Ricci tensor of real hypersurfaces

Mayuko Kon

Vol. 281 (2016), No. 1, 103–123

DOI: 10.2140/pjm.2016.281.103

Abstract

Let $M$ be a real hypersurface of a complex space form $M^n\left(c\right)$, $c\ne 0$, and suppose that the structure vector field $\xi$ is an eigen vector field of the Ricci tensor $S$, which satisfies $S\xi =\beta \xi$ where $\beta$ is a function. We show that if $\left({\nabla }_{X}S\right)Y$ is proportional to $\xi$ for any vector fields $X$ and $Y$ orthogonal to $\xi$, then $M$ is a Hopf hypersurface, and if it is perpendicular to $\xi$, then $M$ is a ruled real hypersurface.

Keywords

real hypersurface, Ricci tensor

Mathematical Subject Classification 2010

Primary: 53C40

Secondary: 53C55, 53C25

Milestones

Received: 7 May 2015

Revised: 21 July 2015

Accepted: 24 July 2015

Published: 9 February 2016