Lagrange geometry on tangent manifolds

Izu Vaisman

doi:10.1155/s0161171203303059

Abstract

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.

Highlights

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
Lagrange geometry [3, 6, 7] is the extension of Finsler geometry (e.g., [1]) to transversal “metrics” of the vertical foliation of a tangent bundle, which are defined as the Hessian of a nondegenerate Lagrangian function
We study the generalization of Lagrange geometry to arbitrary tangent manifolds [2]

Summary

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. It follows that the leafwise locally affine foliation Ᏺ has a vector bundle-type structure if and only if [E](Ᏺ) = 0 [10]. The local equations xa = const define a vector bundle-type foliation, which has the global Euler field E =. 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 = LX 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. Define compatible local Lagrangians with the corresponding Lagrange metric n i=1

Lagrange symplectic form

An easy computation yields

We also mention the inclusion σ