Izu Vaisman, "Lagrange geometry on tangent manifolds", International Journal of Mathematics and Mathematical Sciences, vol. 2003, Article ID 974096, 26 pages, 2003. https://doi.org/10.1155/S0161171203303059
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.
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