Cartan connection of transversally Finsler foliation

The purpose of this paper is to define transversal Cartan connection of Finsler foliation and to prove its existence and uniqueness.

1. Introduction.Let (M, F) be a smooth n-dimensional manifold equipped with a foliation F of codimension q.We put n = p + q.We denote by (x i , y α ), i = 1, 2, . . ., p, α = 1, 2, . . ., q the foliated (or distinguished) coordinates with respect to the foliation F. If (x i , y α ), i = 1, 2, . . ., p, α = 1, 2, . . ., q is another foliated coordinate system, then Let T M be a tangent bundle of M .We consider an induced coordinate system (x i , y α , a i , b α ) in T M , where (a i , b α ) are coordinates of the vector a i ∂ ∂x i + b α ∂ ∂y α tangent to M at the point p = (x, y).Let Q(M ) denotes the normal bundle of the foliation F with the projection δ : T M −→ Q(M ).In Q(M ) we have the coordinates (x i , y α , b α ), where b α are coordinates of the vector b α ∂ ∂y α .Here ∂ ∂y 1 , . . ., ∂ ∂y q is a local frame of Q.The coordinates in Q transform as follows x i = x i (x, y), y α = y α (y), b α = J α α (y), where J α α (y) = ∂y α ∂y α (y).If ∂ ∂y α , α , = 1, . . ., q are the vectors of a local frame in new coordinates in Q, then Let us recall some basic facts from the theory of Riemannian foliations ( [5]).Let g T be a Riemannian metric in the normal bundle Q.The metric g T is called adapted to the foliation F if for any vector field X tangent to the leaves of F and any sections Y, Z of the normal bundle where Y , Z are any vector fields on M such that δ( Y ) = Y , δ( Z) = Z.
The vector field V on M is called foliated if for any vector field X tangent to the leaves of F the vector field [X, V ] is also tangent to the leaves.Locally in the foliated coordinate system foliated vector fields are of the form The section Y of the normal bundle is called a transverse vector field if Y = δ(V ) with V foliated.It is clear that the metric g T is adapted if the function g T (Y, Z) is constant along the leaves for any transverse vector fields Y, Z.
Let (x i , y a ) be a foliated coordinate system on U ⊂ M .Denote by U the local quotient manifold and let π : U −→ U be a local projection π(x i , y a ) = (y a ).The adapted metric g T induces the metric g on U such that for each point u ∈ U , π is an isometry between the transversal space at u and the tangent space at π(u).
Let B T (M ) be the bundle of transversal frames of M and θ T be the canonical form on B T (M ) with values in R q .P. Molino ([5]) has proved that p-dimensional distribution P T on B T (M ) such that Using the metric g T , we can define the bundle E 1 T of the orthonormal transversal frames.The bundle E 1 T is saturated by the leaves of the lifted foliation.The connection H in E 1 T is called transverse if the distributions tangent to the leaves of the lifted foliation are horizontal with respect to H.The following theorem is fundamental in the theory of transversally Riemannian foliations.
Theorem ([5]).For any transversal metric g T there exists in E 1 T exactly one torsion-free transverse connection.
A. Spiro in [6] has given the characterization of Cartan connection of Finsler manifold (M, F ) in terms of a bundle of Chern frames.The purpose of this paper is to prove the similar theorem for the transversally Finsler foliation.

Transversally Finsler foliations.
We start with the definition of the transverse Finsler metric.Definition 2.1.A Finsler metric F T in the normal bundle of the foliation F is called transverse if for any transverse vector field X the function F T (X) is basic.
Consider a foliated coordinate system (x i , y α ), where y 1 , . . ., y q are transverse coordinates.If V = a i (x, y) ∂ ∂x i +b α (y) ∂ ∂y α is a foliated vector field and b α (y) ∂ ∂y α is a corresponding transverse vector field, then F T is a transverse Finsler metric if and only if the function F T (x, y, b) does not depend on x.Let π : U −→ U be a local projection.Then we have the Finsler metric F on U defined by the formula F (y, b) = F T (y, b) such that π induces an isometry between Q u and T π(u) U , for any u ∈ U .
A. Spiro in [6] has defined the bundle of spheres of the Finsler metric F .In our case we define the bundle of the transversal spheres.

Definition 2.2. The set S
In this way we obtained a bilinear form g T on the tangent space , where S u is a sphere at u with respect to F .
A. Spiro in [6] has constructed a bundle of Chern orthogonal frames for the Finsler space.In the case of the transverse Finsler metric we can define in a similar way a bundle of transverse orthogonal Chern frames.
For fixed V ∈ Q u there is the natural identification of the vector space (2) The vectors E 1 , . . ., E q−1 are tangent to Denote by O g T (S T (M )) the bundle of the transverse orthogonal Chern frames.For a distinguished open set U the bundle O g T (S T (U )) is a pull-back of the bundle O g (S(U )) of orthogonal Chern frames of the local quotient manifold U under the restriction π T of π T to O g T (S T (U )).There is a natural right action of the group O(R, q − 1) on O g T (S T (M )).
Proposition 2.1.The bundle O g T (S T (M )) is saturated by the leaves of the lifted foliation and foliation of O g T (S T (M )) is invariant under the action of O(R, q − 1).

Proof.
Let X e ∈ P T (e) be a vector tangent at e to the leave of the lifted foliation.Consider distinguished open set U and the projection which means that dim ker π T = p.From (1.1) it follows that the foliation of O g T (S T (M )) is invariant under the action of O(R, q − 1).

Definition 2.5. A local section
Equivalently σ is a foliated section if it sends locally the leaves of F into the leaves of the lifted foliation.
Let p T : O g T (S T (M )) −→ M be the natural projection.

Proposition 2.2.
For any e ∈ O g T (S T (M )) there exists a local foliated section σ : Proof.Let u 0 = π(u 0 ), where π is a projection onto a local quotient manifold U .The projection π T : B T (U ) −→ B(U ) induces an isometry of the transversal sphere S T p T (e) onto the sphere S π(u) of the Finsler space (U , F ).The image of the transversal orthonormal frame e is an orthonormal frame e at u with respect to F .Let e = (e 0 , e 1 , . . ., e q−1 ), where F (u 0 , e 0 ) = 1 and e 1 , . . ., e q−1 is an orthonormal basis of T e 0 S u 0 .Denote the transversal coordinates by (y 0 , . . ., y q−1 ).We can suppose that ) ∂ ∂y 0 .We can use the scalar product g(u, z 0 ) to project the vectors ∂ ∂y 1 , . . ., ∂ ∂y q−1 onto the tangent space T z 0 S u and next applying the Gram-Schmidt orthonormalization process, we obtain an orthonormal frame z 1 , . . ., z q−1 of T z 0 S u .In this way we obtain a section σ : U −→ O g (S(U )) such that σ(u 0 ) = (z 0 , . . ., z q−1 ) and z 0 = e 0 .Using an appropriate element g ∈ O(R, q − 1) we get a section σ : U −→ O g (S(U )) such that σ(u 0 ) = e = (e 0 , . . ., e q−1 ).The section σ : U −→ 0 g T (S T (U )) is the unique section defined by the condition p T • σ = σ • π.
The fibre V T u = p −1 T (u) consists of the orthonormal frames of the transversal sphere S T u .Denote by V T e the subspace of T u O g T (S T (M )) of the vectors tangent at e to the V T u .Let A be the fundamental vector field on B T (M, F) associated to the element A ∈ gl(q, R).We put For any open U ∈ M adapted to the foliation F and any g ∈ Gl(R, q) where R g (resp.R g ) denotes the right translation of B T (U ) (resp.B(U )).q (U ) of S(U ) such that V ∈ S q (U ) if and only if w q > 0. Then (y β , z α ), z α = w α w q , are coordinates in S q (U ), (x i , y β , z α ) are coordinates in S q (U ) = π −1 (S q (U )), π(x i , y β , z α ) = (y β , z α ) and (x i ) are coordinates along the plaques of the lifted foliation.Let e ∈ O T g (S T (U ), e = (x i , y β , z α , A α γ ) where e is constant along the plaques of the lifted foliation restricted to O g T (S T (U )) and g T e = g e , where g e corresponds to the vertical subspace at e of the bundle O g (S(U )) of the Finsler space (U , F ).
Proof.Proposition 2.3 is a direct consequence of (2.1).
Then A ∈ g e and from [6] we know that (2.2) where H is the Hessian at the point (u, e 0 ) of the transversal Finsler metric.

Definition 2.6 ([6]
).A non-linear connection in O T g (S T (M )) is a distribution H such that H is complementary to the vertical distribution and for any h ∈ O(q − 1, R), Equivalently a non-linear connection is defined by a g e -valued 1-form ω on O T g (S T (M )) which vanishes on H and ω(A e ) = A for any A ∈ g e .A non-linear connection H is called adapted to the transverse Finsler sphere bundle if the vectors tangent to the lifted foliation are horizontal.The R q valued 2-form Σ T = dθ T + ω ∧ θ T is called the torsion of H.In the similar way as in we can prove the following proposition.Proof.We need to prove the uniqueness of torsion-free non-linear connection.Let ω and ω be the torsion-free non-linear connections.It is enough to prove that for any e ∈ O T g (S T (M )) ω and ω agreed on on the section σ : U −→ O T g (S T (U )) such as in Proposition 2.2.Let σ (θ T ) = (θ 0 , θ 1 , . . ., θ q−1 ) and σ (ω) = A α γβ θ γ , σ ( ω) = B α γβ θ γ , where ω γ = (A α γβ ) and ω γ = (B α γβ ) are the elements of g e .Since ω and ω are torsion-free it follows that (ω − ω) ∧ θ T = 0. We have From (2.2) and (2.3) we get Example 2.2.Let F be a transversal Finsler metric in Q and g an arbitrary Riemannian metric on M .Denote by (T u L) ⊥ an orthogonal complement of T u M with respect to g.The projection ρ u : T u M −→ Q u induces an isomorphism of (T u L) ⊥ onto Q u .Put for X ∈ T u M , X = X L + X ⊥ L , X L ∈ T u L, X ⊥ L ∈ (T u L) ⊥ F (u, X) = g u (X L , X L ) + F 2 (u, ρ u (X)).
Then F is a Finsler metric on M adapted to F in the sense of [3], [4], (T u L) ⊥ is its transversal cone at u ( [3]) and the metric F induces the metric F on the bundle Q.

(1. 1 )
P T (e) = {X e ∈ T e B T : i Xe θ T = i Xe dθ T = 0} is completely integrable and the associated foliation (the lifted foliation) is invariant by the right translations.Let B T (U ) be the bundle of transversal frames on U and denote by B(U ) the bundle of linear frames of local quotient manifold U .Let π T : B T (U ) −→ B(U ) be the natural projection.Then locally X e ∈ P T (e) ⊂ T e B T (U ) if and only if π T (X) = 0.

Example 2 . 1 .
Let U be a distinguished open set and π : U −→ U .Denote by (x i , y β ) the coordinates of u ∈ U .For any non-zero vectors V, W ∈ T u U put V ≡ W if and only if there exists λ > 0 such that V = λW .Let P u = T u U / ≡ and P U = u∈U P u .Then the bundle of spheres S(U ) is diffeomorphic to P U and we can use the positively homogeneous coordinates to get the coordinates in S(U ).For V ∈ S u (U ), V = v β ∂ ∂y β , (y β , w β ), where w β = λv β , λ > 0 are called homogeneous coordinates of V .Let π : S T (U ) −→ S(U ) be a natural projection.Consider an open subset S

Proposition 2 . 4 .Theorem 2 . 1 .
A non-linear connection H is adapted to the transverse Finsler sphere bundle if and only if i Xe Σ T = 0 for any X e tangent to the lifted foliation and e ∈ O T g (S T (M )).Proposition 2.5.Let F be a transverse Finsler metric on a foliated manifold (M, F).Then there exists on O T g (S T (M )) an adapted non-linear connection with zero torsion.Proof.Let U be a distinguished open set andπ T : O T g (S T (U )) −→ O g (S(U )).There exists in O g (S(U )) a unique torsion free connection ω.Then π T (ω) is a torsion free connection in O T g (S T (U )) adapted to the lifted foliation to O T g (S T (U )).Consider a covering {U i : i ∈ I} of M by the distinguished open sets and let π i : U i −→ U i be a local projection and ω i denotes a unique torsion free connection on O g (S(U i )).Let {f i : i ∈ I} be a partition of unity subordinate to the covering {U i : i ∈ I}.Then ω = f i •pπ T (ω i ) is a torsion free connection adapted to the lifted foliation.On the bundle O T g (S T (M )) of the transversal Chern orthonormal frames there exists a unique torsion-free non-linear connection.