Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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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.

The unit tangent sphere bundle whose characteristic Jacobi operator is pseudo-parallel

Jong Taek Cho and Sun Hyang Chun

Chonnam National University, Chosun University

Abstract

\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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