Conformal vector fields and totally umbilic hypersurfaces

Dong-Soo Kim; Seon-Bu Kim; Young Ho Kim; Seong-Hee Park

Bulletin of the
Korean Mathematical Society BKMS

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

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Bull. Korean Math. Soc. 2002; 39(4): 671-680

Printed December 1, 2002

Conformal vector fields and totally umbilic hypersurfaces

Dong-Soo Kim, Seon-Bu Kim, Young Ho Kim, and Seong-Hee Park

Chonnam National University, Chonnam National University, Kyungpook National University, Chonnam National University

Abstract

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Abstract

In this article, we show that if a semi-Riemannian space form carries a conformal vector field $V$ of which the tangential part $V^T$ on a connected hypersurface $M^n$ becomes a conformal vector field and the normal part $V^N$ on $M^n$ does not vanish identically, then $M^n$ is totally umbilic. Furthermore, we give a complete description of conformal vector fields on semi-Riemannian space forms.

Keywords: totally umbilic hypersurface, space form, conformal vector field

MSC numbers: 53B25, 53B30