Geodesics on tangent bundles with horizontal Sasaki gradient metric

Let (Mm, g) be an m-dimensional Riemannian manifold and TM be its tangent bundle. In this paper, we introduce a new class of natural metrics with respect to g non-rigid on TM, called the horizontal Sasaki gradient metric. we investigate geodesics of the horizontal Sasaki gradient metric. Afterward establish a necessary and sufficient conditions under which a curve be a geodesic respect