Convex Neighbourhoods and Complete Finsler Spaces
aa r X i v : . [ m a t h . DG ] J un Convex Neighbourhoods and CompleteFinsler Spaces ∗ O. M. Amici and B. C. CasciaroDipartimento di MatematicaUniversit`a di BariCampus UniversitarioVia Orabona 4, 70125 Bari, [email protected]@dm.uniba.it
November 2, 2018
Abstract
In this paper, it is shown that a large set of connections on asuitable sub–bundle of the tangent bundle of a Finsler Manifold can beused to study all the properties of convex neighbourhoods with respectto the Finsler Metric, which are needed to see that any CompleteFinsler Space is Geodesically Connected. ∗ Key words and phrases:
Finsler spaces, geodesic connectivity
Research supported by the Italian MURST 60% and GNSAGA Introduction
Let M be a C ∞ –differentiable n –dimensional manifold endowed with a Fins-lerian metric function F : T M → R , being T M the tangent bundle of M .The following properties of the convex neighbourhoods of M with respectto F are well known (see, [1], [2], see also [3]).In order to quote them, let us denote by B ρ (0 x ) = { X ∈ T x ( M ) /F ( x, X ) < ρ } the open indicatrix having ρ as its radius and the zero vector 0 x ∈ T x ( M ) asits center, for any ρ > x ∈ M .Moreover, we denote by Exp x the exponential mapping of the Finslermetric and Exp x is defined on an open neighbourhood of 0 x ∈ T x M into T x ( M ), with x ∈ M . We recall that the mapping Exp x is defined by set-ting Exp x ( X ) = c X (1), being c X the geodesic of F defined by the initialconditions c X (0) = x and ˙ c X (0) = X , with X belonging to a suitable openneighbourhood of 0 x in T x M .With these notations, we have: Proposition 1.1.
For each x ∈ M , there exist two positive real numbers ε = ε ( x ) and η = η ( x ) with < ε < η such that:i). Exp x : B ε (0 x ) → Exp x ( B ε (0 x )) = B ε ( x ) is a diffeomorphism definedon B ε (0 x ) , whose degree of differentiability is C ∞ on B ε (0 x ) − { x } and C on B ε (0 x ) . Moreover, B ε ( x ) is an open neighbourhood of x .ii). For each y, z ∈ B ε ( x ) , there exists a unique geodesic c : [0 , → M lying entirely in B η ( y ) = Exp y ( B η (0 y )) having length lesser than η and suchthat c (0) = y and c (1) = z .iii). For y ∈ B ε ( x ) , the mapping Exp y : B η (0 y ) → B η ( y ) is a dif-feomorphism defined on B η (0 y ) , whose degree of differentiability is C ∞ on B η (0 y ) − { y } and C on B η (0 y ) and B η ( y ) is an open neighbourhood of y containing B ε ( x ) . As in the Riemannian case (see, e.g., [4]), we consider the canonical identi-fication T X T x M = T x M , for any X ∈ T x M and x ∈ M and we fix an element x of M . Then, T ′ x M = T x M − { x } is endowed by the Riemannian Metric g p ( X, Y ) = g ( x,p ) ( X, Y ), for each p ∈ T ′ x M and X, Y ∈ T p T x M = T x M , being g the metric tensor which is induced by F . Moreover, let e T M ⊆ T M be theopen neighbourhood of the zero section where the exponential map of theFinsler Metric is defined. Then, we have:2 roposition 1.2.
Let x ∈ M and X ∈ e U x = e T M ∩ T x M . Moreover, let e b : [0 , s ] → e U x be any differentiable curve such that e b (0) = 0 x and e b ( s ) = X .Finally let us put b ( s ) = Exp x e b ( s ) , for any s ∈ [0 , s ] and c ( t ) = Exp x ( tX ) ,for any t ∈ [0 , . Then:i). L ( c ) ≤ L ( b ) , where L denotes the Finslerian length of curves.ii). If e b ( s ) = τ ( s ) X , for any s ∈ [0 , s ] , being τ : [0 , s ] → [0 , a strictlyincreasing differentiable mapping, then L ( c ) = L ( b ) .iii). If L ( c ) = L ( b ) and the total differential ( DExp x ) t e b ( s ) has maximalrank, for any s ∈ [0 , s ] and t ∈ [0 , , then there exists a differentiable map τ : [0 , s ] → [0 , such that e b ( s ) = τ ( s ) X , for each s ∈ [0 , s ] and τ is strictlyincreasing. Proposition 1.3.
Let e ε = ε/ , and ε as in Proposition 1.1. Then, every y, z ∈ B e ε ( x ) can be joined by a geodesic c , lying entirely in B e ε ( x ) and havingFinslerian length equal to the Finslerian distance between y and z . Moreover,any further geodesic joining y and z (if it exists) has points outside of B η ( y ) . Finally, we recall the following definition: we say that F is a CompleteFinsler Metric , if and only if the distance which is induced by F on M iscomplete. Then it results: Proposition 1.4.
Let F be a Finslerian metric on M . Then the followingproperties are equivalent:i). F is a complete Finslerian metric.ii). There exists a point p ∈ M such that any geodesic starting from p can be extended to the whole R .iii). Assertion ii) holds, for any p ∈ M .Moreover, it results:iv). If Assertion i) holds, any two points of M can be joined by a geodesicof F . In 1967, B. T. Hassan (see [3]) proved Proposition 1.1 by using the
Car-tan Connection relative to F . Proofs of more or less complete versions ofProposition 1.1 are given by several authors by means of connections differ-ent from the Cartan one. In 1993, D. Bao and S. S. Chern proved the sameproposition by using a new connection, called the Chern Connection (see [5]),which coincides with the
Rund Connection (see [6]).Finally, in [1] the previous proposition was proved by means of the
CartanForm . The proofs of all the previous propositions can be found in this book.3enerally, all these proofs are different from the corresponding proofs ofthe Riemannian case, more difficult than them and this fact explains thenumber of the proofs of Proposition 1.1.Some questions arise for the previous observations. For example: Whythere exist so much proofs for Proposition 1.1? Why they are so differentfrom the proof needed in the Riemannian case? Does the used connectionplay any role?In [8] and [7], by using the bundle T ′ M = T M − σ ( M ), where σ : M → T M is the zero section, we determined all the connections by means ofwhich the Euler–Lagrange assumes the simplest form. These connectionswere called
Finslerian Connections and they are a large class of connections.In this paper we prove:
Proposition 1.5.
Any Finslerian Connection can be used to prove all theprevious propositions.
We also show that all the proofs of the previous propositions can followin the closest way the corresponding proofs used in the Riemannian case andthis fact can be useful for the proof of further results. We choose to follow[4]. In our opinion, the problems previously listed raise because classical meth-ods used in this context mix up the Finsler Geometry with the geometry ofthe tangent bundle of the manifold under consideration.By using the previous considerations and the methods introduced here,in [9] we determine all the properties corresponding to the ones consideredhere for a large class of Lagrangian Functions and in [10] we show that theresults of [9] give a new sufficient condition for the geodesic connectedness ofthe generalized Bolza problem.
Let M be a C ∞ –differentiable n –dimensional manifold.The notations, which are more frequently used in the following, are: α ). T M = ∪ x ∈ M T x M is the tangent bundle of M and π : T M → M its natural projection. Moreover, for each x ∈ M , either X x or equivalently( x, X ) will denote the same element of T x M according to this element isconsidered as a tangent vector at x ∈ M or as a point of T M ; hence X x =( x, X ). 4 ). I ( M ) = L ( r,s ) ∈ N I rs ( M ) is the algebra of tensor fields on M , where I ( M ) = X ( M ) and I ( M ) = F ( M ) are the Lie algebra of vector fields andthe ring of C ∞ –differentiable real valued functions, respectively. γ ). Let σ : M → T M be the zero section and we set T ′ M = T M − σ ( M ). T ′ M is the open sub–bundle of T M of non–zero tangent vectors of M andwe shall denote by π ′ : T ′ M → M its canonical projection. δ ). I π ′ = L I rsπ ′ is the F ( T ′ M )–module ( F ( M )–module) of differentiabletensor fields along π ′ , with I π ′ = X π ′ and I π ′ = F ( T ′ M ). θ ). If ( U, ϕ ) is a local chart of M , we denote by ( T U, T ϕ ) the local chartcanonically induced on
T M , with
T U = π − U . We set ϕ = ( x , . . . , x n ), T ϕ = ( x , . . . , x n , X , . . . , X n ), e i = ∂∂x i , ε ˆ i = ∂∂X ˆ i , e j = dx j , ε ˆ j = dX ˆ j , forany i, j, ˆ i, ˆ j ∈ { , . . . , n } .We recall that:A non–linear connection ∇ ( with three indices ) can be regarded as an R –bilinear mapping ∇ : X π ′ × I ( M ) → I π ′ such that:i). ∇ fX ( kY ) = f X ( k ) Y + f k ∇ X Y, ∀ X ∈ X π ′ , ∀ Y ∈ X ( M ) , ∀ f, k ∈F ( M ) ;ii). ∇ commutes with all the contractions;iii). for each X ∈ X π ′ , with X positively homogeneous of degree ρ , ∇ X Y is positively homogeneous of degree ρ , for any Y ∈ X ( M ).Moreover, ∇ defines a horizontal lift h : X π ′ → X ( T ′ M ) which can beextended in a trivial way to the whole tensor algebra I π ′ . This extensionis called Matsumoto lift (see, e.g., [3]). We also recall that the identity id : T ′ M → T ′ M can be considered as a vector field along π ′ ( id is theso–called fundamental vector field along π ′ ). The vector field id h will be saidto be associated to ∇ and one of its properties is: Proposition 2.1. id h can be extended to a spray defined on the whole M ,which will be denoted again with id h . This extension is C –differentiable on T M and C ∞ –differentiable on T ′ M . Moreover, a curve γ : [0 , → M is anon constant geodesic of id h if and only if γ is a path of ∇ . The spray id h defines an exponential map Exp : e T M → M , being e T M an open neighbourhood of σ ( M ). We have Exp ( x, X ) = c ( x,X ) (1), being c ( x,X ) : [0 , ε ] → M , with ε >
1, the geodesic of id h having ( x, X ) as initialcondition, for all ( x, X ) ∈ e T M . The exponential map verifies the followingproperties: 51).
Exp is C –differentiable on e T M and C ∞ –differentiable on e T M − σ ( M ) (see [11], pg. 72).(2). Exp has maximal rank in the zero vector 0 x ∈ T x M , for any x ∈ M (see [12], 2.8, Satz. (a), pg. 61).(3). The map ( π, Exp ) : e T M → M × M has maximal rank in 0 x ∈ T x M ,for any x ∈ M (see [12], 2.8, Satz. (c), pg. 61).(4). The total differential of Exp , when it is restricted to e T M ∩ T x M andit is calculated in 0 x , coincides with the identity map, for any x ∈ M (see[11], Th. 8, pg. 72).Moreover, as in the Riemannian case (see, e.g., [4]) from the previousproperties, it follows: Proposition 2.2.
There exists an open neighbourhood f W of σ ( M ) , such that Exp ( f W ) = W is open and:i). For each y, z ∈ W there exists a unique geodesic c : [0 , → M suchthat c (0) = y, c (1) = z , ˙ c (0) ∈ f W and c ( t ) ∈ W , for any t ∈ [0 , .ii). For each y ∈ W , the map Exp y : f W ∩ T y M → Exp ( f W ∩ T y M ) = W ( y ) is a diffeomorphism of class C , of class C ∞ on ( f W ∩ T y M ) − { y } and if y ∈ W ( x ) , with x ∈ M , then W ( x ) ⊆ W ( y ) .iii). The mapping ( π, Exp ) | f W : f W → W × W is a diffeomorphism of class C and of class C ∞ on f W − σ ( M ) . Taking into account the previous proposition, the set W (0 x ) = f W ∩ T x M is an open neighbourhood of 0 x in T x M and W ( x ) = Exp x ( W (0 x )) is anopen neighbourhood of x , for any x ∈ M .Now, let F : T M → R be a Finsler metric and g its metric tensor. Then g is a family of Riemannian metrics on M depending on ( x, X ) ∈ T ′ M andsuch that g ( x,X ) ( X x , X x ) = F ( x, X ), for any x ∈ M and ( x, X ) = X x ∈ T M .Locally, we set g = g ij e i ⊗ e j , with g ij = 12 ∂ F ∂X i ∂X j . Let V = V ( T M ) be the vertical sub bundle of
T T M ; i.e., V is the subset of T T M containing all the vectors tangent to the fibres of the tangent bundle of M . We fix a further sub bundle H of T T M such that T ( x,X ) T M = H ( x,X ) ⊕ V ( x,X ) , for any ( x, X ) ∈ T M . Two canonical projections P : T T M → H and Q : T T M → V are associated to the previous splitting into direct sum of6 T M . These projections are determined by two tensor fields of type (1,1)denoted, by an abuse of notation, again by P and Q . Locally, P and Q aredefined by: P = δ ij e i ⊗ e j − P ˆ ij ε ˆ i ⊗ e j and Q = δ ˆ i ˆ j ε ˆ i ⊗ ε ˆ j + P ˆ ij ε ˆ i ⊗ e j . We recall that the distribution H is said to be a non linear–connection withtwo indices if the functions P ˆ ij are positively homogeneous of degree one (see,e.g.,[2]). Moreover, any connection on M determines a couple of tensor fieldsof the previous kind.Now, we turn to the general case and we set G = g v = g ij e i ⊗ e j , (2.1)being v the extension to the tensor fields along π ′ of the usual vertical lift oftensors (see [13]).Then, from [7], it follows: Proposition 2.3.
There exists a connection ∇ on T ′ M such thati). C ( P ⊗ T ) = 0 , where T is the torsion tensor field of ∇ and C is thecontraction of the first contra–variant index with the second covariant indexof P ⊗ T .ii). Let us denote by π ∗ : T T M → T M the total differential of thecanonical projection π : T M → M (see [14]). Then, for each vector field X ∈ X ( T ′ ( M )) , having π ∗ ( X ( x,Z ) ) = Z x , for any Z x = ( x, Z ) ∈ T ′ ( M ) itresults: ( ∇ X G )( X, Y ) = ( ∇ Y G )( X, X ) = 0 , ∀ Y ∈ X ( T ′ M ) . iii). The mapping ∇ ′ : X π ′ × I ( M ) → I π ′ defined by ∇ ′ X Y = π ∗ ( ∇ X c Y c ) , ∀ X, Y ∈ X ( M ) being c the complete lift (see [13]), is a non–linear connection with threeindices. A connection ∇ verifying i), ii) and iii) of the previous proposition is called Finslerian Connection . Moreover, the non–linear connection ∇ ′ , defined byiii) of the same proposition, is called Berwald Connection deduced from ∇ .From [7], we also get: 7 roposition 2.4. Let us denote ∇ by a Finslerian Connection and let ∇ ′ be the Berwald Connection deduced from ∇ . Let Y = ( γ, ˙ γ ) : [ a, b ] → T ′ M be a curve. Then, γ is a path of ∇ ′ if and only if there exists a vector field Z ∈ X ( T ′ M ) such that: ( ∇ ˙ Y ˙ Y )( t ) = Q ( γ ( t ) , ˙ γ ( t )) ( Z ( γ ( t ) , ˙ γ ( t )) ) , ∀ t ∈ [ a, b ] . In [7], it was also proved that the set of all the Finslerian Connectionscan be obtained in the following way.Let e ∇ be any connection on T ′ M and ( U, ϕ ) be a chart of M . Withrespect to the chart ( T ′ U, T ′ ϕ ), induced by ( U, ϕ ) on T ′ M , we set: ∇ e j e k = e Γ ijk e i + e Γ ijk ε ˆ i , ∇ ε ˆ j e k = e Γ i ˆ jk e i + e Γ i ˆ jk ε ˆ i ∇ e j ε ˆ k = e Γ ij ˆ k e i + e Γ ij ˆ k ε ˆ i , ∇ ε ˆ j ε ˆ k = e Γ i ˆ j ˆ k e i + e Γ i ˆ j ˆ k ε ˆ i . The 8 n functions ( e Γ ijk , e Γ i ˆ jk , e Γ ij ˆ k , e Γ i ˆ j ˆ k , e Γ ijk , e Γ i ˆ jk , e Γ ij ˆ k , e Γ i ˆ j ˆ k ) are the local compo-nents of e ∇ . By using the local components of G and P and the previous 8 n functions, we obtain the following new 8 n functions defined on T ′ U :2Γ ijk = g im ( ∂ j g mk + ∂ k g mj − ∂ m g jk + P ˆ hm ∂ ˆ h g jk − P ˆ hj ∂ ˆ h g mk − P ˆ hk ∂ ˆ h g mj ) , Γ i ˆ jk = 0 , Γ ij ˆ k = 0 , Γ i ˆ j ˆ k = 0 , Γ ijk = P ˆ it e Γ tjk − P ˆ it Γ tjk + e Γ ijk , Γ i ˆ jk = P ˆ it e Γ t ˆ jk + e Γ i ˆ jk , Γ ij ˆ k = P ˆ it e Γ tj ˆ k + e Γ ij ˆ k , Γ i ˆ j ˆ k = P ˆ it e Γ t ˆ j ˆ k + e Γ i ˆ j ˆ k , where we used the notations ∂ i f = ∂f∂x i and ∂ ˆ i f = ∂f∂X ˆ i , for any f ∈ F ( T ′ M )and any i, ˆ i ∈ { , . . . , n } .The above functions are the local components of a connection ∇ of T ′ M .Since ∇ verifies i), ii) and iii) of Proposition 2.3, it is a Finslerian Connec-tion. The connection ∇ can be used in order to determine all the FinslerianConnections. In fact, let ∇ be a further connection and let us denote by N ∈ I ( T ′ M ) the tensor field defined by setting: N ( X, Y ) = ∇ X Y − ∇ X Y , ∀ X, Y ∈ X ( T ′ M ) . There exists a unique tensor field, e N , of type (1 ,
2) along π ′ , such that: e N ( X, Y ) = π ∗ N ( X c , Y c ) , ∀ X, Y ∈ X ( M )8 N is called the tensor field associated to N . Now, we denote by d ′ the socalled ”derivation along the fibres”, which is defined by setting d ′ f = ( ∂ ˆ i f ) e ˆ i ,for any f ∈ F ( T ′ M ) and is extended to the whole tensor algebra I π ′ in thewell–known way. Then, we have: Proposition 2.5.
Under the previous assumptions, the connection ∇ is aFinslerian Connection, if and only if the following conditions hold for N :i). e N ( x,Z ) ( Z x , Z x , ) = 0 , ∀ ( x, Z ) = Z x ∈ T ′ M .ii). C ( Z x ⊗ C (( d ′ F ⊗ e N ) ( x,Z ) )) = 0 , ∀ ( x, Z ) = Z x ∈ T ′ M .iii). P N ( X, QY ) = 0 , ∀ X, Y ∈ X ( T ′ M ) .iv). P N is symmetric with respect to the two lower indices and it ispositive homogeneous of degree 0.
In the literature on Finsler spaces, one can find a lot of tensor fieldshaving the properties i)–iv) (see, e.g., [15] and [2]) and by following theknown examples many other tensors of the same kind can be constructed.From e N one can obtain many tensors like N by following methods, which areanalogous to the ones previously used for the construction of the connection ∇ . Hence, we omit them for the sake of brevity.Now, let us consider a point x ∈ M . Then the ( closed ) indicatrix having x as its center and ρ > as its radius is the set B ρ (0 x ) = { ( x, X ) ∈ T x M/F ( x, X ) ≤ ρ } and it is the closure of the open indicatrix B ρ (0 x ), definedin the Introduction. B ρ (0 x ) and B ρ (0 x ) are both convex, that is for any X x , Y x elements of B ρ (0 x ) ( B ρ (0 x )) the vector tX x + (1 − t ) Y x belongs to B ρ (0 x ) ( B ρ (0 x )), for any t ∈ [0 , Proposition 2.6.
Let x ∈ M , the following assertions are true:i). The family { B ρ (0 x ) } ρ> is a basic system of open neighbourhoods of x in T x M .ii). The sets e A = ∪ z ∈ A B ρ (0 z ) , with A open neighbourhood of x in M and ρ > form a basic system of open neighbourhoods of x in T M . Now, we are in position to prove Proposition 1.1:
Proof.
Let ∇ be a Finslerian Connection and let x ∈ M . Since the BerwaldConnection ∇ ′ induced by ∇ induces a spray, by Proposition 2.2, there existsan open neighbourhood f W of the zero section σ ( M ) in T M and an opensubset W of M such that ( π, Exp ) : f W → W × W is a diffeomorphism of class9 , where Exp is the exponential map of the spray defined by ∇ ′ . Moreover,ii) of Proposition 2.6 ensures us that there exists a positive real number η andan open neighbourhood A of x in M , such that e A = ∪ z ∈ A B η (0 z ) ⊆ f W . Then,by using ii) of Proposition 2.3 and i) of Proposition 2.6, there also exists a realnumber ε , with 0 < ε ≤ η such that B ε ( x ) × B ε ( x ) ⊆ ( π, Exp )( e A ) ⊆ W × W ,where B ε ( x ) = Exp x ( B ε (0 x )) is an open neighbourhood of x . Finally, i)follows, since Exp x : B ε (0 x ) → B ε ( x ) is a C –diffeomorphism.To prove ii), we consider y, z ∈ B ε ( x ) with y = z . It results:0 y = ( π, Exp ) − ( y, z ) = X zy ∈ B η (0 y ) ⊆ f W .
Then, i) of Proposition 1.1, implies that the curve c ( t ) = Exp y ( tX zy ), with t ∈ [0 , W and joining y and z . Thecurve c is also the unique geodesic that joins y and z lying entirely in B η ( y ). Infact, we shall see that c has length lesser than η . Let β = ( c, ˙ c ) : [0 , → T ′ M the complete lift of c to T M , for any t ∈ [0 , β = ˙ c i e i + ¨ c ˆ i ε ˆ i .Hence it follows: g ( c ( t ) , ˙ c ( t )) ( ˙ c ( t ) , ˙ c ( t )) = G β ( t ) ( ˙ β ( t ) , ˙ β ( t )) , ∀ t ∈ [0 , . Moreover, being c a path of ∇ ′ , we have:( ∇ ˙ β ˙ β )( t ) = Q β ( t ) ( Z β ( t ) ) , ∀ t ∈ [0 , Z is a suitable vector field defined on an open neighbourhood of β ([0 , T ′ M . From the previous identities and ii) of the previous propo-sition we obtain: ddt g ( c ( t ) ˙ c ( t ) ( ˙ c ( t ) , ˙ c ( t )) = ( ∇ ˙ Y G ) β ( t )) ( ˙ β ( t ) , ˙ β ( t )) +2 G β ( t ) ( ˙ β ( t ) , Q β ( t ) ( Z ( β ( t )) )) = 0 , ∀ t ∈ [0 , . Hence, we have: F ( c ( t ) , ˙ c ( t )) = g ( c ( t ) , ˙ c ( t )) ( ˙ c ( t ) , ˙ c ( t )) = F ( y, X zy ) < η , ∀ t ∈ [0 ,
1] ; (2.2)that is c has length lesser than η . Moreover, for y = z ∈ B ε ( x ) the uniquegeodesic joining y and z in W is the constant curve. Now, we observe that,the mapping Exp y : B η (0 y ) → B η ( y ) is a C –diffeomorphism, because ofii) of Proposition 2.2, by using the inclusion B η (0 y ) ⊆ f W ∩ T y M . Finally,we have iii), since the inclusion ( π, Exp ) − ( { y } × B η ( x )) ⊆ B η (0 y ) implies B ε ( x ) ⊆ B η ( y ). 10ow, we recall that the standard identification T X T x M = T x M , inducesa Riemannian Metric on T ′ x M , for any X ∈ T x M and any x ∈ M . By meansof this identification, we prove the corresponding of the Gauss Lemma. Proposition 2.7.
Let x ∈ M and X ∈ B ε (0 x ) , with ε as in Proposition 1.1.Then:i). It results F ( x, X ) = F ( Exp x ( X ) , ( DExp x ) X X ) , where we denoted by ( DExp x ) X : T X T x M = T x M → T Exp x ( X ) M the total differential of Exp x .ii). For any Y ∈ T x M , such that g ( x,X ) ( X, Y ) = 0 , it results: g ( Exp x ( X ) , ( DExp x ) X X ) (( DExp x ) X X, ( DExp x ) X Y ) = 0 . Proof. If X = 0 x , then the assertion is trivially true. Hence, we suppose X = 0 x .In order to prove i) let us denote by c X : [0 , → M the geodesic of theBerwald connection ∇ ′ , then we have:( DExp x ) tX X = ˙ c X ( t ) = β ( t ) ∈ T ′ M , ∀ t ∈ [01] ;and ddt g ( c X ( t ) , ˙ c X ( t )) ( ˙ c X ( t ) , ˙ c X ( t )) = 0Hence, from Equation (2.2), it follows: g ( c x ( t ) , ˙ c X ( t )) ( ˙ c X ( t ) , ˙ c X ( t )) = F ( x, X ) , ∀ t ∈ [0 , . Consequently F ( c X ( t ) , ˙ c X ( t )) = F ( Exp x ( tX ) , ( DExp x ) tX X ) = F ( x, X ) , ∀ t ∈ [0 ,
1] ; (2.3)and the assertion follows for t = 1.Now, we prove ii). Let Y ∈ T ′ x M , such that g ( x,X ) ( X, Y ) = 0. Considera curve, denoted again by X : ( − a, a ) → B ε (0 x ), with a > ε as inProposition 1.1, having X (0) = X , ˙ X (0) = Y and F ( x, X ( s )) = F ( x, X ) = r < ε , for each s ∈ ( − a, a ). Being tX ( s ) ∈ B ε (0 x ), we can define themapping: λ ( s, t ) = Exp x ( tX ( s )) = c X ( s ) ( t ) , ( s, t ) ∈ ( − a, a ) × [0 , . ∂λ∂s ) ( s,t ) = ( DExp x ) tX ( s ) t ˙ X ( s )and ( ∂λ∂t ) ( s,t ) = ( DExp x ) tX ( s ) X ( s ) = ˙ c X ( s ) ( t ) , for any ( s, t ) ∈ ( − a, a ) × [0 , F ( λ ( s, t ) , ( ∂λ∂t ) ( s,t ) ) = F ( x, X ( s )) = r , ∀ ( s, t ) ∈ ( − a, a ) × [0 , . (2.4)Now, we set: β ( s, t ) = ( λ ( s, t ) , ( ∂λ∂t ) ( s,t ) ) ∈ T ′ M, ∀ ( s, t ) ∈ (0 , a ) × [0 , . Then, we have: (cid:0) ∂∂t g ( λ, ∂λ∂t ) ( ∂λ∂s , ∂λ∂t ) (cid:1) | ( s,t ) = G β (cid:0) ∇ ∂β∂s ∂β∂t , ∂β∂t ) | ( s,t ) =12 (cid:0) ∂∂s G β ( ∂β∂t , ∂β∂t ) (cid:1) | ( s,t ) − (cid:0) ∇ ∂β∂s G )( ∂β∂t , ∂β∂t ) (cid:1) | ( s,t ) . Hence, because of ii) of Proposition 2.3, recalling the definition of β , we have: ∂∂s F ( λ ( s, t ) , ( ∂λ∂t ) ( s,t ) ) = 0 , ∀ ( s, t ) ∈ ( − a, a ) × [0 , . Consequently: g ( Exp x ( tX ( s )) , ( DExp x ) tX ( s ) X ( s )) (( DExp x ) tX ( s ) t ˙ X ( s ) , ( DExp x ) tX ( s ) X ( s )) = 0 ∀ ( s, t ) ∈ ( − a, a ) × [0 , . (2.5)Finally, we obtain ii) by putting t = 1 and s = 0 in the previous identity.For the sequel, we need the following lemma, which is proved in [3]. Lemma 2.1.
Let x ∈ M and Z ∈ T ′ x M . Then for each Y ∈ T x M such that g ( x,Z ) ( Z, Y ) = 0 it results F ( x, Y + Z ) ≥ F ( x, Z ) and equality holds if andonly if Y = 0 x . Proof.
First, we prove i).We can suppose e b ( s ) = 0 x , for any s ∈ (0 , s ]. Then, the function r ( s ) = F ( x, e b ( s )) is not zero, for each s ∈ (0 , s ]. Hence, we can put: X ( s ) = 1 r ( s ) e b ( s ) , ∀ s ∈ (0 , s ] . With these notations, it is easy to see that: e b ( s ) = r ( s ) X ( s ) , (2.6) g ( x,X ( s )) ( X ( s ) , X ( s )) = 1 (2.7)˙ b ( s ) = ( DExp x ) e b ( s ) ˙ r ( s ) X ( s ) + ( DExp x ) e b ( s ) r ( s ) ˙ X ( s )) , (2.8) g ( x,X ( s ) ( X ( s ) , ˙ X ( s )) = 0 , (2.9)for any s ∈ (0 , s ].Then, from ii) of Proposition 2.7 and from (2.4), it follows: g ( b ( s ) , ( DExp x ) e b ( s ) e b ( s )) (cid:0) ( DExp x ) e b ( s ) ˙ r ( s ) X ( s ) , ( DExp x ) e b ( s ) r ( s ) ˙ X ( s ) (cid:1) = 0 (2.10)for any s ∈ (0 , s ]. Moreover, i) of Proposition 2.7 and equations (2.4) and(2.5) imply: F ( b ( s ) , ( DExp x ) e b ( s ) ˙ r ( s ) X ( s )) = ˙ r ( s ) , ∀ s ∈ (0 , s ] . (2.11)Then, from (2.7), (2.9), (2.10) and from the Lemma 2.1, we obtain: F ( b ( s ) , ˙ b ( s )) ≥ ˙ r ( s ) , ∀ s ∈ (0 , s ] , (2.12)hence: L ( b ) = Z σ ρ | ˙ r ( s ) | ds ≥ | r ( s ) − r ( ρ ) | , ∀ ρ ∈ (0 , s ]13nd i) is true.Assertion ii) trivially holds.Under the assumptions of iii), by using the previous notations, we have L ( c X ) = L ( b ) = r ( s ). Moreover, being i) true, it results L ( b s ) ≥ L ( c e b ( s ) ) = r ( s ), with b s = b | [0 ,s ] , for any s ∈ (0 , s ]. Hence, it follows: Z s ρ ( F ( b ( s ) , ˙ b ( s )) − | ˙ r ( s ) | ) ds ≤ , ∀ ρ ∈ (0 , s ] . Since (2.12) holds, the previous inequality implies: Z s ρ ( F ( b ( s ) , ˙ b ( s )) − | ˙ r ( s ) | ) ds = 0 , ∀ ρ ∈ (0 , s ]and, for continuity reasons, we have F ( b ( s ) , ˙ b ( s )) = | ˙ r ( s ) | , ∀ s ∈ [0 , s ] . Consequently, by using (2.8), (2.10), (2.11) and the previous Lemma we have:(
DExp x ) e b ( s ) r ( s ) ˙ X ( s ) = 0 , ∀ s ∈ [0 , s ] . Then, ˙ X ( s ) = 0, for any s ∈ [0 , s ], because Exp x has maximal rank alongthe curve e b and r ( s ) = 0 for any s ∈ (0 , s ]. Hence, iii) is true.From the above proposition, it trivially follows: Proposition 2.8.
Let x ∈ M . Moreover, let us consider ε = ε ( x ) and η = η ( x ) as in Proposition 1.1. Then, a geodesic of length lesser than η starting from an arbitrary point y ∈ B ε ( x ) is a curve of minimal lengthbetween its end points. Let x ∈ M and B ρ (0 x ) ⊆ f W ∩ T x M . Moreover, let u = ( u i ) ≤ i ≤ n be abasis of T x M . Then, u defines a mapping, which by an abuse of notations,we denote again by u : R n → T x M obtained by putting u ( ξ ) = ξ i u i , for any ξ = ( ξ i ) ≤ i ≤ n ∈ R n . Furthermore, the mapping ψ = u − ◦ Exp − x : B ρ ( x ) → u − ( B ρ (0 x )) is a C ∞ –diffeomorphism on B ρ ( x ) − { x } and C –differentiableon x . Consequently, ( B ρ ( x ) , ψ ) is a chart of the C –differentiable mani-fold structure canonically induced on M by the considered structure of C ∞ –differentiable manifold on M .Then, we can prove: 14 roposition 2.9. Let x ∈ M and ε = ε ( x ) as in Proposition 1.1. Thereexists ε = ε ( x ) ∈ (0 , η ) such that for any non constant geodesic c : [0 , a ] → M , with a > , satisfying the conditions c ( t ) ∈ ∂B ε (( x ) = Exp x ( ∂B ε (0 x )) and ˙ c ( t ) ∈ T ∂B ε ( x ) , for some t ∈ (0 , a ) and ε ∈ (0 , ε ) and for each µ ∈ (0 , there exists ρ = ρ ( µ, ε ) with the following property: d ( x, c ( t )) ≥ d ( x, c ( t )) + µ ( t − t ) , ∀ t ∈ [ t − ρ, t + ρ ] , where d is the Finslerian distance function.Proof. We fix a basis of T x M and consider the C –differentiable chart( B η ( x ) , ψ ), with ψ = u − ◦ Exp − x . Moreover, we can suppose x = c ( t ) ∈ B η ( x ), for any t ∈ (0 , a ). Then all the derivatives of the function f ( t ) = d ( x, c ( t )) = F ( x, γ ( t )), being γ ( t ) = ψ ( c ( t )), for any t ∈ (0 , a ), are defined.Hence by using the homogeneity conditions, we get: f ( t ) = ε ; ˙ f ( t ) = g ( x,γ ( t )) ( γ ( t ) , ˙ γ ( t )) ;¨ f ( t ) = g ( x,γ ( t )) ( ˙ γ ( t ) , ˙ γ ( t )) + g ( x,γ ( t )) ( γ ( t ) , ¨ γ ( t )) . Therefore, the proof follows as in [4] (cf. [3], too).The proof of Proposition 1.3 follows from the previous proposition in atrivial way.We can also omit the proof of the following Lemma, because it needsonly the compactness of boundary of convex neighbourhoods, which holds,because an indicatrix is always compact and the exponential map is a diffeo-morphism near to any point. Hence, its proof follows as in [4].
Lemma 2.2.
Let p ∈ M and ρ > be a positive real number such that Exp p is defined on the ball B ρ (0 p ) ⊆ T p M . Then, every q ∈ M , with d ( p, q ) < ρ ,can be joined to p by a minimizing geodesic. Finally, the proof of Proposition 1.4 is the same as in the Riemanniancase (see, e.g., [4]).
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