Geodesics, Weighted Sasaki Metric and the Tangent Bundle

A geodesic on the tangent bundle of a Riemannian manifold M , equipped with the Weighted Sasaki metric, can be considered as a curve on M together with a vector field along it. Following the results of Sasaki, we investigated the curvature properties and geodesics of the tangent bundle with Weighted Sasaki metric on Riemannian manifolds. The purpose of the present article is to obtain sufficient conditions for the geodesics of the tangent bundle with Weighted Sasaki metric to be flat. 2000 MSC: 53C07, 53C22, 53C25