Survey on the Biharmonic Hypersurfaces in Terms of the Induced Metric of Tensor Ricci

TL;DR

This survey explores biharmonic hypersurfaces within Sasakian space forms, focusing on conditions for their existence and properties related to the induced Ricci tensor metric, highlighting minimality and non-existence results.

Contribution

It provides necessary and sufficient conditions for biharmonic hypersurfaces in this setting and clarifies when they are minimal or do not exist.

Findings

Biharmonic Hopf hypersurfaces are minimal when the gradient of mean curvature aligns with structural vector fields.

No biharmonic Hopf hypersurfaces exist when the gradient of mean curvature is a principal direction.

The paper establishes conditions for the existence of biharmonic hypersurfaces in Sasakian space forms.

Abstract

In this article, we study the biharmonic hypersurfaces in the Sasakian space form with the induced metric of tensor Ricci. We find the existence necessary and sufficient condition of the biharmonic hypersurfaces there. We show that the biharmonic Hopf hypersurfaces are minimal where gradient of the mean curvature is in direction of the structural vector fields. Furthermore, we prove that does not exist any biharmonic Hopf hypersurfaces when the gradient of the mean curvature is a principal direction.