Some properties of geodesics and F-geodesics on tangent bundle with gradient Sasaki metric

Abstract

In this paper, we investigate the properties of geodesics and $F$-geodesics on the tangent bundle equipped with the gradient Sasaki metric. First, we establish necessary and sufficient conditions for a curve to be a geodesic. We then study the behavior of $F$-geodesics and $F$-planar curves on the tangent bundle with respect to the induced Levi-Civita connection. Our theoretical results are supported by explicit examples that illustrate the behavior of these curves.