Quasi-Einstein Hypersurfaces of Complex Space Forms

Xiaomin Chen,Xuehui Cui,Guozhen Lu

doi:10.1155/2020/8891658

Abstract

Based on a well-known fact that there are no Einstein hypersurfaces in a nonflat complex space form, in this article, we study the quasi-Einstein condition, which is a generalization of an Einstein metric, on the real hypersurface of a nonflat complex space form. For the real hypersurface with quasi-Einstein metric of a complex Euclidean space, we also give a classification. Since a gradient Ricci soliton is a special quasi-Einstein metric, our results improve some conclusions of Cho and Kimura.

Highlights

Denote by M~ n the complex space form, i.e., a complex n-dimensional Kähler manifold with constant holomorphic sectional curvature c
Since there are no Einstein real hypersurfaces in M~ nðcÞ ([1, 2]), a natural question is whether there is a generalization of an Einstein metric in the real hypersurface of M~ nðcÞ
Cho and Kimura [4, 5] proved that a Hopf hypersurface and a non-Hopf hypersurface in a nonflat complex space form do not admit a gradient Ricci soliton

Summary

Introduction

Denote by M~ n the complex space form, i.e., a complex n-dimensional Kähler manifold with constant holomorphic sectional curvature c. Cho and Kimura [4, 5] proved that a Hopf hypersurface and a non-Hopf hypersurface in a nonflat complex space form do not admit a gradient Ricci soliton. Advances in Mathematical Physics with m = ∞ From this observation, we are inspired to improve the results of [4] and study the quasi-Einstein condition for the real hypersurface of a complex space form. Let M2n−1 be a complete contact hypersurface of complex Euclidean space Cn. If M admits a quasi-Einstein metric, M is a sphere S2n−1 or a generalized cylinder Rn × Sn−1. Let M2n−1 be a complete real hypersurface with Aξ = 0 of complex Euclidean space Cn. If M admits a nonsteady quasi-Einstein metric, it is a hypersphere, hyperplane, or developable hypersurface. In Proof of Theorem 1 and Proof of Theorem 2, we give, respectively, the proofs of Theorem 1 and Theorem 2, and the real hypersurface with a quasiEinstein metric of complex Euclidean spaces is presented in Proofs of Theorem 4 and Corollary 5

Some Basic Concepts and Related Results

Proof of Theorem 1

Proof of Theorem 2

Proofs of Theorem 4 and Corollary 5