Derivatives of the Operator $$\phi A-A\phi $$ϕA-Aϕ on a Real Hypersurface in Non-flat Complex Space Forms

Abstract Let M be a real hypersurface in nonflat complex space forms of complex dimension two. In this paper, we prove that the shape operator of M is transversally Killing with respect to the generalized Tanaka-Webster connection if and only if M is locally congruent to a type (A) or (B) real hypersurface. We also prove that shape operator of M commutes with Cho operator on holomorphic distribution if and only if M is locally congruent to a ruled real hypersurface.

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Quasi-Einstein Hypersurfaces of Complex Space Forms

Advances in Mathematical Physics ◽

10.1155/2020/8891658 ◽

2020 ◽

Vol 2020 ◽

pp. 1-9

Author(s):

Xiaomin Chen ◽

Xuehui Cui

Keyword(s):

Real Hypersurface ◽

Complex Space ◽

Space Form ◽

Ricci Soliton ◽

Complex Space Form ◽

Einstein Metric ◽

Space Forms ◽

The Real ◽

Complex Euclidean Space ◽

Complex Space Forms

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.

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𝔻-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms

Mathematica Slovaca ◽

10.1515/ms-2017-0402 ◽

2020 ◽

Vol 70 (4) ◽

pp. 903-908

Author(s):

Yaning Wang

Keyword(s):

Ricci Tensor ◽

Real Hypersurface ◽

Complex Space ◽

Short Note ◽

Three Dimensional ◽

Affirmative Answer ◽

Real Hypersurfaces ◽

Space Forms ◽

Nonflat Complex Space Forms ◽

Complex Space Forms

AbstractKaimakamis and Panagiotidou in [Taiwanese J. Math. 18(6) (2014), 1991–1998] proposed an open question: are there real hypersurfaces in nonflat complex space forms whose ∗-Ricci tensor satisfies the condition of 𝔻-parallelism? In this short note, we present an affirmative answer and prove that a three-dimensional real hypersurface in a nonflat complex space form has 𝔻-parallel ∗-Ricci tensor if and only if it is locally congruent to either a geodesic hypersphere of radius r in ℂ H2(c) with $\begin{array}{} \displaystyle \tanh(\frac{\sqrt{|c|}}{2}r) = \frac{1}{2} \end{array}$ or a ruled real hypersurface.

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A characterization of ruled hypersurfaces in complex space forms in terms of the Lie derivative of shape operator

AIMS Mathematics ◽

10.3934/math.2021813 ◽

2021 ◽

Vol 6 (12) ◽

pp. 14054-14063

Author(s):

Wenjie Wang ◽

Keyword(s):

Real Hypersurface ◽

Complex Space ◽

Space Form ◽

Shape Operator ◽

Real Hypersurfaces ◽

Lie Derivative ◽

Space Forms ◽

Complex Dimension ◽

Complex Space Forms

<abstract><p>In this paper, it is proved that if a non-Hopf real hypersurface in a nonflat complex space form of complex dimension two satisfies Ki and Suh's condition (J. Korean Math. Soc., 32 (1995), 161–170), then it is locally congruent to a ruled hypersurface or a strongly $ 2 $-Hopf hypersurface. This extends Ki and Suh's theorem to real hypersurfaces of dimension greater than or equal to three.</p></abstract>

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