Some Notes on Geodesics of Vertical Rescaled Berger Deformation Metric in Tangent Bundle

Abstract

In this paper, we study the geodesics on the tangent bundle $TM$ with respect to the vertical rescaled Berger deformation metric over an anti-paraK\"{a}hler manifold $(M, \varphi, g)$. In this case, we establish the necessary and sufficient conditions under which a curve be geodesic with respect to this. Finally, we also present certain examples of geodesic.

On the Geometry of Tangent Bundle and Unit Tangent Bundle with Deformed-Sasaki Metric

“…The deformations of the Sasaki metric on the tangent bundle are not limited to those mentioned above. Also, we refer to [1,3,10,15,16,17].…”

A study on the tangent bundle with the vertical generalized Berger type deformed Sasaki metric

“…In this direction, some authors defined and studied some natural metrics which are called as Cheeger-Gromoll metric [7,8,12] or Kaluza-Klein metric [3] or Berger type deformed Sasaki metric [4,13] or more generally g-natural metrics [1,2]. For deformations of the Sasaki metric, we also refer to [4,5,[16][17][18][19][20][21][22].…”

Geodesics on tangent bundles with horizontal Sasaki gradient metric

“…Yampolsky [25] also did the same studies on the tangent bundle and on unit the tangent bundle with the Berger-type deformed Sasaki metric over Kählerian manifold, in the cases of locally symmetric base manifold and of the constant holomorphic curvature base manifold. Also, we refer to [4,17,19,29].…”