On Pseudo-Einstein Real Hypersurfaces

TL;DR

This paper characterizes pseudo-Einstein real hypersurfaces in complex space forms by showing the Ricci tensor's proportionality to the metric on the holomorphic distribution, establishing a precise geometric condition.

Contribution

It provides a necessary and sufficient condition for a real hypersurface to be pseudo-Einstein based on Ricci tensor properties in complex space forms.

Findings

Ricci tensor is proportional to the metric on the holomorphic distribution

Characterization of pseudo-Einstein hypersurfaces in complex space forms

Condition holds if and only if the hypersurface is pseudo-Einstein

Abstract

Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\neq0$, $n\geq 3$. We show that the Ricci tensor $S$ of $M$ satisfies $S(X,Y)=ag(X,Y)$ for any vector fields $X$ and $Y$ on the holomorphic distribution, $a$ being a constant, if and only if $M$ is a pseudo-Einstein real hypersurface.