The Riemannian Metric on Tangent Bundle

Abstract

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM
have well-known physical interpretations and have been studied by physicists and geometricians. In this paper we define a
Riemannian metric g~ on TM, which is in some senses more general than other metrics previously defined on TM and also study the
conformal vector fields on (TM, g~) that every complete conformal vector filed on TM is homothetic and moreover every
horizontal or vertical conformal vector filed on TM is a killing vector.