j 1 · · · · · δ g β S x j S , 1 ≤ j i ≤ dim H 1 , suc h that v v = δ g α 1 x j 1 · · · · · δ g α S x j S · δ g β 1 x j 1 · · · · · δ g β S x j S ∈ G g and L ( v ) · L ( v ) ∈ G f ( g ) . (3.3.5) 74 By (3.3.4), the v alue L ( v v ) is independen t of the represen tation of an elemen t v v as the pro duct (3.3.5). Henc e, applying the conclusions of the previous step to v v and its r eprese n tation (3.3.5), w e see tha t L ( v v ) = δ f ( g ) α 1 a j 1 · · · · · δ f ( g ) α S a j S · δ f ( g ) β 1 a j 1 · · · · · δ f ( g ) β S a j S = L ( v ) · L ( v ) . Th us, the mapping L : G g → G f ( g ) is a contin uous group homomorphism. By the w ell-kno w n prop erties of the Lie gro up theory [137], the mapping L is a homomorphism of the lo cal Lie groups. No w, from (3.3.4) it can b e deduced that L comm utes with a dilation, L ◦ δ g t = δ f ( g ) t ◦ L , t > 0. F urthermore, since X i f ( g ) ∈ H f ( g ) M , the homo- morphism L is the hc -diﬀe r ential of the mapp ing f : M → N at g . The Lie algebra homomorphism cor responding to L is a mapping of horizontal subspaces. Corollary 3.3.2 ([12 9]) . Assume that we have a b asi s { X i } , i = 1 , . . . , N , on a Carnot m anifold M fo r which Assumption 2.1.4 or c onditions of R e- mark 2.7.2 hold with some α ∈ (0 , 1] . Supp os e that f : M → N is a mapping of Carnot manifolds such that, at e ach p oint g ∈ M , ther e exist horizon- tal derivatives X i f ( g ) ∈ H f ( g ) N c on tinuous on M , i = 1 , . . . , dim H 1 . Then f is hc -di ﬀer entiable at e v e ry p oint of M . The Lie algeb r a homomo rp h ism c orr esp ond ing to the hc -diﬀer ential is uniq uely deﬁne d by the mapping H g M ∋ X i ( g ) 7→ X i f ( g ) = d dt f (exp tX i ( g )) | t =0 = dim e H 1 X j =1 b ij Y j ( f ( g )) ∈ H f ( g ) N of the b as i s horizontal ve ctors X i ( g ) , i = 1 , . . . , dim H 1 , to horizontal ve ctors in H f ( g ) N : H G g M ∋ b X g i 7→ dim e H 1 X j =1 b ij b Y f ( g ) j ∈ H G f ( g ) N . Pr o of. The h yp othesis implie s that f is a lo cally Lipsch itz mapping: ˜ d ∞ ( f ( x ) , f ( y )) ≤ C d ∞ ( x, y ), x , y b elong to some compact neigh b orho o d of U . T o v erify this, it suﬃces to join p oints x, y ∈ U b y the horizon tal curv e γ of Subsection 2.8 whose length is con trolled by the hc - distance d ∞ ( x, y ) and observ e tha t f ◦ γ is a horizon tal curve whose length is con trolled b y the length of the initial curv e. F rom this, Corollary 2.8.6 and Remark 2.8.7 w e infer ˜ d ∞ ( f ( x ) , f ( y )) ≤ C 1 L ( f ◦ γ ) ≤ C 2 L ( γ ) ≤ C 3 d ∞ ( x, y ). 75 3.3.2 F unctorial prop ert y of tangen t cones The deﬁnition of the tangent cone dep ends o n the lo cal basis. The ques - tion arises on the connec tion b et w ee n t w o tangen t cones found from t w o diﬀeren t bases . The last Theorem 3.3 implies: Corollary 3.3.3 ( [128, 129]) . Supp ose that we have two lo c a l b ases { X i } and { Y i } , i = 1 , . . . , N , on a Carnot manifold for b oth of which Assumption 2.1.4 or c onditions of R emark 2.7.2 hold with some α ∈ (0 , 1] , and that two c ol le ctions X 1 , . . . , X dim H 1 and Y 1 , . . . , Y dim H 1 gener ate the same horizontal subbund le H 1 . Then the tangent c one G g deﬁne d by the { X i } ’s is i s omorphic to the lo c al Carno t gr oup e G g , determine d by the { Y i } ’s: ( e δ g t − 1 ◦ δ g t )( v ) c onver ges to an iso m orphism of lo c al Carnot gr oups G g and e G g as t → 0 uniformly in v ∈ G g . ( Her e e δ g t is the one- p ar am e ter dilation gr oup deﬁne d by the ve ctor ﬁelds { Y i } . ) The isomorphism of the Lie algebr as c o rr esp ond ing to the hc -diﬀ er ential is deﬁne d uniquely by giving the mappin g H g M ∋ X i ( g ) 7→ X i ( g ) = dim H 1 X j =1 b ij Y j ( g ) ∈ H g M of the b asis ve ctors X i ( g ) , i = 1 , . . . , dim H 1 , of the horizontal sp ac e H g M to horizontal ve ctors of the sp ac e H g M : H G g M ∋ b X g i 7→ dim H 1 X j =1 b ij b Y g j ∈ H e G g M . Pr o of. Denote b y M X the Carnot manifo ld M with lo cal basis { X i } and denote by M Y the Carnot manifold M with lo cal basis { Y i } , i = 1 , . . . , N . Let also the sym b ol i : M X → M Y stand for the iden tit y mapping from M in to M . Clearly , i meets the conditio ns o f Corolla ry 3.3.2. Then i is hc - diﬀeren tiable at g and, b y Coro llary 3.3.2, the “diﬀerence r atios” e δ g t − 1 ( δ g t ( w )) con v erge uniformly to a homomorphism D i ( g ) : G g → e G g as t → 0. Applying the same argumen t to the in v ers e mapping i − 1 and Theorem 3.1.3, w e infer that D i ( g ) is an isomorphism of the lo cal Carnot groups (of the local tangent cones at g with resp ect to diﬀeren t lo cal bases). Remark 3.3.4. In [4, 15, 66, 100] ab o v e stateme n t is prov ed b y other meth- o ds under additional a ssumptions on the smo othness of the basis v ector ﬁelds. 76 3.3.3 Rademac her Theorem The aim of this part is to form ulate Rademac her ty p e theorems on the dif- feren tiabilit y of Lipsc hitz mappings of Carnot manifolds. This theorem w as pro v ed in [129] b y means of the theory expounded a b o v e. The w a y of pro v ing this result is based on the methods of [125], where the P - diﬀeren tiabilit y of Lipsc hitz mappings of Carnot groups deﬁned o n me asurable sets w as prov ed in details. Let M , N b e t w o Carnot manifolds and let E ⊂ M b e an arbitrary set. A ma pping f : E → N is called a Lipsch itz mapping if ˜ d ∞ ( f ( x ) , f ( y )) ≤ C d ∞ ( x, y ) , x, y ∈ E , for some constan t C indep enden t o f x and y . The least constan t in this relation is denoted b y Lip f . The f ollo wing result extends the theorems on the P -diﬀeren tiabilit y on Carnot groups [115, 125, 13 5] (see also [93]) to Carnot manifolds. Theorem 3.3.5 ([129]) . L et E b e a set in M an d let f : E → N b e a Lips c h itz mapping fr om E into N . Then f is hc -diﬀer entiable on E . The homomo rphism o f the Lie a lgebr as c orr esp onding to the hc -diﬀer ential is deﬁne d uniquely by the mapping H g M ∋ X i ( g ) 7→ X i f ( g ) = d dt f (exp tX i ( g )) | t =0 = dim e H 1 X j =1 a ij Y j ( f ( g )) ∈ H f ( g ) N of the horizontal b asis ve ctors X i ( g ) , i = 1 , . . . , dim H 1 , to horizontal ve ctors of the sp ac e H f ( g ) N : H G g M ∋ b X g i 7→ dim e H 1 X j =1 a ij b Y f ( g ) j ∈ H G f ( g ) N . 3.3.4 Stepano v Theorem As a corolla ry to The orem 3.3.5, w e obtain a generalization of Stepano v’s theorem: Theorem 3.3.6 ([129]) . L et E ⊂ M b e a set in M and let f : E → N b e a mapping such that lim x → a,x ∈ E ˜ d ∞ ( f ( a ) , f ( x )) d ∞ ( a, x ) < ∞ 77 for alm o st a l l a ∈ E . T h en f is hc -diﬀer entiable almost e v erywher e on E and the hc -di ﬀ er ential is unique. The homomo rphism o f the Lie a lgebr as c orr esp onding to the hc -diﬀer ential is deﬁne d uniquely by the mapping H g M ∋ X i ( g ) 7→ X i f ( g ) = d dt f (exp tX i ( g )) | t =0 = dim e H 1 X j =1 a ij Y j ( f ( g )) ∈ H f ( g ) N of the b as i s horizontal ve ctors X i ( g ) , i = 1 , . . . , dim H 1 , to horizontal ve ctors of the sp ac e H f ( g ) N : H G g M ∋ b X g i 7→ dim e H 1 X j =1 a ij b Y f ( g ) j ∈ H G f ( g ) N . 4 Applicatio n : The Coarea F orm ula 4.1 Notations All the ab ov e results o n geometry and diﬀeren tiabilit y ar e applied in pro ving the sub-R iemannian analog of the w ell-known coarea formula fo r some class es of con tact mappings o f Carnot — Carath´ eo dory spaces. Notation 4.1.1. Denote by N i the to po logical dimensions of M i and denote b y ν i the Hausdorﬀ dimensions of M i , i = 1 , 2. Assume that T M 1 = M 1 M j =1 ( H j /H j − 1 ) , H 0 = { 0 } , and T M 2 = M 2 M j =1 ( e H j / e H j − 1 ) , e H 0 = { 0 } , where H 1 ⊂ T M 1 and e H 1 ⊂ T M 2 are hori z o ntal subbundles. The subspace H j ⊂ T M 1 ( e H j ⊂ T M 2 ) is spanned b y H 1 ( e H 1 ) and all comm uta tors of or der not exc eeding j − 1, j = 2 , . . . , M 1 ( M 2 ). Denote the dimension of H j /H j − 1 ( e H j / e H j − 1 ) by n j ( ˜ n j ), j = 1 , . . . , M 1 ( M 2 ). Here the nu m b er M 1 ( M 2 ) are suc h tha t H M 1 /H M 1 − 1 6 = 0 ( e H M 2 / e H M 2 − 1 6 = 0), and H M 1 +1 /H M 1 = 0 ( e H M 2 +1 / e H M 2 = 0). The n um ber M 1 ( M 2 ) is called the de pth of M 1 ( M 2 ). Assumption 4.1.2. Supp ose that 1. N 1 ≥ N 2 ; 2. dim H i ≥ dim e H i , i = 1 , . . . , M 1 ; 78 3. the basis v ector ﬁelds X 1 , . . . , X N 1 (in the preimage) are C 1 ,α -smo oth, α > 0, and e X 1 , . . . , e X N 2 (in the image) a re C 1 ,ς -smo oth, ς > 0, or conditions of Remark 2.7.2 hold for α > 0 in the preimage and ς > 0 in t he image. Remark 4.1.3. Note that, if there exists at least one p oin t where the hc - diﬀeren tial b D ϕ is non-degenerate, then the condition dim H 1 ≥ dim e H 1 im- plies n i ≥ ˜ n i , i = 2 , . . . , M 1 (compare with the ab o v e assumption). Notation 4.1.4. Denote b y Z the set o f p oints x ∈ M 1 suc h that rank( D ϕ ( x )) < N 2 . 4.2 La y-out of the Pro of The k ey p oint in pro v ing the non-holonomic coarea for m ula is to in v esti- gate the in terrelation of ”Riemannian“ a nd Hausdorﬀ measures on lev el sets (see b elow ). The researc h on the compar ison of ”Riemannian“ and Haus- dorﬀ dimensions of submanifolds of Carnot groups can b e found in pap er b y Z. M. Balogh, J. T. T y son and B. W arh urst [1 4]. See other results on sub- Riemannian geometric measure theory in w orks b y L. Ambrosio, F. Serra Cassano and D. Vittone [12], L. Cap og na, D. Danielli, S. D. P auls and J. T. T yson [28], D. Danielli, N. Garof alo and D.- M. Nhieu [34 ], B. F ranc hi, R. Serapioni and F. Serra Cassano [55 , 56], B. Kirchheim and F. Serra Cas- sano [87], V. Magnani [94], S. D. P auls [11 6] and man y other. The purp o se of Section 4 is t o explain the ideas of pro of of the coarea form ula for suﬃcien tly smo oth con tact mappings ϕ : M 1 → M 2 of Carnot manifolds. Note that, all the obtained results are new ev en for the particular case of a mappings of Carnot groups. Remark 4.2.1. F or pro ving Theorems 4.2.6, 4.2.8, 4 .2.9, a nd 4.2.13, the smo othness C 1 (in Riemannian sense) for mappings ϕ : M 1 → M 2 is suf- ﬁcien t. F o r pro ving The orem 4.2.12, the ( Riemannian) smoothness C 2 , ,  > 0, of ϕ is suﬃcien t. As it is men tioned ab ov e, for the ﬁrst time, a non-holonomic analog ue of the coarea formula is prov ed in pap er of P . P ansu [112]. The main idea of this w ork (whic h is used in man y o ther ones) is to prov e the coarea fo rm ula 79 via t he Riemannian one: (1.0.3) = ⇒ Z U J S b N 2 ( ϕ, x ) d H ν 1 ( x ) = Z M 2 d H ν 2 ( z ) Z ϕ − 1 ( z ) J S b N 2 ( ϕ, u ) J N 2 ( ϕ, x ) d H N 1 − N 2 ( u ) ? = Z M 2 d H ν 2 ( z ) Z ϕ − 1 ( z ) d H ν 1 − ν 2 ( u ) (4.2.1) Here N 1 , N 2 are top olo gical dimensions, and ν 1 , ν 2 are Hausdorﬀ dimens ions of preimage and image, respectiv ely; it is w ell-kno wn t hat, in sub-Riemannian case, top ological and Hausdorﬀ dimensions diﬀer. It easily follows from (4.2.1), that the k ey point in this pro blem is to in v estigate the in terrelation of ”Riemannian“ and Hausdorﬀ measures on Carnot manifolds theirselv es and on lev e l sets of ϕ , and of Riemannian and sub-Riemannian coarea factors. It is w ell known t hat the question on interrelation o f measures on Carnot man- ifolds is quite easy , while b oth t he inv estigatio n of geometry of lev el sets and the calculation of sub-Riemannian coarea factor are non-trivial. The main problems are connected with p eculiarities of a sub-Riemannian metric. In particular, the non-equiv alenc e of Riemannian and sub-Riemannian metrics can b e seen in the fact that ”R iemannian“ radius of a sub-Riemannian ball of a radius r v aries from r to r M , M > 1, where the constan t M dep ends on the Carnot manifo ld structure. Th us, a question arises immediately on ho w ”sharp“ the appro x imation of a lev el b y its tangen t plain is (since the ”usual“ order of tangency o ( r ) is ob viously insuﬃcien t here: a lev e l may ”jump“ from a ball earlier then it is exp ected). Also a question arises on existence of a suc h sub-Riemannian metric suitable for the description of the geometry of an in tersection of a ball and a lev el set. But ev en if w e answe r thes e q uestions, one more question app ears: w hat is the relation of the Hausdorﬀ dimension of the image and measure of the in tersec tion of a ball and a lev el set. W e hav e solv e d all the ab o v e problems. First of a ll, the p oints in whic h the diﬀeren tial is non-degenerate, are divided in to tw o sets: regular and c haracteristic. Deﬁnition 4.2.2. The set χ = { x ∈ M 1 \ Z : rank b Dϕ ( x ) < N 2 } is called the char acteristic set. The p oin ts o f χ ar e called char acteristic . Deﬁnition 4.2.3. The set D = { x ∈ M 1 : rank b D ϕ ( x ) = N 2 } is called the r e gular set. If x ∈ D , then w e say that, x is a r e gular p oint. 80 W e deﬁne a n um b er ν 0 ( x ) dep ending on x ∈ M that sho ws whether a p oin t is regular or c haracteristic. Deﬁnition 4.2.4. Consider the n um b er ν 0 suc h tha t ν 0 ( x ) = min n ν : ∃{ X i 1 , . . . , X i N 2 }  rank([ X i j ϕ ]( x )) N 2 j =1 = N 2  ⇒  N 2 X j =1 deg X i j = ν o . It is clear that ν 0 | χ > ν 2 and ν 0 | D = ν 2 . W e also deﬁne suc h sub-Riemannian quasimetric d 2 , that mak es the cal- culation of measure of the in tersection of a sub-Riemannian and a t angen t plain to a lev el set p ossible: Deﬁnition 4.2.5. Let M b e a Carnot manifold of top ological dimension N and of depth M , a nd let x = exp  N 1 P i =1 x i X i  ( g ). D eﬁne the distance d 2 ( x, g ) as f ollo ws: d 2 ( x, g ) = max n n 1 X j =1 | x j | 2  1 2 ,  n 1 + n 2 X j = n 1 +1 | x j | 2  1 2 · deg X n 1 1 +1 , . . . ,  N X j = N − n M +1 | x j | 2  1 2 · deg X N o . The similar metric d u 2 is intro duced in the lo cal Carnot group G u M . The cons truction of d 2 is based on the f act that a ball in this quasimetric Bo x 2 asymptotically equals a Cartesian pro duct of Euclidean balls: Bo x 2 ( x, r ) ≈ B n 1 ( x, r ) × B n 2 ( x, r 2 ) × . . . × B n M ( x, r M ) , M > 1 , where N , n i , i = 1 , . . . , M , are (top ological) dimens ions of balls. The latter fact mak es the calculation of ab o v e-men tioned measure p ossible (while in the case when w e replace balls by cub es, it is quite complicated since cub es ha v e diﬀeren t shapes of sections). Using properties of this quasimetric, we calculate the H N 1 − N 2 -measure of the inters ection of a tangen t plain to a leve l set and a sub-Riemannian ball in t he quasimetric d 2 . Theorem 4.2.6. Fix x ∈ ϕ − 1 ( t ) . Then, the H N 1 − N 2 -me asur e of the inter- se c tion T 0 [( ϕ ◦ θ x ) − 1 ( t )] ∩ Bo x 2 (0 , r ) is e quivalent to C (1 + o (1)) r ν 1 − ν 0 ( x ) wher e C do es not d e p end on r , and o (1) → 0 as r → 0 . 81 While inv estigating the approximation of a surface by its tangent plain, w e in tro duce a ”mixed“ metric p ossessing some Riemannian a nd sub-Riemannian prop erties. Deﬁnition 4.2.7. F or v , w ∈ Box 2 (0 , r ) put d 0 2 E ( v , w ) = d 0 2 (0 , w − v ), where w − v denotes the Euclide an diﬀerence. This deﬁnition implies that Bo x 2 (0 , r ) coincides with a ba ll Bo x 2 E (0 , r ) cen tered at 0 o f radius r in the metric d 0 2 E . W e prov e that in regular p oints the tangen t plain approxim ates the lev el set quite sharp with respect to this metric, and f rom here we deduce the p ossibilit y of calculation of the Riemannian measure of a lev el set and a sub-Riemannian ball in tersection. Notable is the fact that this measure can b e expresse d via Hausdorﬀ dimensions of the preimage and the image: it is equiv alen t to r ν 1 − ν 2 (see b elow ): Theorem 4.2.8. Supp ose that x ∈ ϕ − 1 ( t ) is a r e g ular p oint. Then : (I) In the neighb orho o d of 0 = θ − 1 x ( x ) , t her e exists a mapping fr om T 0 [( ϕ ◦ θ x ) − 1 ( t )] ∩ Box 2 (0 , r (1 + o (1))) to ψ − 1 ( t ) ∩ Box 2 (0 , r ) , such that b oth d 2 - and ρ -distortions with r esp e ct to 0 e q ual 1 + o (1) , wher e o (1) is uniform on Bo x 2 (0 , r ); (I I) The H N 1 − N 2 -me asur e of the interse ction ϕ − 1 ( t ) ∩ Bo x 2 ( x, r ) e quals   g | k er D ϕ ( x )   · M 1 Y k =1 ω n 1 k − n 2 k · | D ϕ ( x ) | | b D ϕ ( x ) | r ν 1 − ν 2 (1 + o (1)) , wher e g is a Riemann tensor, b D ϕ is the hc -diﬀer ential of ϕ , and o (1 ) → 0 as r → 0 . F rom t hese results and obtained prop erties, using a result of [136], w e deduce the inte rrelation of tw o measures in regular p oin ts of a leve l sets. Theorem 4.2.9 (Measure Deriv ativ e on Lev el Sets) . Hausdo rﬀ me asur e H ν 1 − ν 2 of the interse ction Bo x 2 ( x, r ) ∩ ϕ − 1 ( ϕ ( x )) , whe r e x is a r e gular p o i n t, and dist(Box 2 ( x, r ) ∩ ϕ − 1 ( ϕ ( x )) , χ ) > 0 , asymptotic al ly e quals ω ν 1 − ν 2 r ν 1 − ν 2 . The derivative D H N 1 − N 2 H ν 1 − ν 2 ( x ) e quals 1   g | k er D ϕ ( x )   · ω ν 1 − ν 2 M 1 Q k =1 ω n k − ˜ n k · | b D ϕ ( x ) | | D ϕ ( x ) | . 82 Finally , we in troduce the notion of the sub-Riemannian coarea fa ctor via the v alues of the hc -diﬀerential of ϕ . Deﬁnition 4.2.10. The sub-R iemannian c o ar e a factor equals J S R N 2 ( ϕ, x ) = | b Dϕ ( x ) | · ω N 1 ω ν 1 ω ν 2 ω N 2 ω ν 1 − ν 2 M 1 Q k =1 ω n k − ˜ n k . W e consider and solve problems connected with the c haracteristic set. The case of c hara cteristic p oin ts is a little more complicated since in c hara c- teristic p oin ts a surface ma y jump from a sub-Riemannian ball, conseq uen tly , w e cannot e stimate the meas ure of the in tersection of the surface and the ball via the o ne of the tangen t plain and the ball. Note also that in all the other w orks on sub-Riemannian coarea formula, the preimage has a g roup struc- ture, which is essen tially used in pro ving the fact that the Hausdorﬀ measure of c haracteristic points on eac h lev el s et e quals zero (see also the pap er [13] b y Z. M. Balogh, dedicated to properties of the characteristic set). In the case of a mapping of t w o Carnot manifolds, there is no group structure neither in image, nor in preimage. Moreo v er, the approximation of Carnot manifold b y its lo cal Carnot group is insuﬃc ien t f or generalization of methods dev elop ed b efore. That is wh y w e construct new ”intrinsic“ metho d of in v e stigation of prop erties of the c haracteristic se t. F irst of all, in all the c haracteristic p oin ts the hc - diﬀeren tial is degenerate. W e solv e this problem with t he followin g assumption. Prop erty 4.2.11. Supp ose that x ∈ χ , and ra nk b Dϕ ( x ) = N 2 − m . Let also b D ϕ ( x ) equals zero on n 1 − ˜ n 1 + m 1 horizon tal (linearly indep enden t) v ectors, n 2 − ˜ n 2 + m 2 (linearly indep enden t) ve ctors from H 2 /H 1 , n k − ˜ n k + m k (lin- early independen t) v ectors from H k /H k − 1 , k = 3 , . . . , M 2 . Then, on the one hand, since r ank b D ϕ ( x ) = N 2 − m , we ha v e M 1 P i =1 m k = m . On the other hand, rank D ϕ ( x ) = N 2 . Consequen tly , there exist m (linearly indep enden t) v ec - tors Y 1 , . . . , Y m of degrees l 1 , . . . , l M 2 (whic h a re minimal) from the k ernel of the hc - diﬀeren tial b Dϕ , suc h that D ϕ ( x )(span { H M 2 , Y 1 , . . . , Y m } ) = T ϕ ( x ) M 2 . In this subsection, w e will a ssume that, amo ng the vec tors Y 1 , . . . , Y m , m 1 of them of the degree l 1 ha v e the horizontal image, m 2 of them of the degree l 2 ≥ l 1 ha v e image b elonging to e H 2 , and m k of them of the degree l k , l k ≥ l k − 1 , hav e image b elonging to e H k , k = 3 , . . . , M 2 . By another words , the ” extra“ vector ﬁelds on whic h the hc - diﬀeren tial of ϕ is degenerate in characteristic p oints, p ossess the following prop erty : if in H k /H k − 1 ( x ) the quan tit y of suc h ”extra“ v ectors equals m k > 0, then 83 there exist m k v ectors from H l k /H l k − 1 ( x ) suc h that their images ha v e the degree k , they are linearly independen t with each o ther and with t he imag es of H l k − 1 ( x ), l k ≥ l k − 1 . W e dev elop new ”in trinsic“ method of in ves tigation of the prop erties of the c haracteris tic set. Example. The condition described in Assumption 4.2 .11, is alwa y s v alid for the fo llo wing M 1 and M 2 : 1. M 1 is an arbitrary Carnot–Carath ´ eo dory space, and M 2 = R ; 2. M 1 is an arbitrary Carnot–Carath´ eo dory space of the to po logical di- mension 2 m + 1, G u M 1 = H m for all u ∈ M 1 , M 2 = R k , k ≤ 2 m ; 3. M 1 = M 2 , dim H 1 ≥ dim e H 1 , dim( H i /H i − 1 ) = dim( e H i / e H i − 1 ), i = 2 , . . . , M 1 ; 4. M 1 = M 2 + 1, dim H i = dim e H i , i = 1 , . . . , M 2 . In pa rticular, in Theorem 4.2.6 it is sho w n, that in the c haracteristic p oin ts H N 1 − N 2 -measure of the intersec tion o f a sub-Riemannian ball and the tangen t plain to the lev el set is equiv alen t to r to the p ow er ν 1 − ν 0 ( x ) < ν 1 − ν 2 . Next, w e sho w, that H N 1 − N 2 -measure of the in tersection of the lev el set and the sub-Riemannian ball cen tered at a c haracteristic p oin t is inﬁnitesimally big in comparison with r ν 1 − ν 2 , i. e., is eq uiv alent to r ν 1 − ν 2 o (1) (but it is not nec essarily equiv alen t t o r ν 1 − ν 0 ( x ) ). F rom here w e deduce that, the in tersec tion of the c haracteristic set with eac h lev el set has zero H ν 1 − ν 2 - measure. Theorem 4.2.12 (Size of the Characteristic Set) . T h e Hausdorﬀ me asur e H ν 1 − ν 2 ( χ ∩ ϕ − 1 ( t )) = 0 for al l z ∈ M 2 . W e also sho w that the degenerate set of the diﬀerential do es not inﬂuence b oth parts of the coarea form ula. Theorem 4.2.13. F or H ν 2 -almost al l t ∈ M 2 , we have H ν 1 − ν 2 ( ϕ − 1 ( t ) ∩ Z ) = 0 . Finally , w e deduce the sub-Riemannian coarea form ula. Theorem 4.2.14. F o r any smo oth c ontact m a pping ϕ : M 1 → M 2 p os s essing Pr op erty 4.2.11, the c o ar e a formula holds: Z M 1 J S b N 2 ( ϕ, x ) d H ν 1 ( x ) = Z M 2 d H ν 2 ( t ) Z ϕ − 1 ( t ) d H ν 1 − ν 2 ( u ) . 84 As an application, us ing the result of the paper b y R. Monti and F. Serra Cassano [107, T heorem 4.2 ] for Lip-functions deﬁned on a Carno t–Carath ´ eo dory space M of the Hausdorﬀ dimen sion ν , w e deduce that the De Giorgi p erime- ter coincide s with H ν − 1 -measure on almost ev ery lev el of a smooth function ϕ : M : Theorem 4.2.15. F or C 2 ,α -functions ϕ : M → R , α > 0 , wher e dim H M = ν , the De Gior gi p erimeter c oin c ides with H ν − 1 -me asur e on almost every level. 5 App e ndix 5.1 Pro of of Lemma 2.1.13 Pr o of. It is w ell know n, that the solution y ( t, u ) of the ODE (2.1.6) equals y ( t, u ) = lim n →∞ y n ( t, u ), where y 0 ( t, u ) = t Z 0 f ( y (0) , u ) dτ , and y n ( t, u ) = t Z 0 f ( y n − 1 ( τ , u ) , u ) dτ . This con v ergence is uniform in u , if u b elongs to some compact set. F rom the deﬁnition of this sequence it fo llo ws, that y n ( t ) → y ( t ) a s n → ∞ in C 1 -norm. 85 1. W e show, that ev ery y n ( t, u ) ∈ C α ( u ) f or eac h t ∈ [0 , 1]. W e ha v e max t | y n ( t, u 1 ) − y n ( t, u 2 ) | ≤ 1 Z 0 | f ( y n − 1 ( τ , u 1 ) , u 1 ) − f ( y n − 1 ( τ , u 2 ) , u 2 ) | dτ ≤ 1 Z 0 | f ( y n − 1 ( τ , u 1 ) , u 1 ) − f ( y n − 1 ( τ , u 1 ) , u 2 ) | dτ + 1 Z 0 | f ( y n − 1 ( τ , u 1 ) , u 2 ) − f ( y n − 1 ( τ , u 2 ) , u 2 ) | dτ ≤ H ( f ) | u 1 − u 2 | α + L max t | y n − 1 ( t, u 1 ) − y n − 1 ( t, u 2 ) | ≤ H ( f ) n − 1 X m =0 L m | u 1 − u 2 | α + L n max t | y 0 ( t, u 1 ) − y 0 ( t, u 2 ) | ≤ H ( f ) ∞ X m =0 L m | u 1 − u 2 | α , where H ( f ) is a constan t, suc h that | f ( u 1 ) − f ( u 2 ) | ≤ H ( f ) | u 1 − u 2 | α . Note that the constan t H = H ( f ) ∞ P m =0 L m < ∞ since L < 1, and it do es not dep end on n ∈ N . Supp ose that u b elongs to some compact set U . Then | y ( t, u 1 ) − y ( t, u 2 ) | ≤ | y ( t, u 1 ) − y n ( t, u 1 ) | + | y n ( t, u 1 ) − y n ( t, u 2 ) | + | y ( t, u 2 ) − y n ( t, u 2 ) | ≤ H | u 1 − u 2 | α + 2 ε for eve ry ε = ε ( n ) > 0 . Since the conv ergence is uniform in u ∈ U , and ε ( n ) → 0 as n → ∞ , t hen | y ( t, u 1 ) − y ( t, u 2 ) | ≤ H | u 1 − u 2 | α , and y ∈ C α ( u ) lo cally . T o show , that ∂ y ∂ v i ( t, v , u ) ∈ C α ( u ) lo cally , i = 1 , . . . , N , w e obtain our estimates in the simple st case o f N = 1. 2. Note that the mappings { y n } n ∈ N con v erge to y in C 1 -norm, and this con v ergence is uniform, if u b elongs to some compact set U . Let u ∈ U , v ∈ W (0) ⊂ R N . Then similarly to the case 1 , w e see, that if 86 the H¨ older constan t of y ′ n do es not depend on n ∈ N , then y ′ ∈ C α ( u ). max t,v    dy n dv ( t, v , u 1 ) − dy n dv ( t, v , u 2 )    ≤ max t,v    d dv t Z 0 f ( y n − 1 ( τ , v , u 1 ) , v , u 1 ) − f ( y n − 1 ( τ , v , u 2 ) , v , u 2 ) dτ    ≤ max t,v    d dv t Z 0 f ( y n − 1 ( τ , v , u 1 ) , v , u 1 ) − f ( y n − 1 ( τ , v , u 2 ) , v , u 1 ) dτ    + max t,v    d dv t Z 0 f ( y n − 1 ( τ , v , u 2 ) , v , u 1 ) − f ( y n − 1 ( τ , v , u 2 ) , v , u 2 ) dτ    . (5.1.1) F or the ﬁrst summand we hav e max t,v    d dv 1 Z 0 f ( y n − 1 ( τ , v , u 1 ) , v , u 1 ) − f ( y n − 1 ( τ , v , u 2 ) , v , u 1 ) dτ    ≤ max v 1 Z 0    d dv ( f ( y n − 1 ( τ , v , u 1 ) , v , u 1 ) − f ( y n − 1 ( τ , v , u 2 ) , v , u 1 ))    dτ ≤ max v 1 Z 0    d f dy dy n − 1 dv ( τ , v , u 1 ) − d f dy dy n − 1 dv ( τ , v , u 2 )    dτ + max v 1 Z 0    ∂ f ∂ v ( y n − 1 ( τ , v , u 1 )) − ∂ f ∂ v ( y n − 1 ( τ , v , u 2 ))    dτ . (5.1.2) Then, w e get max v 1 Z 0    ∂ f ∂ v ( y n − 1 ( τ , v , u 1 )) − ∂ f ∂ v ( y n − 1 ( τ , v , u 2 ))    dτ ≤ C ( f ) H ( y ) | u 1 − u 2 | α , since eac h y m is H¨ older. The ﬁrst summand in (5.1.2) is ev aluated in the 87 follo wing w ay: max v 1 Z 0    d f dy dy n − 1 dv ( τ , v , u 1 ) − d f dy dy n − 1 dv ( τ , v , u 2 )    dτ ≤ max v 1 Z 0    d f dy ( u 1 ) dy n − 1 dv ( τ , v , u 1 ) − d f dy ( u 1 ) dy n − 1 dv ( τ , v , u 2 )    dτ + max v 1 Z 0    d f dy ( u 1 ) dy n − 1 dv ( τ , v , u 2 ) − d f dy ( u 2 ) dy n − 1 dv ( τ , v , u 2 )    dτ ≤ L max t,v    dy n − 1 dv ( t, v , u 1 ) − dy n − 1 dv ( t, v , u 2 )    + max u,v 1 Z 0    dy n − 1 dv ( τ , v , u )    dτ · H ( D f ) | u 1 − u 2 | α . (5.1.3) Next, w e estimate max u,v 1 Z 0    dy m dv ( τ , v , u )    dτ ≤ max t,u,v    dy m dv ( t, v , u )    = max t,u,v h L    dy m − 1 dv    +    ∂ f ∂ v    i ≤ max t,u,v    ∂ f ∂ v    h ∞ X k =0 L k i < ∞ . Th us, in the ﬁr st summand of (5.1.1) w e hav e L max t,v    dy n − 1 dv ( t, v , u 1 ) − dy n − 1 dv ( t, v , u 2 )    + C | u 1 − u 2 | α , where 0 < C < ∞ do es not dep end on n ∈ N . The second summand in (5.1.1) is max t,v    d dv t Z 0 f ( y n − 1 ( τ , v , u 2 ) , v , u 1 ) − f ( y n − 1 ( τ , v , u 2 ) , v , u 2 ) dτ    max v 1 Z 0    ∂ f ∂ v ( y n − 1 , v , u 1 ) − ∂ f ∂ v ( y n − 1 , v , u 2 )    dτ ≤ C ( f ) | u 1 − u 2 | . 88 Th us, max t,v    dy n dv ( t, v , u 1 ) − dy n dv ( t, v , u 2 )    ≤ L max t,v    dy n − 1 dv ( t, v , u 1 ) − dy n − 1 dv ( t, v , u 2 )    + K | u 1 − u 2 | α ≤ k ∞ X k =0 L k | u 1 − u 2 | α , and dy n dv ∈ C α ( u ) lo cally . Hence, dy dv ∈ C α ( u ) lo cally . Ac kno wl edgmen t W e are g rateful to V a leri ˇ ı Beres to vski ˇ ı, for his collab o ration in pro ving of Gromov ’s theorem on nilp oten tization of v ector ﬁelds (see Remark 2.2.12). W e also w an t t o thank Pie rre P ansu for fruitful discussion of our results con- cerning the geometry of Carnot–Carath ´ eo dory spaces and the sub-Rieman- nian coa rea form ula, and H. Martin Reimann for in teresting discussions of diﬀeren tiabilit y theorems. The researc h was partially supp orted b y the Commission of the Euro- p ean Comm unities (Sp eciﬁc T argeted Pro ject “Geometrical Analysis in Lie groups a nd Applications”, Con tract num b er 0287 66), the Russian F ounda- tion for Basic Res earc h (Gran t 06-01-0 0735), the State Maintenance Program for Y oung R ussian Scien t ists and the Leading Scien tiﬁc Sc ho ols of Russian F ederation (Grant NSh-5682.2008.1) . References [1] A. A. Agrac hev, Comp actness for sub-Riemannian length minimizers and sub an alyticity. Rend. Semin. Mat. T orino, 56 (199 8). [2] A. A. Agr ac hev and R. Gamkrelidze. Exp onential r e p r esentation of ﬂows and chr onolo gic al c alculus. Math. USSR-Sb. 107 (4) (1978), 487– 532 ( in Russian). 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Geometry of Carnot--Carath{e}odory Spaces, Differentiability and Coarea Formula

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