Lagrange geometry on tangent manifolds by Izu Vaisman

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate 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. Lagrange geometry [3, 6, 7] is the extension of Finsler geometry (e.g., [1]) to transversal " metrics " (non degenerate quadratic forms) of the vertical foli-ation (the foliation by fibers) of a tangent bundle, which are defined as the Hessian of a non degenerate 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 * Mathematics Subject Classification: 53C15, 53C60.