Symplectic Structures and Lagrange Geometry

Abstract

On the tangent bundle TM of a manifold M we have the vertical bundle as the kernel of the differential of the projection TM -→ M. We take a supplement of it, that is, a horizontal bundle or a nonlinear connection and we consider the natural almost complex F on TM associated to these bundles. We show in the first section of this paper that the symplectic structures on TM having the horizontal and vertical bundles as Lagrangian subbundles and being compatible with F are essentially induced by a Lagrangian structure on M. In the second section we state precisely the symplectic structure ΩL induced by a Lagrangian structure L. In the third section some properties of ΩL are pointed out. The fourth section is devoted to the Finslerian symplectic structures, that is, to ΩL when L = F2 for F the fundamental function of a Finsler space.