Properties Of The Complete Lift Of Riemannian Connection For Flat Manifolds

Abstract

Here, we deals with a special lift $\tilde{g}$ of a Riemannian metric $g$ on a manifold $M$ to the tangent

bundle $TM$ of $M$. This lift is defined as a linear combination of certain well-known lifts of $g$. The main

results of the paper are proved under the condition that the Riemannian manifold $(M,g)$ is flat, in fact

the Riemannian connection of the metric $\tilde{g}$ coincides with the complete lift of the Riemannian

connection of the metric $g$.

In addition, the main objectives of this study is to find the

necessary and sufficient conditions such that some of the lift vector fields with this general metric

to be $\emph{parallel}$.