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

Abstract

In this paper, we introduce a vertical generalized Berger type deformed Sasaki metric on the tangent bundle $TM$ over an anti-paraK\"{a}hler manifold as a new natural metric. Firstly, we investigate the Levi-Civita connection of this metric and then we calculate all forms of the Riemannian curvature tensors. Also, we present some results concerning curvature properties. Finally, we study the geometry of $\varphi $-unit tangent bundle equipped with the vertical generalized Berger type deformed Sasaki metric.