International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 51, Pages 3241-3266
doi:10.1155/S0161171203303059
Lagrange geometry on tangent manifolds
Department of Mathematics, University of Haifa, Haifa 31905, Israel
Received 13 March 2003
Copyright © 2003 Izu Vaisman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.