Reeb Flow Invariant Unit Tangent Sphere Bundles with the Kaluza-Klein Metric

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 $h=\frac{1}{% 2}\mathfrak{L}_{\xi }\phi $ be the structural operator and $l=\bar{R}(\cdot ,\xi )\xi $ be the characteristic Jacobi operator on $T_{1}M.\ $In this paper, we find some conditions for the Reeb flow invariancy of the $(0,2)-$ type tensors $L$ and $H$ defined by $L(\tilde{X},\tilde{Y})=g(l\tilde{X},\tilde{Y})$ and $H(\tilde{X},\tilde{Y})=g(h\tilde{X},\tilde{Y})$ for all vector fields $% \tilde{X}$ and $\tilde{Y}$ on $T_{1}M.$

Keywords

• tangent sphere bundle
• Kaluza-Klein metric
• Jacobi operator
• contact metric structure

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