Unit tangent sphere bundles with the Kaluza-Klein metric satisfying some commutative conditions

Abstract

Let $(M,g)$ be an $n-$dimensional Riemannian manifold and $T_{1}M$ its tangent sphere bundle with the contact metric structure $(\tilde{G},\eta ,\phi ,\xi )$, where $\tilde{G}$ is the Kaluza-Klein metric. Let $\tilde{S}$ be the Ricci operator and $h$ be the structural operator on $T_{1}M$. In this paper, we find some conditions for the relations $\tilde{S}h=h\tilde{S}$ and $\tilde{S}\phi h=\phi h\tilde{S}$ to be satisfied.