$\eta$-*Einstein Hopf Real Hypersurfaces in the Complex Space Forms

Abstract

Einstein's metrics and their generalizations have attracted the attention of Mathematicians due to their applications in physics and other natural sciences. The generalization of Einstein metrics is Ricci solitons, $\eta$-Einstein metrics, pseudo-Einstein metrics, and Miao-Tam critical metrics. Given the established non-existence of Einstein real hypersurfaces in a non-flat complex space form $\hat{M}_n(c)$ \cite{2, 4}, motivated our investigation into the properties of $\eta$-*Einstein real hypersurface in $\hat{M}_n(c)$.

In this paper, we examine the $\eta$-*Einstein Hopf real hypersurface in the complex space form. We prove that there exist $\eta$-*Einstein Hopf real hypersurfaces.