Bull. Korean Math. Soc. 2016; 53(6): 1715-1723
Online first article September 22, 2016 Printed November 30, 2016
https://doi.org/10.4134/BKMS.b150897
Copyright © The Korean Mathematical Society.
Jong Taek Cho and Sun Hyang Chun
Chonnam National University, Chosun University
\noindent We study the characteristic Jacobi operator $\ell=\bar R(\cdot,\xi)\xi$ (along the Reeb flow $\xi$) on the unit tangent sphere bundle $T_1 M$ over a Riemannian manifold $(M^n,g)$. We prove that if $\ell$ is pseudo-parallel, i.e., $\bar R\cdot \ell=L \mathcal{Q}(\bar g,\ell)$, by a non-positive function $L$, then $M$ is locally flat. Moreover, when $L$ is a constant and $n\neq 16$, $M$ is of constant curvature $0$ or $1$.
Keywords: unit tangent sphere bundle, contact metric structure, characteristic Jacobi operator
MSC numbers: 53C15, 53C25, 53D10
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