Geometry of Twisted Sasaki Metric

Abstract

Let $(M,g)$ be a $n$-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by $G^{f,h}$ and called twisted Sasaki matric on the tangent bundle $TM$. We studied the geometry of $(TM,G^{f,h})$ by giving a relationships of the curvatures, Einstein structure, scalar and sectional curvatures between $(TM,G^{f,h})$ and $(M,g)$.