scholarly journals The Euler characteristic of hypersurfaces in space forms and applications to isoparametric hypersurfaces

2021 ◽  
Vol 312 (2) ◽  
pp. 259-277
Author(s):  
Rui Albuquerque
Keyword(s):  
Euler Characteristic ◽  
Space Forms ◽  
Isoparametric Hypersurfaces

Classifications of Isoparametric Hypersurfaces in Randers Space Forms

2020 ◽  
Vol 36 (9) ◽  
pp. 1049-1060
Author(s):  
Qun He ◽  
Pei Long Dong ◽  
Song Ting Yin
Keyword(s):  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Randers Space

scholarly journals A Class of Special Hypersurfaces in Real Space Forms

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Yan Zhao ◽  
Ximin Liu
Keyword(s):  
Real Space ◽  
Principal Curvatures ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Real Space Forms ◽  
Constant Principal Curvatures

We define the generalized golden- and product-shaped hypersurfaces in real space forms. A hypersurfaceMin real space formsRn+1,Sn+1, andHn+1is isoparametric if it has constant principal curvatures. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the generalized golden- and product-shaped hypersurfaces in real space forms.

Classification of the Blaschke Isoparametric Hypersurfaces in Lorentzian Space Forms

2015 ◽  
Vol 65 (3) ◽  
Author(s):  
Fengyun Zhang ◽  
Huafei Sun
Keyword(s):  
Conformal Transformation ◽  
Transformation Group ◽  
Second Fundamental Form ◽  
Conformal Metric ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Lorentzian Space ◽  
Blaschke Tensor ◽  
Lorentzian Space Forms

AbstractIn this paper, we study regular immersed hypersurfaces in Lorentzian space forms with a conformal metric, a conformal second fundamental form, the conformal Blaschke tensor and a conformal form, which are invariants under the conformal transformation group. We classify all the immersed hypersurfaces in Lorentzian space forms with two distinct constant Blaschke eigenvalues and vanishing conformal form.

scholarly journals A classification of Riemannian 3-manifolds with constant principal ricci curvaturesρ1= ρ2≠ ρ3

1993 ◽  
Vol 132 ◽  
pp. 1-36 ◽  
Author(s):  
Oldřich Kowalski
Keyword(s):  
Differential Geometry ◽  
Homogeneous Spaces ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Main Motivation ◽  
Locally Homogeneous ◽  
Riemannian Spaces

This paper has been motivated by various problems and results in differential geometry. The main motivation is the study of curvature homogeneous Riemannian spaces initiated in 1960 by I.M. Singer (see Section 9—Appendix for the precise definitions and references). Up to recently, only sporadic classes of examples have been known of curvature homogeneous spaces which are not locally homogeneous. For instance, isoparametric hypersurfaces in space forms give nice examples of nontrivial curvature homogeneous spaces (see [FKM]). To study the topography of curvature homogeneous spaces more systematically, it is natural to start with the dimension n = 3. The following results and problems have been particularly inspiring.

GOLDEN- AND PRODUCT-SHAPED HYPERSURFACES IN REAL SPACE FORMS

2013 ◽  
Vol 10 (04) ◽  
pp. 1320006 ◽  
Author(s):  
MIRCEA CRASMAREANU ◽  
CRISTINA-ELENA HREŢCANU ◽  
MARIAN-IOAN MUNTEANU
Keyword(s):  
Hyperbolic Space ◽  
Space Form ◽  
Shape Operator ◽  
Real Space ◽  
Clifford Torus ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Real Space Forms ◽  
Unit Hypersphere

We define two classes of hypersurfaces in real space forms, golden- and product-shaped, respectively, by imposing the shape operator to be of golden or product type. We obtain the whole families of above hypersurfaces, based on the classification of isoparametric hypersurfaces, as follows: in the golden case all are hyperspheres, a hyperbolic space and a generalized Clifford torus, while for the product case we obtain the unit hypersphere, the hyperplane, a hypersphere and its associated Clifford torus, respectively, according to the type of the ambient space form namely parabolic, hyperbolic or elliptic, respectively.

scholarly journals Classifications of 2-complexes whose finite fundamental group is that of a 3-manifold

1997 ◽  
Vol 40 (1) ◽  
pp. 69-84 ◽  
Author(s):  
F. Rudolf Beyl ◽  
M. Paul Latiolais ◽  
Nancy Waller
Keyword(s):  
Fundamental Group ◽  
Euler Characteristic ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Universal Cover ◽  
Homotopy Equivalent ◽  
Spherical Space ◽  
Space Forms ◽  
Simple Homotopy ◽  
Spherical Space Forms

We consider spines of spherical space forms; i.e., spines of closed oriented 3-manifolds whose universal cover is the 3-sphere. We give sufficient conditions for such spines to be homotopy or simple homotopy equivalent to 2-complexes with the same fundamental group G and minimal Euler characteristic 1. If the group ring ℤG satisfies stably-free cancellation, then any such 2-complex is homotopy equivalent to a spine of a 3-manifold. If K1(ℤG) is represented by units and K is homotopy equivalent to a spine X, then K and X are simple homotopy equivalent. We exhibit several infinite families of non-abelian groups G for which these conditions apply.

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.

scholarly journals Classification of product shaped hypersurfaces in Lorentz space forms

2017 ◽  
Vol 101 (115) ◽  
pp. 223-230
Author(s):  
Dan Yang ◽  
Hao Le ◽  
Bingren Chen
Keyword(s):  
Lorentz Space ◽  
Shape Operator ◽  
Product Type ◽  
De Sitter ◽  
Space Forms ◽  
Isoparametric Hypersurfaces

We define the product shaped hypersurfaces in Lorentz space forms by imposing the shape operator to be product type. Based on the classification of the isoparametric hypersurfaces, we obtain the whole families of the product shaped hypersurfaces in Minkowski, de Sitter and anti-de Sitter spaces.

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