On The Existence of Globally Solvable Vector Fields in Smooth Manifolds

TRANSACTIONS OF THE AMERICAN MA THEMA TICAL SOCIETY V olume 00, N umber 0, Pages 000– 000 S 0002-9947(XX)0000 -0 ON THE EXISTENCE OF GLOBALL Y SOL V ABLE VECTOR FIELDS IN SMOOTH MANIF OLDS JOS ´ E RUIDI V AL DO S SANTOS FILHO AND JOA QU IM T A V ARES Abstract. Let (M , ω 0 ) be a connecte d paracompact smooth orien ted mani- fold. W e establish a necessary and sufficient conditions on an inv olutive sub- bundle of T M suc h that M b ecomes simply connected. 1. Introduction It is well k nown that there exist an obstruction to the ex istence of dim M − k + 1 real linearly independent v ector fields o n a n manifo ld M in the k th coho mology group of M, the so called k th Stiefel-Whitney clas s. W e mean that the k th Stiefel- Whitney cla ss b e ing nonzero implies that there do not exis t everywhere linearly independent vector fields. In particular, the dim M th Stiefel-Whitney clas s is the obstruction to the existence of an everywhere nonzer o vector field, and the first Stiefel-Whitney class of a manifold is the obstruction to orientabilit y . Thus if one wishes to ass ume a hypotheses of exis tence k linea rly indepe ndent vector fie lds in an orientable manifold it certainly impo s es the v anishing of such Stiefel-Whitney classes. But it is not ev iden t that additional hypothesis of integrability and global solv ability of k sections of Γ(T M) the tr iviality of the all l th cohomology groups of M for l > dim M − k + 1. This fact is enco ded in a fundamental theo r em o f Hor ma nder and Duistermaat (Theorem 6.4 .2 [HD], pp 30) which characterizes glo bal so lv ability of first orde r r eal differen tial op erator s in a ori e nted smo o th manifold. But its pro of is not tr iv ial bec a use it pass tr oughs a ge ner alization of the theore m ab ov e. The gener alization lies in a induction o v er the conv exi ty condition stated in c ) of Theor e m 6 .4.2 mentioned ab ove. In the reading of pr o of one re alize that the 1991 Mathematics Subje ct Classific ation. Pr imary 35A05; Secondary 54C25. Key wor ds and phr ases. i njectiv e m aps, real ve ctor fields. c  XXXX American Mathematical S ociety 1 2 JOS ´ E RUIDIV AL DOS SANTOS FILHO AND J O AQUIM T A V ARES condition of co n vexit y is needed o nly inside the orbits of the vector field as well as compactness is needed only for the intersection o f the sublevels { x ∈ M : u ( x ) ≤ c } with the orbits in c ) o f Theorem 6.4.2. T aking this p ointview we will extend the concept o f convexit y ab ov e b y taking a finitely g enerated Lie algebra a E Γ(T M) and for a compact set K ⊂ M we define b K to b e the set obtained taking a ll smo oth regular paths γ with with endp oints in K a nd γ ′ ∈ a . W e will show that in this context the analog ous condition c ) of Theorem 6 .4.2 is equiv alent to one of the tw o condition stated be low for the orbits O a ( p ) of a (in the sense of Sussmann ([Su])). ( C ) ther e exists at le ast dim O a ( p ) − 1 line arly indep endent se ctions of Γ(T O a ( p )) which ar e glob al ly solvable for every p ∈ M . ( C ′ ) ther e exist at le ast dim O a ( p ) − 1 line arly indep endent se ctions of Γ(T O a ( p )) and O a ( p ) is simply c onne cte d for every p ∈ M. The condition ( C ′ ) is a further res tr iction on the Stiefel-Whitney classes of T O a ( p )) since we thro ughout this pa per we will assume (M , ω 0 ) will be a para- compact manifold with pos itive o r ientation ω 0 . Then the condition ( C ′ ) is ent irely determined b y the class of homotopy of the C W c omplex asso ciated to M. Also we will assume the following hypo thesis on the a finite set of ge ner ators { X 1 , ..., X n } of a D Γ(T M); s et d l to b e the forma l degree of X l and assume; [ X j , X k ] = X d l ≤ d j + d k c l j k X l in the same s ense as defined in the bas ic work of Nag el et al ([NSW]). W e also will denote by ∼ a the equiv a lence rela tion of being in a same orbit O a ( p ) and H 1 d O a ( p ) the first De Rham co homologica l group. T o avoid intro duction of a metric in M we will adopt the following definitio n found in page 111 of the reference [NSW]. Definitions. Let a a Lie subalgebra of Γ(T M) f ini tel y gener ated by { X 1 , ..., X n } and as sume that Denote by C ( δ ) the c lass of smo oth paths γ : [0 , 1 ] : → M such GLOBALL Y SOL V ABLE VECTOR FIELDS IN SMOOTH MANIFOLDS 3 that γ ′ ( s ) = n X l =1 c l X l ( γ ( s )) with | c l | ≤ δ d l and a ll c l constants. Define the pseudo-distanc e ρ ( p, q ) = inf { δ > 0 : ∃ γ ∈ C ( δ ) with γ (0) = p, γ (1) = q } and denote b y b K a the set of all paths γ ∈ ∪ δ> 0 C ( δ ) with γ (0) , γ (1 ) ∈ K . W e say that the triplet (M , a , ω 0 ) is a − co nvex if for every compact subset K ⊂ M and p ∈ M a pair of conditions holds; a) b K a c omp act and b) ther e exists a set { X 1 , ..., X n ( p ) − 1 } ∈ Γ(T O a ( p )) of line arly indep endent se c- tions such that al l its inte gr al orbits ar e non c omp act. Our main result is; Theorem.A. Let (M , ω 0 ) b e a smo oth or ien ted paraco mpact co nnec ted manifold and a E Γ(T M) b e a finitely ge ne r ated Lie a lgebra. Then three statements are equiv alent; A) M is a − c onv ex , B) a v erifies ( C ), C) a v erifies ( C ′ ). 2. Proof of Theorem A. W e start the pro of with a lemmma; Lemma. L et a ⊂ Γ(TM) b e a Lie sub algebr a c ontaining by k − line arly indep en- dent glob al ly solvable ve ctor fields. Then ther e exist a c ommutative su b algebr a g E a gener ate d by k − line arly indep endent glob al ly solvable ve ctor fields. Pr o of. W e p erfor m induction on the num ber k . F or k = 1 we just apply the equiv alence b etw een b) and f ) in the Theo rem 6.4.2 in [DH]. Supp ose that the Lemma is true for k and let a an Lie algebra co ntaining k + 1linearly independent set { X 1 , ..., X n , X n +1 } globally solv able vector fields. Then by induction hypothesis we may assume that [ X l , X k ] = 0 if 1 ≤ l , k ≤ n . Linear indep endence implies that X n +1 = a 1 X 1 + · · · + a n X n + Y with Y ∈ a a non v anishing vector field. W e define 4 JOS ´ E RUIDIV AL DOS SANTOS FILHO AND J O AQUIM T A V ARES the a diffeomorphism Φ : M → R n × N n taking a s the n − first co or dinates of Φ the solutions X k t k = 1 with its natur al o rdering and the trivialization comes again from the equiv alence betw een a) a nd f ) in Theor e m 6.4.2 in [DH] just by a straightforw ard inductive argument ov er the num b er k . It follows that D Φ( Y ( p )) ∈ T p ( N k ) fo r every p ∈ M, where Π − 1 (Π( p )) = R n × Π( p ) s tands for the n − k − las t co o rdinates of Φ. Then by hypothesis for every orbit γ of X n +1 with endpo ints q 0 , q 1 ∈ K there exist another compact s et K ′ ⊃ K such that γ ⊂ K ′ . The n Π( q 0 ) , Π( q 1 ) ∈ Π( K ) are endp oints of Π ◦ γ ⊂ Π( K ′ ) and DΠ( γ ′ ) = DΠ( Y ( γ )) 6 = 0 . If a integral or bit γ of X n +1 is not contained in a compact set K then the same sho uld happ ens to Π( γ ) with res pect to the compa c t Π( K ). W e now co nsider the orbits O a n ( p ) where a n E a is the Lie subalg ebra genera ted b y { X 1 , ..., X n } . All the o rbits O a n ( p ) are diffeomorphic to R n . W e can solve X n +1 u = 1 a nd consequently if γ a n integra l tra jectory of X n +1 parameterize d by [0 , ∞ ) with γ (0) = p then lim t →∞ u ( γ ( t )) = ∞ and conse q uen tly u will no t be smo oth in M if Π( γ ) has co m- pact clo sure. (Another argument: Deno te b y ω n the po sitive or ie n tation induced from (M , ω 0 ). Since ω n ( X 1 , ..., X n ) > 0 we can solve X n +1 u = ω n ( X 1 , ..., X n ) − 1 ent ailing that d u ∧ ω n ( X n +1 , X 1 , ..., X n ) ≡ 1. Let K ⊂ M s uch that K ∩ O a n ( γ ( t )) is compact a nd 1 d t = Z Π − 1 (Π( γ ( t )) ) χ K ω n for ev ery p ∈ M. If γ a n integral tra jecto ry of X n +1 parameterize d b y [0 , ∞ ) with γ (0) = p then Z Π( γ )  Z Π − 1 (Π( p )) χ K ω n  d u = Z Π − 1 (Π( γ )) d u ∧ χ K ω n = ∞ consequently if Π( γ ) if compact closure b ecause then the quantit y on the left is finite). Th us a ny integral tra jector y of Y is un bo unded and Y verify the condition d) in Theo rem 6.4.2 in [DH] is globa lly so lv able in N n indeed and the equiv alence betw een f ) and ) in Theor em 6.4.2 in [DH] applies to conclude the induction.  Now if B) is true we apply the Lemma to Γ(T O L ( p )) to find o ut that is dif- feomorphic to R n ( p ) − 1 × γ for so me smo oth reg ular connected cur ve γ and with GLOBALL Y SOL V ABLE VECTOR FIELDS IN SMOOTH MANIFOLDS 5 n ( p ) = dim O L ( p ). W e consider a no n v anishing section o f γ ′ ∈ Γ(T γ ) which verifies ω O L ( p ) ( γ ′ ( p ) , X 1 ( γ ( p )) , ..., X n ( p ) − 1 ( p )) > 0 and define u ( t ) = Z t 0 ω O L ( s ) ( γ ′ ( s ) , X 1 ( γ ( s )) , ..., X n ( p ) − 1 ( γ ( s )) ) d s is a strictly monotonous function o f t ∈ R a nd consequently prop er showing that γ is diffeomorphic to an o pen interv al and the vector field X dim O L ( p ) ( p ) = γ ′ ( p ) is indeed glo bally s olv able in M b y d) in Theorem 6.4.2 in [DH]. Then w e apply again the Lemma to find out that O a ( p ) ≃ R dim O a ( p ) . O bserve that O a ( p ) ∩ b K a = \ O a ( p ) ∩ K a comes from a result o f Sussmann ([Su]) and consequently compactness of b K a is e q uiv alent to compactness of its in tersection with an ar bitrary or bit O a ( p ). On the other hand the condition ( C ) implies that O a ( p ) ≃ R dim O a ( p ) with b K a ∩ O a ( p ) cor r esp onding to the conv ex env elope o f K ∩ O a ( p ) in R dim O a ( p ) . Since the Lie a lgebra a is finitely generated and O a ( p ) ≃ R dim O a ( p ) the metric ρ is w ell defined by a in O a ( p ). Denote b y dia m the s et function induced in O a ( p ) by the Euclidean metr ic in R dim O a ( p ) with the inherited o r ientation fr om M (see Theor em 1. & 3, page 110 in [NSW]). It follows that if p, q ∈ K ∩ O a ( p ) then C 1 diam( K ∩ O a ( p )) ≤ ρ ( q , p ) ≤ C 2 diam 1 / max d j ( K ∩ O a ( p )) from Pro po sition 1.1 pag e 1 07 in [NSW]. Consequently b K a ∩ O a ( p ) ⊂ { q ∈ O a ( p ) : ρ ( q , p ) ≤ C 3 diam 1 / max d j ( K ∩ O a ( p )) } and { ρ ( · , p ) ≤ c }∩O a ( p ) is alwa ys compact for every c ∈ R . Mor eov er if ( x 1 , ..., x n ( p ) ) are the co o r dinates o f R dim O a ( p ) then the inverse image of ( ∂ / ∂ x 1 , ..., ∂ /∂ x n ( p ) ) in Γ(T O a ( p )) will verify the pro pe rty b ) for a − c onv ex it y , completing the first part of the pro o f. Now if A) is tr ue , since O a ( p ) is a manifold and lo cally we may assume that V ≃ R dim O a ( p ) for a n op en conv ex neighborho o d V ⊂ M of p ∈ M . Then ( V , a , ω 0 ) verifies ( C ) and { ρ ( · , p ) ≤ c } ∩ O a ( p ) remains compact for small c ∈ R . On the other hand by b esides the existence of global linearly indep endent set { X 1 , ..., X n ( p ) − 1 } in Γ(T O a ( p )) by the pro per t y b ) of a − conv exit y , the pseudo-distances defined by 6 JOS ´ E RUIDIV AL DOS SANTOS FILHO AND J O AQUIM T A V ARES ρ but taking only smoo th paths tangent to the linear span of the set { X 1 , ..., X n } generating a is equiv alent to the former one by Theo rem 3. &4. E quiv alent pseudo- distances, page 111 in [NSW]. As a cons e quence the set { X 1 , ..., X n ( p ) − 1 } will b e a set of glo bally s olv able vector fields of Γ(T O a ( p )) by the pr op erty a ) of a − conv exit y and the condition ( C ) is verified and A) implies B ). That B) implies C) is just application of the Lemma tog e ther the equiv alenc e f ) in in Theore m 6 .4.2 in [DH]. Then it is left to pr ov e that C) implies B). Since a is finitely generated we may select a set { X 1 , ..., X n } ⊂ Γ(T M ) generating a and define the second order op erato r H = X 2 1 + · · · + X 2 n which is well defined in e very or bit O a ( p ) wher e it is hypo elliptic by a result of Hormander[Ho]) . It is a consequence of the Bony’s max im um principle ([Bo]) tha t a twice differentiable function u verifying H u ≥ 0 hav e upp erlevel sets { u ≥ c } ∩ O a ( p ) is r elatively non compa ct in O a . On the o ther hand if we assume also that the sublevel se ts { u ≤ c } ∩ O a ( p ) are co mpact and the 2th Stiefel- Whitney cla s s of T O a ( p ) is trivial there e xist n ( p ) − 1 linearly independent sections { X 1 , ..., X n ( p ) − 1 } ⊂ Γ(T O a ( p )). Since the or bits O a ( p ) are p ositively oriented with the inherited n ( p ) − different ial form ω O a ( p ) , the 1 − differential form ω O a ( p )) ( · , X 1 , ..., X n ( p ) − 1 ) is non v anishing. But it v anishes in the tange nt space of the orbit generated b y the the linearly indep endent set o f vector fields { X 1 , ..., X n ( p ) − 1 } which dimension is n ( p ) − 1 or n ( p ), the latter inco mpatible with the p os itivit y of the or ient ation in O a ( p ). If the fir st De Rham cohomolo gy gr oup o f O a ( p ) is trivial then lo c ally we can find smo oth function u with d u = ω O a ( p ) ( · , X 1 , ..., X n ( p ) − 1 ) which is co n- stant in the comp onents of o rbits gener ated by { X 1 , ..., X n ( p ) − 1 } . Then for any choice of X n ( p ) ∈ Γ(T O a ( p )) linearly independent from { X 1 , ..., X n ( p ) − 1 } suc h that ω O a ( p ) ( X n ( p ) , X 1 , ..., X n ( p ) − 1 ) > 0 will verify X 2 n ( p ) e − κ u 2 ( p ) > 0 for large po sitive κ ( K ) and a ll p ∈ K , a co mpact set. The parac ompactness o f M allows one to find sequence o f compacts K n such tha t int K n ⊂ K n +1 and smo o th χ K n with χ K n ( q ) = 1 if q ∈ K n , χ K n ( q ) = 0 if q ∈ M \ K n +1 and 0 ≤ χ K n ( q ) ≤ 1 for all GLOBALL Y SOL V ABLE VECTOR FIELDS IN SMOOTH MANIFOLDS 7 q ∈ M. Denote b y κ ( K n ) a constant verifying κ ( K n ) min p ∈ K n ω O a ( p ) ( X n ( p ) , X 1 , ..., X n ( p ) − 1 ) ≥ sup p ∈ K n − 1 | X 2 n ( p ) χ K n u | . Then by a suitable choice of constants κ ( K n ) we find out that the function u 0 = ∞ X n =1 χ K n e − κ ( K n ) u 2 is smo oth and X 2 n ( p ) u 0 > 0 fo r all p ∈ M. If the integral tra jectory γ of X n ( p ) starting at p ∈ O a ( p )) r emains in a compact set K then at s o me p oint q of this compact ω O a ( p ) ( X n ( p ) ( q ) , X 1 ( q ) , ..., X n ( p ) − 1 ( q )) = 0 which is a contradicts the p os itivit y of the o rientation ω O a ( p )) ( q ). It follows that the condition c ) in in Theo rem 6.4.2 in [DH] is verified for X n ( p ) and it is g lobally int egrable and we apply f ) in the same theor em to wr ite O a ( p )) = N n ( p ) − 1 × R . Since the subma nifolds N n ( p ) − 1 × t inherits same pr o pe rties o f O a ( p ) we apply induction to conclude that O a ( p ) ≃ R n ( p ) , finishing the pr o of.  Before finish this pape r w e must r emark the the extrao rdinary sem blance b etw een this form o f presenting global solv a bilit y for a real vector field in a ma nifold with the par allelization Theorem A in a work o f Greene & Shiohama ([GS]), but in the absence of a Riemannian structure or the necessity of emptiness o f the critica l set of u . The Theor em B in [GS] says that when the ma nifold is not simply co nnected the obs truction to global so lv ability will be lo cated in the sing ular set o f the conv ex function pointing to further inv e stigation of this phenomena. Also one also may apply the generaliz ed Tietze- Nak a jima theorem in [K B] together the Lemma a bove to show that the manifold M has a s pro jection o f the fir s t n − co ordina tes of Φ a conv ex subset o f R n when Φ is prop er. References [B] Bony , J.M. Princip e du maximum, in´ galit´ e de Harnack e t unicit´ e du pr obl` eme de Cauchy p our les op ´ er ate urs el liptiques d ´ eg´ en´ er ´ es , Annales de l’ institut F ourier, 19 , 277-304, (1969). [DH] Duistermaa t, J.J., Hormander, L. F ourier Inte gr al Op er ators II , Acta Math., 128 , 183- 269, (1972) . [GS] Greene, R.E., Shioham a, K. Convex functions on c omplete nonc omp act manifolds : dif- fer e nt iable structur e , Annales scientifiques de l’cole Nor male Suprieure 14 , 357-367, (1981). 8 JOS ´ E RUIDIV AL DOS SANTOS FILHO AND J O AQUIM T A V ARES [Ho] Hormander, L. Hyp o el liptic se co nd or der differ ential e quations Act. Math. , 11 9 ,147-171 (1968). [KB] Karshon, Y., Bjorndhal, C. R evisiting Tietze-N akajima L o ca l and Glob al Convexity for maps T o app ear in the Canadian Journal of Mathematics, preprin t: arXiv:math.CO/0701745 25 , 1-15 ,(2007). [NSW] Nagel, A.,Stein, E., and W a igner, S. Bal ls and metrics define d by ve ctor file ds:Basic Pr op e rt ies Acta Math. , 155 , 103-147, (1985) . Acta Math. 155 (1985) , 103-147 [Su] Sussmann, H. Orbits of families of ve ctor fields and inte gr ability of distributions T ransac- tions of the Amer icam Mathematical So ciet y , 180 ,171-188, (1973). Dep ar t amen to de Ma tem ´ atica, Universidade Federal de S ˜ ao Carlos, S ˜ ao Carlos, 13565-90 5, S P, Brazil E-mail addr ess : santos@dm.ufs car.br Dep ar t amen to de Matem ´ atica, Universidade Federal de Perna m buco, Recife, 50 7 40- 540, PE, Brazil E-mail addr ess : joaquim@dmat. ufpe.br