Geometry of the Total Space of a Tangent Bundle

Abstract

The geometry of the manifold TM, the total space of the tangent bundle over a smooth manifold M is very rich. This manifold carries a lot of interesting geometrical structures, which were intensively studied (see Yano, K. and Ishihara, S.[168] and the references therein). The development of the Finsler geometry brought in this field new ideas especially that of using systematically a non-linear connection in the tangent bundle(TM,T,M). Also, a possibility to think the Finsler geometry as a subgeometry of the geometry of TM has appeared. This point of view on Finsler geometry, that allowed new approaches of its generalisations and applications, was clarified by R. Miron [103,104], M. Anastasiei [9], I. Popovici and M. Anastasiei [129].