Finsler conformal Lichnerowicz-Obata conjecture

Finsler Conformal Lic hnero wicz-Obata conjecture Vladim ir S. Mat v eev Hans-Bert Ra demac her Marc T roy ano v Ab delgha ni Zeghi b Abstract W e pro v e the Finsler analog of th e conform al Lic hnero wicz-Obata conjecture sho wing that a complete and essen tial conform al ve ctor field on a non -Riemann ian Finsler m ani- fold is a homothetic v ector field of a Mink o wski m etric. MSC 2000: 53A30; 53C60 Key wor ds: essen tia l conformal v ector field, conformal transformations, Finsler metrics, Lic hnero wicz-Obata conjecture 1 Definiti o ns and re sults In this pap er a Fin sler metric on a smo oth manifold M is a f unction F : T M → R ≥ 0 satisfying the following prop erties: 1. It is smo oth on T M \ T M 0 , where T M 0 denotes the zero section of T M , 2. F o r ev ery x ∈ M , the restriction F | T x M is a norm on T x M , i.e., for ev ery ξ , η ∈ T x M and for ev ery nonnegativ e λ ∈ R ≥ 0 w e hav e (a) F ( λ · ξ ) = λ · F ( ξ ), (b) F ( ξ + η ) ≤ F ( ξ ) + F ( η ), (c) F ( ξ ) = 0 = ⇒ ξ = 0. W e do not require t ha t (the restriction of ) the function F is strictly conv ex . In this point our definition is more general tha n the usual definition. In addition w e do not assume the metric to b e r evers ible, i.e., we do not assume that F ( − ξ ) = F ( ξ ) . Geometrically sp eaking a Finsler metric is c haracterized by a smoo th fa mily x ∈ M 7→ { ξ ∈ T x M | F ( ξ ) = 1 } ⊂ T x M of con v ex h yp ersurfaces (sometimes called indic atric es , cf. [Br]) con taining the zero section in 1 the tangent bundle. Recen t refere nces for Finsler geometry include [BCS, Sh, BBI, Alv]. P articular classes of Finsler metrics whic h o ccur in our results are the following: Example 1 (Riemannian metric) . F or ev ery R iemannian metric g on M the function F ( x, ξ ) := p g ( x ) ( ξ , ξ ) is a Finsler metric. G eometrically t he Finsler metric is a s mo oth family of ellipsoids. Example 2 (Mink ows ki metric) . Consider a norm on R n , i.e., a function p : R n → R ≥ 0 satisfy- ing 2a, 2b, 2c. W e canonically ide ntify T R n with R n × R n with co ordinates (( x 1 , ..., x n ) | {z } x ∈ R n , ( ξ 1 , ..., ξ n ) | {z } ξ ∈ T x R n ). Then, F ( x, ξ ) := p ( ξ ) is a Finsler metric. The metric is translation in v a rian t, on the other hand ev ery translation in v ariant Finsler metric is a Mink o wski metric. Due to the translation in v ariance t he Finsler metric is uniquely characterized by a con v ex h yp ersurface in a single tangen t space T x M . Tw o Finsler metrics F and F 1 on an op en subset U ⊆ M are called c onformal ly e quivalent, if F 1 = λ · F f o r a no where v anishing function λ on U . W e sa y that a differen tiable mapping f : ( M 1 , F 1 ) → ( M 2 , F 2 ) is c onformal , if the pullbac k of the metric F 2 is conformally equiv alen t to F 1 , i.e., if for ev ery ξ ∈ T x M w e hav e F 2 ( d f x ( ξ )) = λ ( x ) F 1 ( ξ ) . If the conformal factor λ is constan t the map is called homothetic, for λ = 1 it is isometric. A v ector field is called c onfo rm al ( resp. ho m othetic or isometric ) if its lo cal flow a cts by c onformal (resp. ho mothetic or isometric) lo cal diffeomorphisms. If t he conformal v ector field v is c omplete then the flow φ t : M → M , t ∈ R of v is a one-parameter gr o up of conformal diffeomorphisms o f the manifold M . Ob viously , if a metric F 1 is conformally equiv alen t to F , then ev ery confo rmal v ector field for F is also a confo r mal ve ctor field for F 1 . Example 3. F or the Finsler metric F := p g ( ξ , ξ ) fr o m Example 1, conformal v ector fields for the Riemannian metric g are conformal v ector fields for the Finsler metric F , and vice versa. F o r Euclidean space R n the description of conformal mappings fo r n = 3 is due to Liouville [Lio] and for n ≥ 3 to Lie [Lie], for recen t exp ositions cf. for example [BP, Thm. A.3.7], [Ku] and [KR]. Example 4. F or the Mink ows ki Finsler metric F ( x, ξ ) := p ( ξ ) from Example 2, the mappings x ∈ R n 7→ H t ( x ) = t · x ∈ R n are ho mo t heties f o r all t > 0 . Then, the v ector field v ( x ) = d dt   t =0 H t ( x ) = P n i =1 x i ∂ ∂ x i is the corresp onding homot hetic ve ctor field. No w we can state our ma in result: Theorem 1. Supp ose v is a c onf ormal and c omplete ve ctor field on a c onne cte d Finsler man- ifold ( M , F ) of d i m ension n ≥ 2 . Then, a t le as t one of the fol lowing statement hold s. 2 1. Ther e exists a F insler metric F 1 c onfo rm al ly e quivale n t to F such that the flow of v pr eserve s the Finsler metric F 1 . 2. The manifold M is c onformal ly e quiva lent to the spher e S n with its standar d R iemanni a n metric. 3. The ma n ifold M is diffe omorphic to Euclide an sp ac e R n , and the F i n sler metric F is c onfo rm al ly e quivalent to a Minko w ski metric, cf. Example 2. The v e ctor fi e ld v with r esp e ct to the Minkowski metric is homothetic. F o r Riemannian metrics, the statemen t ab ov e is called the conformal Lichner owicz-Ob ata con- jecture, and w as pro v ed indep enden tly b y D.Alekseevk sii [Al], J.F errand [F e2], M.Obata [Ob] and R.Sc ho en [Sc h], see a lso [La, F T]. Of course, in the Riemannian case, Example 2 corre- sp onds to the Euclidean metric o n R n . A conformal ve ctor field satisfying the assumptions of the first case is also called inesse n tial, otherwise it is called essential. Remark 1. Theorem 1 also implies the follo wing result: If the conformal group is essen tial, i.e. if there is no conforma lly equiv alen t metric suc h that the conformal group becomes the isometry group, then the metric is conformally equ iv a len t to the round sphere, or to a Mink o wski space. Theorem 1 was announced in [Ze] under the following additional assumptions: M is closed, and the Finsler metric F is strictly c onvex , i.e., the second deriv at ive of F 2 | T p M has ra nk n − 1 at ev ery p oint on T p M − T M 0 . The pro of is sk etc hed in [Ze]. It is long and actually is a rep eating of the pro of f r o m [F e2] (whic h is technic ally v ery non trivial) in the Finsler case. Our pro of of Theorem 1 is m uc h shorter. It is based on the followin g observ at io n: for ev ery Finsler metric F w e can canonically construct a Riemannian metric g suc h that if v is a con- formal v ector field for F , then it a lso a conformal v ector field for g . Then, b y the Riemannian v ersion of Theorem 1, the follo wing tw o cases are p ossible: • The flo w of v acts b y isometries o f a certain Riemannian metric g 1 conformally equiv alent to the Riemannian metrics g . This case will b e called “tr ivial case” in the pro of of Theorem 1. In this case, it immediately follo ws, that the flo w of v acts b y the isometries of a particular metric F 1 conformally equiv a len t to F . • The manifold is S n or R n , and the metric g is conformally equiv alent to the standard metric. In this case, all p ossible essen tial conformal ve ctor fields v can b e explicitly described, cf. Example 3. A direct analysis of the flo w of suc h vector field sho ws, tha t the only Finsler metrics for which v is a conformal v ector field are as in Theorem 1. Remark 2. In conformal geometry the case of surfaces n = 2 is sp ecial due to the existence of holomorphic functions. An y holomorphic function defined on an op en subse t in the complex plane C with ev erywhere non-v anishing deriv ativ e is conformal. T his sho ws that the pa rt o f Liouville’s theorem on conformal transformations of Euclidean spaces stating that a conformal 3 diffeomorphism betw een op en su bsets of Euclidean space is the restriction of a conformal diffeomorphism of the standard sphere only holds for dimensions n ≥ 3 . O n the other hand the description of the conformal diffeomorphism of the n - dimensional sphere S n as comp o sitions of homotheties and in v ersions in the Euclidean space R n ∼ = S n − { p } a lso ho lds for n = 2 , as one concludes fr o m the standard classification of (an ti)holomor phic functions on C r esp. C P 1 . It is sho wn b y Alekseevs kii [Al, Thm.8] that an essen tial and complete conformal v ector field on a surface only exis ts on the 2-sphere with the standa r d metric or on Euclidean 2- space. Therefore f o r our main result the case n = 2 is not exceptional. 2 Av eraged Riemannian metric F o r a giv en smoot h norm p on R n w e construct canonically a p ositiv e definite sy mmetric bilinear fo r m g : R n × R n → R . F o r a Finsler metric F , the role of p will play the restriction of F to T x M . W e will see that the constructed g will smo othly dep end o n x , i.e., g ( x ) is a Riemannian metric. Consider t he unit sphere S 1 = { ξ ∈ R n | p ( ξ ) = 1 } of the no rm p. Consider the (unique) v olume form Ω o n R n suc h t ha t the v olume of the 1-ball B 1 = { ξ ∈ R n | p ( ξ ) ≤ 1 } equals 1. Denote by ω the volume form on S 1 , whose v alues on t he v ectors η 1 , ..., η n − 1 tangen t to S 1 at the p oint ξ ∈ S 1 are giv en b y ω ( η 1 , ..., η n − 1 ) := Ω( ξ , η 1 , η 2 , ..., η n − 1 ). No w, for eve ry p oin t ξ ∈ S 1 , consider the symmetric bilinear form b ( ξ ) : R n × R n → R , b ( ξ ) ( η , ν ) = D 2 ( ξ ) p 2 ( η , ν ). In this formu la, D 2 ( ξ ) p 2 is the second differential at the p oint ξ of the function p 2 on R n . The analytic expression for b ( ξ ) in the co ordinates ( ξ 1 , ..., ξ n ) is b ( ξ ) ( η , ν ) = X i,j ∂ 2 p 2 ( ξ ) ∂ ξ j ∂ ξ j η i ν j . (1) Since the norm p is conv ex, t he bilinear form (1) is nonnegative definite: for all η w e ha v e b ( ξ ) ( η , η ) ≥ 0 . (2) Clearly , for ev ery ξ ∈ S 1 , w e ha v e b ( ξ ) ( ξ , ξ ) > 0 (3) No w consider the follo wing bilinear symmetric 2 − form g on R n : for η , ν ∈ R n , w e put g ( η , ν ) = Z S 1 b ( ξ ) ( η , ν ) ω . W e a ssume that the orien tation of S 1 is c hosen in suc h a w ay that R S 1 ω ≥ 0. Because of (2) and (3), g is p ositiv e definite. 4 Remark 3. If the norm p comes from a scalar pro duct, i.e., if p ( ξ ) = p b 1 ( ξ , ξ ) for a certain p ositiv e definite symm etric 2-form b 1 , then b is equal to b 1 m ultiplied by a constan t only dep ending on the dimension. Starting with a Finsler metric F , we can use this construction for ev ery tangen t space T x M of the manifold, the ro le of p is play ed by the restriction F | T x M of the Finsler metric to the tangen t space T x M . Since this construction dep ends smo o thly on the p oin t x ∈ M , w e obtain a Riemannian metric g = g ( F ) o n M . W e call this metric the aver age d Riemannian metric of the Finsler metric F . Remark 4. It is easy to ch ec k that for the metric F 1 := λ ( x ) · F the constructed metric g 1 is conformally equiv alen t to the metric g constructed for F . More precisely , g 1 = λ ( x ) 2 · g . Then, a conformal diffeomorphism (conformal vec tor field, resp ectiv ely) for F is also a conformal diffeomorphism (conformal ve ctor field, resp ectiv ely) for g . Mor eov er, if v is conformal for F a nd is a n isometry (homothety , respectiv ely) for g , t hen it is an isometry (homothet y , resp ectiv ely) for F a s w ell. Remark 5. This a v eraging construction is quite natural and it is v ery po ssible that other researc hers in Finsler geometry a lr eady thought ab o ut it, but we could not find an y reference ab out it in the literature, nor an y significan t result in F insler geometry whose pro of is based on the av eraged metric. It w ould certainly b e w orth wile t o further inv estigate its prop erties. Recen tly , Szab o [Sz] uses a similar av eraging construction to explicitely construct all Finlser Berw ald metrics. There are other canonical constructions of a bilinear form starting from a norm. R. Sc hneider told us, that a for a con v ex geometer the natural bilinear form corresp ond- ing to a conv ex b o dy is one corresp onding to the John ellipsoid of this conv ex b o dy . These constructions ha v e the nice prop erties listed in Remarks 3, 4. W e still prefer our a v er- aged Riemannian metric, since the method of Szabo assumes that t he norm is strictly conv ex, and since it is no t clear whether the John ellipsoid dep ends smo othly on the no r m. In [T o] T orrome suggests another av eraging construction for Finsler metrics. 3 Pro of of The o rem 1 Let v b e a complete conformal v ector field o n a connec ted F insler ma nif o ld ( M , F ). Then, it is also a conformal vector field for the a v eraged Riemannian metric g . Then, by the R iemannian v ersion of our Theorem, whic h, a s w e explained in the in tro duction, w as prov ed in [Ob, Al, Y o, F e2, Sc h], w e hav e the follow ing p ossibilities: ( T rivial case ) v is a Killing v ector field o f a conformally equiv a lent metric λ ( x ) 2 g . ( In teresting case ) F or a certain function λ , the Riemannian manifold ( M n , λ ( x ) 2 g ) is ( R n , g 0 ), or ( S n , g 1 ), where g 0 resp. g 1 is the Euclidean metric o n R n resp. the standard metric of sectional curv ature 1 on S n . 5 In the trivial case , as we explained in Remark 4, for a certain f unction λ , v is a Killing v ector field for the metric F 1 := λ · F , whic h w as one of the p ossibilities in Theorem 1 . No w w e treat the interesting case . Without loss of generality , we can assume that ( M , g ) is ( R n , g 0 ), or ( S n , g 1 ) . 3.1 Case 1: ( M , g ) = ( R n , g 0 ) . Since the vector field is complete, it generates a one parameter group φ t : R n → R n of conformal transformations with resp ect to the Finsler metric F and the av erag ed Riemannian metric g 0 . It follo ws fr o m Liouville’s theorem that f or any t the mapping φ t is a homothet y of the Riemannian metric g 0 . In other words , in an appro priate cartesian co ordinate system ( x 1 , ..., x n ), the conformal diffeomor phism φ = φ 1 has the form φ ( x 1 , ..., x n ) = µ · ( x 1 , ..., x n ) A, (4) where A is a n orthog o nal ( n × n )-matrix. Without loss of generalit y w e can assume that 0 < µ < 1. W e will sho w that in this case the metric F is as in Example 2. W e iden tify T x R n and R n × R n with the help of the cart esian co ordinates x = ( x 1 , ..., x n ). W e assume that the first comp onen t of the pro duct R n × R n corresp onds to our manifold R n , and tha t the second comp onen t of the pro duct R n × R n corresp onds to the tangen t spaces. The co ordinates on the tangent spaces will b e denoted b y ξ , so (( x 1 , ..., x n ) | {z } x ∈ R n , ( ξ 1 , ..., ξ n ) | {z } ξ ∈ T x R n ) is a co ordinate system on T x R n ∼ = R n × R n . Clearly , the differen tial of the mapping φ giv en b y (4) is give n b y dφ x ( ξ ) = ( µ · ( x 1 , ..., x n ) A, µ · ( ξ 1 , ..., ξ n ) A ) . Then, for every ξ , η ∈ T x R n , w e hav e g φ ( x ) ( dφ x ( ξ ) , dφ x ( η )) = µ 2 · g ( x ) ( ξ , η ). Hence , by Remark 4, F ( φ ( x ) , dφ x ( ξ )) = µ · F ( x, ξ ) . Consider the mapping h : T R n ∼ = R n × R n → R n × R n , h ( x 1 , ...x n , ξ 1 , ..., ξ n ) = ( µ · ( x 1 , ..., x n ) A, ( ξ 1 , ..., ξ n ) A ) . By construction, this mapping satisfies F ( h ( x, ξ )) = F ( x, ξ ) . Since the ort ho gonal group O ( n ) is compact, w e can c ho ose a sequence m j → ∞ suc h that A m j → 1 ∈ O ( n ) for j → ∞ . Then, (0 , ξ ) = lim j → ∞ h m j ( x, ξ ). Hence, F ( 0 , ξ ) = F  lim j → ∞ h m j ( x, ξ )  = lim j → ∞ F ( h m j ( x, ξ )) = F ( x, ξ ) . Th us, F is tra nslation in v ariant and therefore a Mink ows ki metric, cf. Example 2. Hence in this case, up to conformal equiv alence, the Finsler metric is a Mink o wski metric, and the conformal v ector field is ho mot hetic. 6 3.2 Case 2: ( M , g ) = ( S n , g 1 ) Then, b y [La, Thm. 1 2 ] any essen tial conformal v ector field v v anishes at exactly one (Case 2a) or exactly t w o (Case 2b) p oin ts. W e denote b y v − 1 (0) = { x ∈ M | v ( x ) = 0 } the set of zero es. If w e assume v ( x ) = 0 w e use the stereographic pro jection s x : S n − { x } → R n and obtain with the push forw ard of the v ector field v a complete and conformal v ector field on R n . 3.2.1 Case 2a: Supp ose v − 1 (0) = { x, y } , x 6 = y Supp ose the conformal v ector field v v anishes precisely at t w o p oints x and y o f t he sphere. W e will show that the Finsler metric F is in fa ct Riemannian. Denote b y s + : ( S n − { x } , g 1 ) → ( R n , g 0 ) the stereographic pro jection from x whic h is confor- mal with respect to the standard Riemannian metrics g 0 , g 1 with conformal factor σ + . Here, R n should b e iden tified with t he h yp erplane through the origin parallel to the tangent spaces T x S n . Then w e define a Finsler metric F + b y s ∗ + F + = σ + F . Then the av eraged Riemannian metric of F + coincides with the Euclidean metric g 0 . The push fow a rd v ector field v + := s ∗ + v is a conformal and complete v ector on R n with resp ect to the Finsler metric F + as w ell with resp ect to the standard metric g 1 . This ve ctor field has exactly one zero on R n . Therefore, b y section 3 .1, the Finsler metric F + is a Mink o wski metric, i.e., translation in v ariant. In par- ticular w e can assume without loss of g enerality that the zero p oin t of v + is the origin of R n . Hence we can assume that the zero p oints of v on S n are antipo da l p o in ts, i.e., v − 1 (0) = {± x } . The stereographic pro jection s − : ( S n − {− x } , g 1 ) → ( R n , g 0 ); s − ( q ) = s + ( − q ) fr o m − x is a conformal mapping with conformal factor σ − with σ + ( − q ) = σ − ( q ) , q ∈ S n i.e., s ∗ ± g 0 = σ 2 ± g 1 . Then w e define also the F insler metric F − on R n b y s ∗ − F − = σ − F . The av eraged Riemannian metrics of F − equals the Euclidean metric g 0 . The push-forw ard v − := ( s − ) ∗ v of the v ector field v is a conformal v ector field on R n with resp ect to the Finsler metric F − and, hence, with resp ect to the standard metric g 0 . Both v ector fields v ± are eviden tly complete and ha v e precisely one z ero at the orig in. Therefore, b y section 3.1, the Finsler metrics F ± are Mink owsk i metrics, i.e., translation in v ariant. It is w ell kno wn that the comp osition s − ◦ s − 1 + : R n − { 0 } → R n equals the inv ersion I ( q ) = q /g 0 ( q , q ) at the unit sphere. Therefore, the in v ersion defines a conforma l transformation I : ( R n − { 0 } , F + ) → ( R n − { 0 } , F − ) b et w een the t w o Mink o wski metrics. The differen tial dI q of the in v ersion at a p oin t q ∈ S n − 1 := { u ∈ R n | g 0 ( u, u ) = 1 } equals t he reflection R q at the h yp erplane normal to q . This implies t ha t d I ∗ q F + = R ∗ q F + = F − for an y q ∈ S n − 1 . Since the reflections generate the o r thogonal group and since the Finsler metrics F ± are translation in v ariant, it f ollo ws that the norms F ± | T 0 M at the origin are inv arian t under the full orthogonal group and hence Euclidean. 7 3.2.2 Case 2b: v − 1 (0) = { x } W e assume that the vec tor field v on S n v anishes precisely a t one p oin t x ∈ S n . W e will again sho w that the metric F is Riemannian. W e again consider the stereographic pro jections s ± : S n − {± x } → R n from the p oin ts x, − x as in tro duced in Section 3 .2 .1, and denote b y F ± := ( s ± ) ∗ F the induced Finsler metrics on R n . The push-forw ard v + of v with resp ect to s + v anishes nowhe re on R n and is complete, let ψ t b e its flo w on R n . Then Liouville’s theorem implies that for an a r bit r a ry t the confor mal diffeomorphism f = ψ t has the form f ( x ) = µAx + b with an orthogonal matrix A and µ > 0 , b ∈ R n . Since the mapping f has no fixed p oint it follo ws that b 6 = 0 ; µ = 1 and Ab = b. W e intro duce the following notation: f A,b ( q ) = Aq + b for an orthogonal matrix A and b ∈ R n with Ab = b. If w e use the stereographic pro jection s − , then the push-forw ard of v has a zero in the origin 0 and the mapping f transforms to f A,b = I ◦ f A,b ◦ I where I = σ − ◦ σ − 1 + is the in v ersion a t the unit sphere. Hence f A,b ( q ) = Aq + b k q k 2 1+2 h Aq ,b i + k b k 2 k q k 2 where < ., . > = g 0 ( ., . ) with related norm k . k . The conformal factor is given b y ψ ( q ) = 1 1+2 h Aq ,b i + k b k 2 k q k 2 . In particular the confor mal mapping f A,b induces at the fixed p oin t 0 the map ξ ∈ T 0 R n 7→ d  f A,b  0 ( ξ ) = Aξ ∈ T 0 R n (5) whic h is an isometry also with resp ect t o the restriction of the Finsler metric F − to 0 since ψ (0) = 1 . F o r a n orthogonal mapping A w e in tro duce the map h A : z ∈ R n → Az ∈ R n with induced mapping ( z , ξ ) ∈ T z R n = R n × R n 7→ dh A ( z , ξ ) = ( Az , Aξ ) ∈ T z R n . W e w an t to show that the map h A is an isometry for the Finsler metric F − . Let v 1 b e t he v ector field on S n whic h corresp onds to the parallel v ector field b on R n with resp ect to the stereographic pro jection s + , i.e., ds + ( v 1 )( q ) = b for all q ∈ S n . The v ector field v 1 is a conformal vec tor field with resp ect to the standard Riemannian metric g 1 with exactly one zero in x. The flo w lines of v 1 consist of the circles passing through x with a common tangent v ector, see the pictures. Hence we obtain the following prop erties of the flow φ t : S n → S n of the conformal v ector field v 1 on S n : Remark 6. The flow φ t of the c onformal ve ctor field v 1 define d ab ove satisfies the fol lowing pr op erties: (a) F or an y p oi n t q ∈ S n : x = lim t →±∞ φ t ( q ) . (b) F or any tangent ve ctor ξ ∈ T x S n , F ( x, ξ ) = 1 ther e is a se q uen c e q i ∈ S n − { x } wi th lim i →∞ q i = x and ξ = lim i →∞ v 1 ( q i ) F ( q i ,v ( q i )) 8 Figure 1 : The vec tor field v 1 / | v 1 | in dimension 2 Figure 2 : The integral curv es of v 1 in dimension 2 9 No w w e sho w that the mapping h A is an isometry a lso for the Finsler metric F − : W e can c ho ose a sequence m i → ∞ with A m i → 1 . In pa rticular for a give n ( z , ξ ) ∈ T z R n ; z 6 = 0 there is a unique (0 , ξ 0 ) ∈ T 0 R n ; F ((0 , ξ 1 )) = 1 suc h that (0 , ξ 1 ) = lim i →∞ d f m i A,b ( z , ξ ) F −  d f m i A,b ( z , ξ )  . (6) Since the mapping f A,b is conformal for F − and since h A and f A,b comm ute it follow s that F − ( dh A ( z , ξ )) F − (( z , ξ )) = lim i →∞ F −  d f m i A,b dh A ( z , ξ )  F −  d f m i A,b ( z , ξ )  = lim i →∞ F −  dh A  d f m i A,b ( z , ξ )  F −  d f m i A,b ( z , ξ )  = F − ( d h A (0 , ξ 0 )) F − ((0 , ξ 0 )) = F −  d  f A,b  0 (0 , ξ 0 )  F − ((0 , ξ 0 )) = 1 as shown ab ov e, cf. Equation 5. Therefore the mapping h A is an isometry of the Finsler metric F − . This implies tha t also the flo w generated b y f 1 ,b = f A,b ◦ h − 1 A is conformal for the Finsler metric F − . Therefore the v ector field v 1 on S n is also a conformal ve ctor field fo r the Finsler metric F on S n . Let us no w consider the follo wing functions m, M : S n → R ≥ 0 : m ( q ) := F 2 ( q , v 1 ( q )) g ( q ) ( v 1 ( q ) , v 1 ( q )) , M ( q ) := max η ∈ T q S n , η 6 =0 F 2 ( q , η ) g ( q ) ( η , η ) − min η ∈ T q S n , η 6 =0 F 2 ( q , η ) g ( q ) ( η , η ) . Both functions are con tin uous functions inv arian t with resp ect to the flo w φ t of v 1 . It fol- lo ws fro m Remark 6(a) t hat the function m is a constan t, i.e., t here exists µ > 0 such that F 2 ( q , v 1 ( q )) = µ g ( q ) ( v 1 ( q ) , v 1 ( q )) . Part (b) of Remark 6 implies that for ev ery 0 6 = η ∈ T x S n w e ha v e F 2 ( x,η ) g ( x ) ( η,η ) = µ. Hence, M ( x ) = max η ∈ T q S n , η 6 =0 F 2 ( q , η ) g ( q ) ( η , η ) − min η ∈ T q S n , η 6 =0 F 2 ( q , η ) g ( q ) ( η , η ) = µ − µ = 0 . But since M is also flow inv arian t b y Remark 6(a) , we ha v e M ( q ) = 0 fo r a ll q ∈ S n , i.e., F is up to a constan t the norm o f the standard metric g . Theorem 1 is prov ed. As a conseq uence of the Pro of of Theorem 1 the in v ersion of the a v eraged Riemannian metric is not a confo r ma l map fo r a non- Euclidean Mink ow ski metric, cf. Section 3.2.1. T herefore one o btains fro m Liouville’s theorem on the conformal transformations of an Euclid ean v ector space the follow ing description of the conformal transformatio ns of a Mink o wski space: 10 Remark 7. Let V b e an n - dimensional v ector space with a Mink o wski norm F whic h is not Euclidean. Denote b y g the corresp onding av eraged Euclidean metric. If f : ( U, F ) → ( V , F ) is a conformal mapping from a n op en subset U and n ≥ 3 then f is a a similarit y with resp ect to the Mink o wski metric F and with respect to the Euclidean metric g . Hence it is of the form x ∈ V 7→ µAx + b ∈ V for some µ > 0 ; b ∈ V and an orthog onal mapping A of ( V , g ) . 4 Conclus ion Theorem 1 describ es complete conformal vec tor fields of F insler metrics; it app ears that no new phenomena (with resp ect to the Riemannian case) app ear. Our pro of is based on the construction of a v eraged metric in Section 2 , and on the description of conformal v ector fields for Riemannian metrics due to [Lie, Lio, Ob, Al, Y o, La, FT, F e2, Sc h]. Let us also note tha t the existence of a conformal v ector field suc h that, for a certain point p 0 , the closure of ev ery tra jectory con tains this p oint is not artificial: as w e kno w now, in view of Theorem 1, it is alw a ys the case, if the conformal transformations are essen tial. F or a closed manifold, one also can sho w it directly b y rep eating the Riemannian pro of of [Al]. As an in teresting and mu c h more inv olved problem in Finsler geometry related to transforma- tion groups we w ould lik e to suggest to generalize t he pro jectiv e Lic hnero wicz-Obata conjecture for Finsler metrics, see [Ma1, Ma2] for the pro of of the Riemannian vers ion, see also [Sh ]. A cknow le dgement: This w ork b egan when the authors met at the 80 ` eme r enc ontr e entr e physiciens th ´ eoriciens et math ´ emathiciens: G´ eom´ etrie de F insler at Stra sb o urg Univ ersit y in Septem b er 2007. W e are gra t eful in part icular to Athanase P apadop oulos for the inv itation to this conference. W e also t ha nk M. East w o o d, D . Hug, Z. Shen and R. Schne ider for useful dis- cussions. W e are also gr a teful to the referee for his suggestions. The first t w o authors thank Deutsc he F orsc h ungsgemeinsc haft (Priority Program 1154 — G lobal Differen tial G eometry) for partia l financial supp ort. References [AZ] H. 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Mat ve ev, Mathematisc hes Institut, F riedrich-Sc hiller Univ ersit¨ at Jena 07737 Jena, Germany , matve ev@minet.u ni-jena.de Hans-Bert Rademac h er , Mathematisc h es Institut, Univ ersit¨ at Leipzig, 04081 Leipzig, German y rademach er@math.un i-leipzig.de Marc T roy anov, Section de Math ´ ematiques, ´ Ecole P olytec h nique F ´ ederale de Lausanne 1015 Lausanne, S witzerland, marc.troyan ov@epfl.ch Ab delghani Zeghib, UMP A, ENS-Lyo n, 46, all ´ ee d ’Italie, 693 64 Ly on Cedex 07, F rance zeghib@u mpa.ens-ly on.fr 13