Connections and M-Tensors on the Tangent Bundle TM

Abstract

After the concept of tangent bundle was discovered, it occurred to few differential geometers, including Professor E. T. Davies, that the proper setting for the Finsler metrics and general paths on a smooth manifold M is the tangent bundle TM and not M itself. This chapter discusses the concept of M-tensor and three types of connections on TM, highlights their properties, and explores the relationship between them. Some of the results help to define the relationship between several related known concepts in the differential geometry of TM, such as the system of general paths of Douglas, the nonlinear connections of Barthel and Yano and Ishihara, and the nonhomogeneous connection of Grifone, while others are generalizations of known results. The chapter describes the structure of the tangent bundle TM and the slit tangent bundle STM. It explores the M-tensors and three types of connections on TM and STM. The chapter highlights a (1, l)-connection on TM as horizontal distribution on TM and discusses the relationship between a vector field on TM and the horizontal distribution associated with a (1, l)-connection. It presents the proofs of two decomposition theorems, one of which is a sharpened version of a theorem of Grifone.