LAGRANGE GEOMETRY ON TANGENT MANIFOLDS

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a nondegenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper, we study a generalization which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples and find the cohomological obstructions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, while giving an index-free presentation of these connections.


Preliminaries. Lagrange geometry
is the extension of Finsler geometry (e.g., [1]) to transversal "metrics" (nondegenerate quadratic forms) of the vertical foliation (the foliation by fibers) of a tangent bundle, which are defined as the Hessian of a nondegenerate Lagrangian function. In the present paper, we study the generalization of Lagrange geometry to arbitrary tangent manifolds [2]. The locally Lagrange-symplectic manifolds [12] are an important particular case. In this section, we recall various facts about the geometric structures that we need for the generalization. Our framework is the C ∞category, and we will use the Einstein summation convention, where convenient.
First, a leafwise locally affine foliation is a foliation such that the leaves have a given locally affine structure that varies smoothly with the leaf. In a different formulation [10], if M is a manifold of dimension m = p + q, a p-dimensional leafwise locally affine foliation Ᏺ on M is defined by a maximal, differential, affine atlas {U α }, with local coordinates (x a α ,y u α ) (a = 1,...,q; u = 1,...,p), and transition functions of the local form on U α ∩ U β . Then, the leaves of Ᏺ are locally defined by x a = const, and their local parallelization is defined by the vector fields ∂/∂y u . Furthermore, if the atlas that defines a leafwise locally affine foliation has a subatlas such that B u (αβ) = 0 for its transition functions, the foliation, with the structure defined by the subatlas, will be called a vector bundle-type foliation. Notice that, if one such subatlas exists, similar ones are obtained by coordinate changes of the local formx a α =x a α x b α ,ỹ α = y a α + ξ a (αβ) x b α . (1.2) For any foliation Ᏺ, geometric objects of M that either project to the space of leaves or, locally, are pullbacks of objects on the latter are said to be projectable or foliated [8,9]. In particular, a foliated bundle is a bundle over M with a locally trivializing atlas with foliated transition functions. The transversal bundle νᏲ = T M/T Ᏺ is foliated. Formulas (1.1) show that for a leafwise locally affine foliation Ᏺ the tangent bundles T Ᏺ and T M are foliated bundles as well. For a foliated bundle, we can define foliated cross sections. Notice that, if Ᏺ is a leafwise locally affine foliation, a vector field on M which is tangent to Ᏺ is foliated as a vector field, since it projects to 0, but it may not be a foliated cross section of T Ᏺ! Furthermore, for a leafwise locally affine foliation, one also has leafwise affine objects which have an affine character with respect to the locally affine structure of the leaves. For instance, a leafwise locally affine function is a function f ∈ C ∞ (M) such that Y f is foliated for any local parallel vector field Y along the leaves of Ᏺ. With respect to the affine atlas, a leafwise locally affine function has the local expression f = p u=1 α u x a y u + β x a . (1.3) A leafwise locally affine k-form is a k-form λ such that i(Z)λ = 0 for all the tangent vector fields Z of Ᏺ and the Lie derivative L Y λ is a foliated k-form for all the parallel fields Y . Then, λ has an expression of the form (1.3) where α u , β are foliated k-forms. A leafwise locally affine vector field is an infinitesimal automorphism of the foliation and of the leafwise affine structure, and has the local expression [10] Any foliated vector bundle V → M produces a sheaf V of germs of differentiable cross sections, and a sheaf V pr of germs of foliated cross sections. The corresponding cohomology spaces H k (M, V pr ) may be computed by a de Rham type theorem [9]. Namely, let NᏲ be a complementary (normal) distribution of T Ᏺ in T M. The decomposition T M = NᏲ ⊕ T Ᏺ yields a bigrading of differential forms and tensor fields, and a decomposition of the exterior differential as d = d (1,0) + d (0,1) + ∂ (2,−1) . (1.5) The operator d is the exterior differential along the leaves of Ᏺ, it has square zero and satisfies the Poincaré lemma. Accordingly, where Ω denotes spaces of differential forms, Ω is the corresponding sheaf of differentiable germs, and Φ is the sheaf of germs of foliated functions, is a fine resolution of V pr . Furthermore, if Ᏺ is leafwise locally affine, one also has the spaces A k (M, Ᏺ) of leafwise locally affine k-forms and the corresponding sheaves of germs A k (M, Ᏺ). These sheaves define interesting cohomology spaces, which may be studied by means of the exact sequences [10] where, for f defined by (1.3), It is important to recognize the vector bundle-type foliations among the leafwise locally affine foliations. First, notice that a vector bundle-type foliation possesses a global vector field which may be seen as the leafwise infinitesimal homothety, namely, where Q α are projectable. E is defined up to the addition of a global, projectable cross section of T Ᏺ, and these vector fields E will be called Euler vector fields. The choice of an Euler vector field E is equivalent with the choice of the vector bundle-type structure of the foliation. We also recall the following result [10]: the vector bundle-type foliation Ᏺ on M is a vector bundle fibration M → N if and only if the leaves are simply connected and the flat connections defined by the locally affine structure of the leaves are complete.
is the generalized Heisenberg group, and Γ (1,p) is the subgroup of matrices with integer entries. The manifold M (1,p) has an affine atlas with the transition functionsx ..,p) are the entries of X, Z, respectively, and a i , b, c i are integers. Accordingly, the local equations x i = const, y = const define a leafwise locally affine foliation Ᏺ of M which, in fact, is a fibration by p-dimensional tori over a (p + 1)-dimensional torus. The manifold M is parallelizable by the global vector fields (1.13) and the global 1-forms and we see that may serve as a normal bundle of Ᏺ. It follows that the linearity obstruction is represented by which is not d -exact. Therefore, Ᏺ is not a vector bundle-type foliation. Example 1.3. Take the real Hopf manifold H (p+q) = S p+q−1 × S 1 seen as (R q × R p \{0})/G λ , where λ ∈ (0, 1) is constant and G λ is the group x a = λ n x a ,ỹ u = λ n y u , n∈ Z, (1.17) where x a , y u are the natural coordinates of R q and R p , respectively. Then, the local equations x a = const define a vector bundle-type foliation, which has the global Euler field E = q u=1 y u (∂/∂y u ). This example shows that compact manifolds may have vector bundle-type foliations.
where λ ∈ (0, 1) and K λ is the cyclic group generated by the transformatioñ It is easy to check that the equality defines a global vector field on M, which has the property of the Euler field for the foliation x i = const Therefore, the latter is a vector bundle-type foliation. The change of coordinates provides a vector bundle-type atlas, and (1.19) becomes This shows that M is the tangent bundle of the Hopf manifold H n defined in Example 1.3. Now, we recall the basics of tangent manifolds [2]. An almost tangent structure on a manifold M is a tensor field S ∈ Γ End(T M) such that In particular, the dimension of M must be even, say 2n, and rank S = n. Furthermore, S is a tangent structure if it is integrable, that is, locally S looks like the vertical twisting homomorphism of a tangent bundle. This means that there exists an atlas with local coordinate (x i ,y i ) (i = 1,...,n) such that The integrability property is equivalent with the annulation of the Nijenhuis tensor where S is a tangent structure, is called a tangent manifold. On a tangent manifold (M, S), the distribution im S is integrable, and defines the vertical foliation ᐂ with T ᐂ = im S. It is easy to see that the transition functions of the local coordinates of (1.24) are of the local form (1.1) with q = p = n and (1.26) Therefore, ᐂ is a leafwise locally affine foliation, and the local parallel vector fields along the leaves are the vector fields of the form SX, where X is a foliated vector field. In particular, a tangent manifold has local Euler fields E α , and a linearity obstruction [E] ∈ H 1 (M, T ᐂ pr ). If [E] = 0, the foliation ᐂ will be a vector bundle-type foliation, and M has global Euler vector fields E defined up to the addition of a foliated cross section of T ᐂ. Furthermore, if we fix the vector-bundle type structure by fixing an Euler vector field E, the triple (M,S,E) will be called a bundle-type tangent manifold.
Using the general result of [10], we see that a tangent manifold is a tangent bundle if and only if it is a bundle-type tangent manifold and the vertical foliation has simply connected, affinely complete leaves.
is a compact, non bundle-type, tangent manifold. The linearity obstruction [E] of this manifold is represented by and [E] ≠ 0.
Tangent bundles posses second order vector fields (semisprays in [6]), so called because they may be locally expressed by a system of second order, ordinary, differential equations. A priori, such vector fields may be defined on any tangent manifold [13], namely, the vector field X ∈ Γ T M (Γ denotes the space of global cross sections) is of the second order if SX| Uα − E α is foliated for all α. But this condition means that SX is a global Euler vector field, hence, only the bundle-type tangent manifolds can have global second order vector fields.
It is important to point out that, just like on tangent bundles (e.g., [3,6,11]), if (M,S,E) is a bundle-type tangent manifold and X is a second order vector field on M, the Lie derivative F = L X S defines an almost product structure on M (F 2 = Id), with the associated projectors such that im V = T ᐂ and im H is a normal distribution Nᐂ of the vertical foliation ᐂ. Finally, we give the following definition.
Definition 1.8. A vector field X on a tangent manifold (M, S) is a tangential infinitesimal automorphism if L X S = 0 (L denotes the Lie derivative).
Obviously, a tangential infinitesimal automorphism X preserves the foliation ᐂ and its leafwise affine structure. Therefore, X is a leafwise affine vector field with respect to ᐂ. Furthermore, in the bundle-type case, if E is an Euler vector field, [X, E] is a foliated cross section of T ᐂ.

Locally Lagrange spaces.
Lagrange geometry is motivated by physics and, essentially, it is the study of geometric objects and constructions that are transversal to the vertical foliation of a tangent bundle and are associated with a Lagrangian (a name taken from Lagrangian mechanics), that is, a function on the total space of the tangent bundle. (See [6] and the d-objects defined there.) Here, we use the same approach for a general tangent manifold (M, S), and we refer to functions on M as global Lagrangians and to functions on open subsets as local Lagrangians.
If ᏸ is a Lagrangian, the derivatives in the vertical directions yield symmetric tensor fields of M defined by (Of course, the result does not depend on the choice of the extensionsX i .) Hess (k) ᏸ is called the k-Hessian of ᏸ. Notice that definition (2.1) may also be replaced by the recurrence formula where the arguments are foliated vector fields. It is worthwhile to notice the following general property.
Proof. Proceed by induction on k, while evaluating the Hessian of Xᏸ on foliated arguments and using the recurrence formula (2.2).
For k = 1, we get a 1-form, say θ ᏸ , and for k = 2, we get the usual Hessian of ᏸ with respect to the affine vertical coordinates y i (see Section 1), hereafter to be denoted by either Hess ᏸ or g ᏸ . Obviously, g ᏸ vanishes whenever one of the arguments is vertical, hence, it yields a well-defined cross section of the symmetric tensor product 2 ν * ᐂ (νᐂ = T M/T ᐂ), which we continue to denote by g ᏸ . If g ᏸ is nondegenerate on the transversal bundle νᏲ, the Lagrangian ᏸ is said to be regular and g ᏸ is called a (local) Lagrangian metric. We note that if the domain of ᏸ is connected, the regularity of ᏸ also implies that g ᏸ is of a constant signature. With respect to the local coordinates of (1.24), one has In Lagrangian mechanics, one also defines another geometric object related to a Lagrangian, namely, the differential 2-form If ᏸ is a regular Lagrangian, ω ᏸ is a symplectic form, called the Lagrangian symplectic form.
In [12,13], we studied particular symplectic forms Ω on a tangent manifold (M, S) that are compatible with the tangent structure S in the sense that If this happens, Ω is called a locally Lagrangian-symplectic form since the compatibility property is equivalent with the existence of an open covering M = ∪U α , and of local regular Lagrangian functions ᏸ α on U α such that Ω| Uα = ω ᏸα for all α. On the intersections U α ∩ U β , the local Lagrangians satisfy a compatibility relation of the form where ϕ (αβ) is a closed, foliated 1-form, b (αβ) is a foliated function, and a(ϕ) = ϕ i y i where the local coordinates and components are taken either in U α or in U β . Furthermore, if it is possible to find a compatible (in the sense of (2.7)) global Lagrangian ᏸ, Ω is a global Lagrangian symplectic form. Conditions for the existence of a global Lagrangian were given in [12,13]. In particular, a globally Lagrangian-symplectic manifold M 2n cannot be compact since it has the exact volume form ω n ᏸ . Following the same idea, we give a new definition.
Definition 2.2. Let (M 2n ,S) be a tangent manifold, and g ∈ Γ 2 ν * ᐂ a nondegenerate tensor field. Then g is a locally Lagrangian metric (structure) on M if there exists an open covering M = ∪U α with local regular Lagrangian functions ᏸ α on U α such that g| Uα = g ᏸα = Hess ᏸ α for all α. The triple (M,S,g) will be called a locally Lagrange space or manifold.
It is easy to see that the local Lagrangians ᏸ α of a locally Lagrange space must again satisfy the compatibility relations (2.7), where the 1-forms ϕ (αβ) may not be closed. In particular, we see that a locally Lagrangian-symplectic manifold is a locally Lagrange space with the metric defined by [12] g [Y ] are the corresponding projections on νᏲ. Furthermore, if there exists a global Lagrangian ᏸ that is related by (2.7) with the local Lagrangians of the structure, (M,S,g,ᏸ) will be a globally Lagrange space.
A globally Lagrange space also is a globally Lagrangian-symplectic manifold, hence, it cannot be compact. We can give a global characterization of the locally Lagrange metrics. First, we notice that the bundles ⊗ k ν * ᐂ of covariant tensors transversal to the vertical foliation ᐂ of a tangent manifold (M, S) may also be seen as the bundles of covariant tensors on M that vanish if evaluated on arguments one of which belongs to im S. (This holds because ν * ᐂ ⊆ T * M). In particular, a transversal metric g of ᐂ may be seen as a symmetric 2-covariant tensor field g on M which is annihilated by im S. With g, one associates a 3-covariant tensor, called the derivative or Cartan tensor [1,6,7] defined by whereX is a foliated extension of X. Obviously, C ∈ Γ ⊗ 3 ν * ᐂ. Then, we get the following proposition. Proof. Since the symmetry of C is equivalent with the existence of the required local Lagrangians ᏸ.
We give a number of examples of locally Lagrange manifolds. is the Lagrange metric of the local compatible Lagrangians is a global, regular Lagrangian, and it produces a positive definite Lagrange metric.
Example 2.7. Consider the Hopf manifold H 2n of Example 1.5 with the tangent structure (1.24), and define the local compatible Lagrangians An easy computation yields (2.16) The determinant of the Hessian (2.16) can be easily computed as a characteristic polynomial, and we get det The compatibility relations (2.7) endow M with an A 0 -valued 1-cocycle defined by any of the members of (2.7), hence, with a cohomology class Ᏻ ∈ H 1 (M, A 0 ), which we call the total Lagrangian obstruction. It is obvious that Ᏻ = 0 if and only if the manifold M with the indicated structure is a globally Lagrange space.
Furthermore, the total Lagrangian obstruction may be decomposed into two components determined by the exact sequence (1.7) with k = 0, which in our case becomes where π is the composition of the projection π of (1.7) by S.
It is easy to see that the connecting homomorphism of the exact cohomology sequence of (2.19) is zero in dimension 0. Accordingly, we get the exact sequence where ι * , π * are induced by the inclusion and the homomorphism π of (2.19). Accordingly, we get the cohomology class Ᏻ 1 = π * (Ᏻ) ∈ H 1 (M, Ω (1,0) pr ), and we call it the first Lagrangian obstruction. The annulation Ᏻ 1 = 0 is a necessary condition for M to be a globally Lagrange space. Furthermore, if Ᏻ 1 = 0, the exact sequence (2.20) tells us that there exists a unique cohomology class Ᏻ 2 ∈ H 1 (M, Φ) such that Ᏻ = ι * (Ᏻ 2 ). We call Ᏻ 2 the second Lagrangian obstruction of the given structure, and Ᏻ = 0 if and only if Ᏻ 1 = 0 and Ᏻ 2 = 0.
We summarize the previous analysis in the following proposition. Assume that a choice of a normal bundle Nᐂ has been made. Then we can use the de Rham theorem associated with the relevant resolution (1.6) in order to get a representation of the Lagrangian obstructions. The definition of Ᏻ 1 shows that the first Lagrangian obstruction is represented by the cocycle {θ ᏸ β − θ ᏸα }. Accordingly, Ᏻ 1 may be seen as the d -cohomology class of the global form Θ of type (1, 1) defined by gluing up the local forms {d θ ᏸα }. If we follow the notation of [9] and take bases with the dual cobases Proof. Essentially, the hypothesis on E means E cannot be a conformal infinitesimal automorphism of g. From (2.23), we get where the local coordinates belong to an affine atlas such that E = y i (∂/∂y i ).
If M is compact, M L E Ψ = 0, and the coefficient of the right-hand side of (2.25) cannot have a fixed sign. But the latter property holds under the hypothesis of the corollary.
For instance, the Hopf manifold H n has no locally Lagrange metric with homogeneous with respect to the coordinates (y i ) Lagrangians ᏸ α . Indeed, homogeneity of degree s ≠ −1 is impossible because of Corollary 2.11, and homogeneity of degree −1 contradicts the transition relations (2.7). Remark 2.12. Because of Corollary 2.11, we conjecture that a compact, bundle-type, tangent manifold cannot have a locally Lagrange metric. Proof. Under the hypothesis, there exists a global form λ of type (1, 0) such that Θ = d θ ᏸα = d λ, therefore, θ ᏸα = λ| Uα +ξ α , with some local foliated 1-forms ξ α = ξ α,i (x j )dx i . Accordingly, we get where a has the same meaning as in (2.7) and b (αβ) are foliated functions.

Now, if we defineᏸ
we are done. The last assertion follows from the definition of Ᏻ 2 .

Corollary 2.14. The locally Lagrange metric of Proposition 2.13 is defined by a global Lagrangian if and only if κ = d k for a function k ∈ C ∞ (M).
In order to give an application of this result we recall the following lemma. Proof. Use a normal bundle Nᐂ, and let λ be a d -closed form of type (p, q) on T N. Since the fibers of T N are contractible, if N = ∪U α is a covering by small enough, T N-trivializing neighborhoods, we have λ| p −1 (Uα) = d µ α (p : T N → N) for some local forms µ α of type (p, q − 1). The local forms µ α can be glued up to a global form µ by means of the pullback to T N of a partition of unity on N, that is, by means of foliated functions. Accordingly, we will have λ = d µ.
From Corollary 2.14 and Lemma 2.15, we get the following proposition.

Proposition 2.16. Any locally Lagrange metric of a tangent bundle T N is a globally Lagrange metric.
Remark 2.17. Propositions 2.3 and 2.16 imply that, in the case of a tangent bundle M = T N, the symmetry of C is a necessary and sufficient condition for g to be a global Lagrangian metric. It was well known that this condition is necessary [6]. On the other hand, the metrics of [6] are usually differentiable only on the complement of the zero section of T N, where Proposition 2.16 does not hold, hence, the condition is not a sufficient one.
We also mention the inclusion σ : Z (1,0) pr → Ω (1,0) pr , where Z denotes spaces of closed forms and the obvious following proposition.

Proposition 2.18. The locally Lagrange metric structure defined by {ᏸ α } is reducible to a locally Lagrangian-symplectic structure if and only if
Other important notions are defined by the following definition. Definition 2.19. Let (M,S,g) be a locally Lagrange space, and let X ∈ Γ T M. Then: (i) X is a Lagrange infinitesimal automorphism if L X g = 0, where g is seen as a 2-covariant tensor field on M; (ii) X is a strong Lagrange infinitesimal automorphism if it is a Lagrange and a tangential infinitesimal automorphism of (M, S), simultaneously.

L X g (Y , SZ) = −g Y ,[X,SZ] (X,Y ,Z ∈ Γ T M).
(2.29) From (2.29) and the nondegeneracy of g on νᐂ, it follows that a Lagrange infinitesimal automorphism is necessarily a ᐂ-projectable vector field. But it may not be leafwise locally affine. Indeed, if g is a foliated metric of νᐂ (Example 2.8), every tangent vector field of ᐂ is a Lagrange infinitesimal automorphism even if it is not leafwise locally affine. We finish this section by considering a more general structure. Lagrangians ᏸ α such that, over the intersections U α ∩U β , the local Lagrangian metrics satisfy a relation of the form where f (αβ) > 0 are foliated functions. A tangent manifold endowed with this type of structure is a locally conformal Lagrange space or manifold.

Clearly, condition (2.30) is equivalent with the transition relations
where the last two terms are like in (2.7). On the other hand, {ln f (αβ) } is a Φ-valued 1-cocycle, and may be written as ln f (αβ) = ψ β − ψ α , where ψ α is a differentiable function on U α (which may be assumed projectable only if the cocycle is a coboundary). Accordingly, the formula g| Uα = e −ψα g ᏸα (2.32) defines a global transversal metric of the vertical foliation which is locally conformal with local Lagrange metrics. As a matter of fact, we have the following proposition. Proof. We still have to prove that the existence of the metric g that satisfies (2.32) implies (2.30), which is clear, except for the fact that the functions f (αβ) = e ψ β −ψα are projectable. This follows from the Lagrangian character of the metrics g ᏸα . Indeed, with the usual local coordinates (x i ,y i ), the symmetry of the derivative tensors C of g ᏸα , g ᏸ β implies ∂f (αβ) ∂y k g ᏸα ij = ∂f (αβ) ∂y i g ᏸα kj , (2.33) and a contraction by (g ᏸα ) ij yields ∂f (αβ) /∂y k = 0.
The cohomology class η = [ln f (αβ) ] ∈ H 1 (M, Φ) will be called the complementary class of the metric g, and the locally conformal Lagrange metric g is a locally Lagrange metric if and only if η = 0. Indeed, if η = 0, we may assume that the functions ψ α are foliated and the derivative tensor C of g = e −ψα g ᏸα is completely symmetric.
Furthermore, using a normal bundle Nᐂ and the leafwise version of the de Rham theorem, the complementary class may be seen as the d -cohomology class of the global, d -closed complementary form τ obtained by gluing up the local forms {d ψ α }. In particular, Lemma 2.15 and Proposition 2.16 imply that any locally conformal Lagrange metric g of a tangent bundle must be a locally, therefore, a globally Lagrange metric. is a corresponding global metric, which, with the previously used notation, corresponds to The corresponding complementary form is  if and only if there exists a d -closed form τ of type (0, 1) such that the ten-sorC = C − (τ • S) ⊗ g, where C is the derivative tensor of g, is a completely symmetric tensor.

Transversal Riemannian geometry.
The aim of this section is to give an index-free presentation of the connections used in Finsler and Lagrange geometry [1,6,7], while also extending these connections to tangent manifolds.
Let (M, S) be a tangent manifold and g a metric of the transversal bundle of the vertical foliation ᐂ (T ᐂ = im S). (The metrics which we consider are nondegenerate, but may be indefinite.) We do not get many interesting differentialgeometric objects on M, unless we fix a normal bundle Nᐂ, also called the horizontal bundle, that is, we decompose (3.1) We will say that Nᐂ is a normalization and (M,S,Nᐂ) is a normalized tangent manifold. Where necessary, we will use the local bases (2.21) and (2.22). The projections on the two terms of (3.1) will be denoted by p N , p T , respectively, and P = p N − p T is an almost product structure tensor that has the horizontal and vertical distribution as ±1-eigendistributions, respectively. For a normalized tangent manifold, the following facts are well known: (i) On a normalized tangent manifold (M,S,Nᐂ), a pseudo-Riemannian metric γ is said to be a compatible metric if the subbundles T ᐂ, Nᐂ are orthogonal with respect to γ and It is easy to see that these conditions imply the compatibility of γ with the structures J and F , that is, Furthermore, if (M, S) is a tangent manifold and γ is a pseudo-Riemannian metric on M, we will say that γ is compatible with the tangent structure S if the γ-orthogonal bundle Nᐂ of im S is a normalization, and γ is compatible for the normalized tangent manifold (M,S,Nᐂ).
The following result is obvious. In what follows, we will refer to the metric γ as the canonical extension of the transversal metric g. On the other hand, a pseudo-Riemannian metric γ of a tangent manifold (M, S) which is the canonical extension of a locally Lagrange metric g will be called a locally Lagrange-Riemann metric. This means that the restriction of γ to the γ-orthogonal subbundle Nᐂ of the vertical foliation ᐂ of S is a locally Lagrange metric g = g ᏸα and γ is compatible with (M,S,Nᐂ). Then, (M,S,γ) will be called a locally Lagrange-Riemann manifold. Notice that, since the induced metric of Nᐂ is nondegenerate, Nᐂ is a normalization of the vertical foliation and the compatibility condition of the definition makes sense. Thus, any normalized locally Lagrange space with the canonical extension γ of the Lagrange metric g is a locally Lagrange-Riemann manifold, and conversely.
is the canonical extension of the locally Lagrange metric defined in Example 2.

on M(1,p)× (R/Z).
Now, let (M,S,Nᐂ,g) be a normalized tangent manifold with a transversal metric of the vertical foliation ᐂ and let ∇ be the Levi-Civita connection of the canonical extension γ of g.
We are going to define a general connection that includes the connections used in Finsler and Lagrange geometry [1,6,7] as particular cases determined by specific normalizations. This will be the so-called second canonical connection D of a foliated, pseudo-Riemannian manifold (M, γ), defined by the following conditions [9]: (i) Nᐂ and T ᐂ are parallel, (ii) the restrictions of the metric to Nᐂ and T ᐂ are preserved by parallel translations along curves that are tangent to Nᐂ, T ᐂ, respectively, (iii) the ᐂ-normal, respectively ᐂ-tangent, component of the torsion T D (X, Y ) vanishes if one of the arguments is normal, respectively tangent, to ᐂ. This connection is given by where Y 1 ,Y 2 ∈ Γ T ᐂ and Z 1 ,Z 2 ∈ Γ Nᐂ. We will say that D is the canonical connection, and the connection induced by D in the normal bundle Nᐂ, or, equivalently, in the transversal bundle νᐂ = T M/T ᐂ, will be called the canonical transversal connection. The canonical, transversal connection is a Bott (basic) connection [8]. The total torsion of the connection D is not zero, namely, one has Proposition 3.4. Let (M,S,g) be a locally Lagrange manifold, and γ the canonical extension of g. Then, the derivative tensor field of g has the following expressions: where X, Y , Z ∈ Γ Nᐂ.
Proof. Of course, in (3.7), g is seen as a 2-covariant tensor field on M (see Section 2). First, we refer to the first two equalities in (3.7). These are pointwise relations, hence, it will be enough to prove these equalities for foliated cross sections of the normal bundle Nᐂ. Indeed, a tangent vector at a point can always be extended to a projectable vector field on a neighborhood of that point. But in this case, the first and second equalities are straightforward consequences of the definitions of the tensor field C and of the connection D. Then, since ∇ has no torsion, (3.5) implies (3.8) and, also using ∇γ = 0, we get the required result.
The first two expressions of C actually hold for any vector fields X, Y , Z ∈ Γ T M.  (3.9) where X, Y , Z ∈ Γ Nᐂ. Corollary 3.6. On a tangent manifold, if γ is a compatible pseudo-Riemannian metric such that ∇S = 0, then γ is a projectable, locally Lagrange-Riemann metric.
Proof. If ∇S = 0, the third equality (3.7) yields C = 0, which is the characterization of this type of metrics. Now, we consider the curvature of D. The curvature is a tensor and it suffices to evaluate it pointwisely. For this reason, whenever we need an evaluation of the curvature (as well as of any other tensor) that involves vector fields, it will suffice to make that evaluation on ᐂ-projectable vector fields.
Proof. Equality (3.10) is in agreement with the fact that D is a Bott connection [8]. Formulas (3.10), (3.11), and (3.12) follow from (3.5) and (3.6). Formula (3.13) is a consequence of (3.11). In the computation, one will take into account the fact that for any foliated vector field X ∈ Γ T M and any vector field Y ∈ Γ T ᐂ one has [X, Y ] ∈ Γ T ᐂ [8]. where X, Y , Z ∈ Γ Nᐂ.
Proof. Write down the general expression of the Bianchi identity of a linear connection with torsion (e.g., [4,5]) for arguments tangent and normal to ᐂ. Then, compute using (3.5), (3.6), and projectable vector fields as arguments. The fourth relation included in the Bianchi identity reduces to (3.12). Proof. This is just a rewriting of the classical second Bianchi identity [4,5] that uses (3.6).
Like Riemannian geometry, we also define a covariant curvature tensor (3.21) In particular, we have the following proposition.
where the arguments are foliated vector fields in Γ Nᐂ, and g is seen as a tensor on M.
where X, Y , Z, U, V ∈ Γ Nᐂ, and so on.
Example 3.13. On the torus T 2n with the metric of Example 3.2, the usual flat connection is both the Levi-Civita connection and the canonical connection D, and it has zero curvature. On the manifold M(1,p)× (R/Z) with the metric of Example 3.3, the connection that parallelizes the orthonormal basis shown by the expression of the metric is not the Levi-Civita connection, since it has torsion, but it follows easily that it has the characteristic properties of the canonical connection D. Accordingly, we are in the case of a locally Lagrange-Riemann manifold with a vanishing curvature R D and a nonvanishing torsion T D .
Proposition 3.14. The Ricci curvature tensor ρ D of the connection D is given by the equalities
Proof. The definition of the Ricci tensor of a linear connection (e.g., [4]), and the use of the bases (2.21) and (2.22) yield Then, the results follow from (3.5) and (3.11).
Remark 3.15. In view of (3.32), we may speak of κ D = tr ρ D on Nᐂ, and call it the transversal scalar curvature.
In the case of a normalized, bundle-type, and tangent manifold (M,S,E,Nᐂ), with a compatible metric γ (E is the Euler vector field), the curvature has some more interesting features, which were studied previously in Finsler geometry [1]. These features follow from the following lemma. Proof. The tensor S is the tensor defined at the beginning of this section, and with local bundle-type coordinates (x i ,y i ) n i=1 and bases (2.21), we have Now, (3.34) follows from (3.5).
Using Lemma 3.16 one can prove the following proposition. On the other hand, we can generalize the notion of flag curvature, which is an important invariant in Finsler geometry [1]. Namely, a flag φ at a point x ∈ M is a 2-dimensional plane φ ⊆ T x M which contains the vector E x . Such a flag is φ = span{E x ,X x }, where X x ∈ N x ᐂ is defined up to a scalar factor, and following [1], the flag curvature is defined by k(φ) = k(X) = R D (X, S E, X, S E) g(S E, S E)g(X, X) − g 2 (S E, X) .
If g is not positive definite, the flag curvature may take infinite values. Remark 3.19. The curvature R D has more interesting properties in the case of a bundle-type, locally Lagrange manifold such that the metric tensor g is homogeneous of degree zero with respect to the coordinates y i . The invariant characterization of this situation is that the derivative tensor C is symmetric, and such that i(S E)C = 0. (3.44) The connection ∇ preserves the vertical and horizontal distributions and the metric, but has a nonzero torsion. Then, we have the connections C D, C ∇ , which can be defined by using formulas (3.5) and (3.44) with the Levi-Civita connection ∇ replaced by the Chern connection C ∇, that is, the γ-metric, Jpreserving connection that has a torsion with no component of J-type (1, 1) (J = S − S) [4,5].
We finish by recalling the well-known fact [3,6,7] that global Finsler and Lagrange structures of tangent bundles have an invariant normalization. This normalization may be defined as follows.
Let ᏸ be the global Lagrangian function. Then the energy function has a Hamiltonian vector field X Ᏹ defined by where ω ᏸ is the Lagrangian symplectic form (2.5), which turns out to be a second order vector field. Accordingly, L XᏱ S is an almost product structure on M (see Section 1), and N Ᏹ ᐂ = im H, with H defined by (1.29) is a canonical normal bundle of ᐂ.
A locally Lagrangian structure {ᏸ α } on a bundle-type tangent manifold (M, S, E) defines a global function (second order energy) but, generally, it has no global Hamiltonian vector field, and, even if such a field exists, is may not be a second order vector field.