Real Hypersurfaces in Nearly Kaehler 6-Sphere

Sharief Deshmukh

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 2

Résumé

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} $.

Détail

Classification : 53C15, 53B25