Gradient pseudo-Ricci solitons of real hypersurfaces

TL;DR

This paper investigates gradient pseudo-Ricci solitons on real hypersurfaces in complex space forms, revealing specific conditions under which such solitons exist, especially in 3-dimensional ruled hypersurfaces of negatively curved complex space forms.

Contribution

It introduces the concept of gradient pseudo-Ricci solitons for real hypersurfaces and characterizes their existence in certain 3D ruled hypersurfaces in complex space forms.

Findings

3D ruled hypersurfaces in complex space forms with negative curvature admit non-trivial gradient pseudo-Ricci solitons.

The structure vector field being an eigenvector of the Ricci tensor is a key condition.

The study extends the understanding of pseudo-Einstein real hypersurfaces and their geometric properties.

Abstract

Let $M$ be a real hypersurface of a complex space form $M^n(c)$, $c\neq 0$. Suppose that the structure vector field $\xi$ of $M$ is an eigen vector field of the Ricci tensor $S$, $S\xi=\beta\xi$, $\beta$ being a function. We study on $M$, a gradient pseudo-Ricci soliton as an extended concept of Ricci soliton, closely related to pseudo-Einstein real hypersurfaces. We show that a $3$-dimensional ruled real hypersurface of $M^2(c), c<0$ admits a non-trivial gradient pseudo-Ricci soliton.