Bulletin

Processing math: 100%

Volume 36 • Number 2 • 2013

• Real Hypersurfaces in Nearly Kaehler 6-Sphere
Sharief Deshmukh

Abstract.
In this paper we characterize Hopf hypersurfaces in the nearly Kaehler

$6$ -Sphere

$S^{6}$ using some restrictions on the characteristic vector field

$\xi =-JN$ , where

$J$ is the almost complex structure on

$S^{6}$ and

$N$ is the unit normal to the hypersurface. It is shown that if the characteristic vector field

$\xi$ of a compact and connected real hypersurface

$M$ of the nearly Kaehler sphere

$S^{6}$ is harmonic and the Ricci curvature in the direction of

$\xi$ is non-negative, then

$M$ is a Hopf hypersurface and therefore congruent to either a totally geodesic hypersphere or a tube over almost complex curve on

$S^{6}$ . It is also observed that similar result holds if

$\xi$ is Jacobi-type vector field (a notion similar to Jacobi fields along geodesics). We also show that if a connected real hypersurface

$M$ is a Ricci soliton with potential vector field

$\xi$ , then

$M$ is congruent to an open piece of either a totally geodesic hypersphere or a tube over an almost complex curve in

$S^{6}$ .

2010 Mathematics Subject Classification: 53C15, 53B25

Full text: PDF