scholarly journals Topological rigidity for closed hypersurfaces of elliptic space forms

2019 ◽  
Vol 62 (4) ◽  
pp. 1063-1072 ◽  
Author(s):  
Eduardo Rosinato Longa ◽  
Jaime Bruck Ripoll
Keyword(s):  
Space Form ◽  
Gauss Map ◽  
Euclidean Sphere ◽  
Spherical Image ◽  
Principal Curvatures ◽  
Space Forms ◽  
Rigidity Result ◽  
Absolute Value ◽  
Elliptic Space ◽  
Closed Hypersurfaces

AbstractWe prove a topological rigidity theorem for closed hypersurfaces of the Euclidean sphere and of an elliptic space form. It asserts that, under a lower bound hypothesis on the absolute value of the principal curvatures, the hypersurface is diffeomorphic to a sphere or to a quotient of a sphere by a group action. We also prove another topological rigidity result for hypersurfaces of the sphere that involves the spherical image of its usual Gauss map.

Remarks on η-parallel real hypersurfaces in ℂP2 and ℂH2

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].

scholarly journals Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chao Yang ◽  
Jiancheng Liu
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

In this paper, we show that biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space form Nsn+1c with constant sectional curvature c and index s have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces Mr2k−1 in even-dimensional pseudo-Euclidean space Es2k, Ms−12k−1 in even-dimensional de Sitter space Ss2kcc>0, and Ms2k−1 in even-dimensional anti-de Sitter space ℍs2kcc<0 are minimal.

Biharmonic hypersurfaces in pseudo-Riemannian space forms

2016 ◽  
Vol 13 (07) ◽  
pp. 1650094 ◽  
Author(s):  
Dan Yang ◽  
Yu Fu
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Constant Sectional Curvature ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

Let [Formula: see text] be a nondegenerate biharmonic pseudo-Riemannian hypersurface in a pseudo-Riemannian space form [Formula: see text] with constant sectional curvature [Formula: see text]. We show that [Formula: see text] has constant mean curvature provided that it has three distinct principal curvatures and the Weingarten operator can be diagonalizable.

scholarly journals On biconservative hypersurfaces in pseudo-Riemannian space forms and their Gauss map

2018 ◽  
Vol 103 (117) ◽  
pp. 223-236 ◽  
Author(s):  
Nurettin Turgay
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Space Form ◽  
Gauss Map ◽  
Space Forms ◽  
Geometrical Properties ◽  
Riemannian Space Forms ◽  
Biconservative Hypersurfaces

We first present a survey about recent results on biconservative hypersurfaces in the Minkowski space E4 1, pseudo-Euclidean space E5 2 and Rieamnnian space-form H4. Then we obtain some geometrical properties of these hypersurface families concerning their mean curvature and Gauss map.

Biharmonic hypersurfaces in 5-dimensional non-flat space forms

2019 ◽  
Vol 19 (2) ◽  
pp. 235-250
Author(s):  
Ram Shankar Gupta ◽  
Deepika ◽  
A. Sharfuddin
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Flat Space ◽  
Higher Order ◽  
Principal Curvatures ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Higher Order Mean Curvature

Abstract We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M5(c) is of constant mean curvature. In particular, every such biharmonic hypersurface in 𝕊5(1) has constant mean curvature. There exist no such compact proper biharmonic isoparametric hypersurfaces M in 𝕊5(1) with four distinct principal curvatures. Moreover, there exist no proper biharmonic hypersurfaces in hyperbolic space ℍ5 or in E5 having constant higher order mean curvature Hr for r > 2.

Triharmonic Riemannian submersions from 3-dimensional space forms

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tomoya Miura ◽  
Shun Maeta
Keyword(s):  
Existence Theorem ◽  
Space Form ◽  
Dimensional Space ◽  
Riemannian Submersion ◽  
Partial Answer ◽  
Riemannian Submersions ◽  
Space Forms ◽  
3 Dimensional

Abstract We show that any triharmonic Riemannian submersion from a 3-dimensional space form into a surface is harmonic. This is an affirmative partial answer to the submersion version of the generalized Chen conjecture. Moreover, a non-existence theorem for f -biharmonic Riemannian submersions is also presented.

Local rigidity of Anosov higher-rank lattice actions

1998 ◽  
Vol 18 (3) ◽  
pp. 687-702 ◽  
Author(s):  
NANTIAN QIAN ◽  
CHENGBO YUE
Keyword(s):  
Abelian Group ◽  
Lyapunov Functional ◽  
Rigidity Result ◽  
Absolute Value ◽  
Local Rigidity ◽  
Higher Rank ◽  
Standard Action ◽  
Lattice Actions

Let $\rho_0$ be the standard action of a higher-rank lattice $\Gamma$ on a torus by automorphisms induced by a homomorphism $\pi_0:\Gamma\to SL(n,{\Bbb Z})$. Assume that there exists an abelian group ${\cal A}\subset \Gamma$ such that $\pi_0({\cal A})$ satisfies the following conditions: (1) ${\cal A}$ is ${\Bbb R}$-diagonalizable; (2) there exists an element $a\in {\cal A}$, such that none of its eigenvalues $\lambda_1,\dots,\lambda_n$ has unit absolute value, and for all $i,j,k=1,\dots,n$, $|\lambda_i\lambda_j|\neq|\lambda_k|$; (3) for each Lyapunov functional $\chi_i$, there exist finitely many elements $a_j\in {\cal A}$ such that $E_{\chi_i}=\cap_{j} E^u(a_j)$ (see \S1 for definitions). Then $\rho_0$ is locally rigid. This local rigidity result differs from earlier ones in that it does not require a certain one-dimensionality condition.

scholarly journals Lightlike Hypersurfaces of Indefinite Generalized Sasakian Space Forms

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Dae Ho Jin
Keyword(s):  
Vector Field ◽  
Space Form ◽  
Sasakian Manifold ◽  
Sasakian Space Form ◽  
Space Forms ◽  
Sasakian Structure ◽  
Sasakian Space Forms ◽  
Generalized Sasakian Space Form ◽  
Characterization Theorems ◽  
Lightlike Hypersurfaces

We study lightlike hypersurfacesMof an indefinite generalized Sasakian space formM-(f1,f2,f3), with indefinite trans-Sasakian structure of type(α,β), subject to the condition that the structure vector field ofM-is tangent toM. First we study the general theory for lightlike hypersurfaces of indefinite trans-Sasakian manifold of type(α,β). Next we prove several characterization theorems for lightlike hypersurfaces of an indefinite generalized Sasakian space form.

scholarly journals A basic inequality for submanifolds in a cosymplectic space form

2003 ◽  
Vol 2003 (9) ◽  
pp. 539-547 ◽  
Author(s):  
Jeong-Sik Kim ◽  
Jaedong Choi
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Sectional Curvature ◽  
Mean Curvature ◽  
Space Form ◽  
The Other ◽  
Space Forms ◽  
Basic Inequality

For submanifolds tangent to the structure vector field in cosymplectic space forms, we establish a basic inequality between the main intrinsic invariants of the submanifold, namely, its sectional curvature and scalar curvature on one side; and its main extrinsic invariant, namely, squared mean curvature on the other side. Some applications, including inequalities between the intrinsic invariantδMand the squared mean curvature, are given. The equality cases are also discussed.

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

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1151 ◽  
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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