Quasi-Einstein Hypersurfaces of 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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On generalized complex space forms satisfying certain curvature conditions

Carpathian Mathematical Publications ◽

10.15330/cmp.8.2.284-294 ◽

2016 ◽

Vol 8 (2) ◽

pp. 284-294

Author(s):

M.M. Praveena ◽

C.S. Bagewadi

Keyword(s):

Scalar Curvature ◽

Curvature Tensor ◽

Complex Space ◽

Space Form ◽

Constant Scalar Curvature ◽

Ricci Soliton ◽

Space Forms ◽

Generalized Complex ◽

Curvature Tensors ◽

Complex Space Forms

We study Ricci soliton $(g,V,\lambda)$ of generalized complex space forms when the Riemannian, Bochner and $W_{2}$ curvature tensors satisfy certain curvature conditions like semi-symmetric, Einstein semi-symmetric, Ricci pseudo symmetric and Ricci generalized pseudo symmetric. In this study it is shown that shrinking, steady and expansion of the generalized complex space forms depends on the solenoidal property of vector $V$. Also we prove that generalized complex space form with conservative Bochner curvature tensor is constant scalar curvature.

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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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CERTAIN RESULTS OF REAL HYPERSURFACES IN A COMPLEX SPACE FORM

Glasgow Mathematical Journal ◽

10.1017/s0017089511000309 ◽

2011 ◽

Vol 54 (1) ◽

pp. 1-8 ◽

Cited By ~ 4

Author(s):

AMALENDU GHOSH

Keyword(s):

Real Hypersurface ◽

Complex Space ◽

Space Form ◽

Ricci Soliton ◽

Complex Space Form ◽

Real Hypersurfaces

AbstractFirst, we classify a real hypersurface of a non-flat complex space form with (i) semi-parallel T(=£ξg), and (ii) recurrent T. Next, we characterise a real hypersurface admitting the generalised η-Ricci soliton in a non-flat complex space form.

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A Characterization of Real Hypersurfaces in Complex Space Forms in Terms of the Ricci Tensor

Canadian Mathematical Bulletin ◽

10.4153/cmb-1997-031-5 ◽

1997 ◽

Vol 40 (3) ◽

pp. 257-265 ◽

Cited By ~ 2

Author(s):

Christos Baikoussis

Keyword(s):

Ricci Tensor ◽

Complex Space ◽

Space Form ◽

Complex Space Form ◽

Real Hypersurfaces ◽

Space Forms ◽

Complex Space Forms ◽

Orthogonal Distribution

AbstractWe study real hypersurfaces of a complex space form Mn(c), c ≠ 0 under certain conditions of the Ricci tensor on the orthogonal distribution T0.

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Ideal Lagrangian immersions in complex space forms

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004199004247 ◽

2000 ◽

Vol 128 (3) ◽

pp. 511-533 ◽

Cited By ~ 12

Author(s):

BANG-YEN CHEN

Keyword(s):

Riemannian Manifold ◽

Isometric Immersion ◽

Complex Space ◽

Space Form ◽

Lagrangian Submanifolds ◽

Ambient Space ◽

Complex Space Form ◽

Space Forms ◽

Complex Space Forms

Roughly speaking, an ideal immersion of a Riemannian manifold into a space form is an isometric immersion which produces the least possible amount of tension from the ambient space at each point of the submanifold. In this paper we study Lagrangian immersions in complex space forms which are ideal. We prove that all Lagrangian ideal immersions in a complex space form are minimal. We also determine ideal Lagrangian submanifolds in complex space forms.

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Some new results on real hypersurfaces with generalized Tanaka-Webster connection

Mathematica Slovaca ◽

10.1515/ms-2017-0255 ◽

2019 ◽

Vol 69 (3) ◽

pp. 665-674

Author(s):

Wenjie Wang ◽

Ximin Liu

Keyword(s):

Real Hypersurface ◽

Complex Space ◽

Shape Operator ◽

Real Hypersurfaces ◽

Space Forms ◽

Complex Dimension ◽

Nonflat Complex Space Forms ◽

Complex Space Forms ◽

Ruled Real Hypersurface

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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The First Fundamental Equation and Generalized Wintgen-Type Inequalities for Submanifolds in Generalized Space Forms

Mathematics ◽

10.3390/math7121151 ◽

2019 ◽

Vol 7 (12) ◽

pp. 1151 ◽

Cited By ~ 3

Author(s):

Mohd. Aquib ◽

Michel Nguiffo Boyom ◽

Mohammad Hasan Shahid ◽

Gabriel-Eduard Vîlcu

Keyword(s):

Complex Space ◽

Space Form ◽

Fundamental Equation ◽

Type Inequality ◽

Lagrangian Submanifold ◽

Space Forms ◽

Slant Submanifolds ◽

Sasakian Space Forms ◽

Generalized Complex ◽

Complex Space Forms

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces.

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Remarks on η-parallel real hypersurfaces in ℂP2 and ℂH2

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820500735 ◽

2020 ◽

Vol 17 (05) ◽

pp. 2050073

Author(s):

Yaning Wang

Keyword(s):

Complex Space ◽

Space Form ◽

Three Dimensional ◽

Shape Operator ◽

Complex Space Form ◽

Real Hypersurfaces ◽

Geodesic Sphere ◽

Principal Curvatures ◽

Space Forms ◽

Complex Dimension

Let [Formula: see text] be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that [Formula: see text] is [Formula: see text]-parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in [Formula: see text] or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in [Formula: see text]. Moreover, [Formula: see text]-parallel real hypersurfaces in [Formula: see text] and [Formula: see text] under some other conditions are classified and these results extend Suh’s in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo’s in [On characterizations of real hypersurfaces in a complex space form with [Formula: see text]-parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126].

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Certain inequalities for submanifolds in (K,μ)-contact space forms

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700039873 ◽

2001 ◽

Vol 64 (2) ◽

pp. 201-212 ◽

Cited By ~ 8

Author(s):

Kadri Arslan ◽

Ridvan Ezentas ◽

Ion Mihai ◽

Cengizhan Murathan ◽

Cihan Özgür

Keyword(s):

Ricci Curvature ◽

Mean Curvature ◽

Complex Space ◽

Space Form ◽

Space Forms ◽

Arbitrary Codimension ◽

Complex Space Forms ◽

Contact Space

Chen (1999) established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemanian space form with arbitrary codimension. Matsumoto (to appear) dealt with similar problems for sub-manifolds in complex space forms.In this article we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in (K, μ)-contact space forms.

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