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.

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 CLASSIFICATION OF LAGRANGIAN SURFACES OF CONSTANT CURVATURE IN THE COMPLEX EUCLIDEAN PLANE

2005 ◽  
Vol 48 (2) ◽  
pp. 337-364 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Constant Curvature ◽  
Euclidean Plane ◽  
Lagrangian Submanifolds ◽  
Real Space ◽  
Point Of View ◽  
Space Forms ◽  
Lagrangian Surfaces ◽  
Real Space Forms ◽  
Geometric Point

AbstractOne of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in $\mathbb{C}^2$ can be obtained locally from the 19 families.

scholarly journals Classification of metallic shaped hypersurfaces in real space forms

2015 ◽  
Vol 39 ◽  
pp. 784-794 ◽  
Author(s):  
Cihan ÖZGÜR ◽  
Nihal YILMAZ ÖZGÜR
Keyword(s):  
Real Space ◽  
Space Forms ◽  
Real Space Forms

scholarly journals Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaces

2021 ◽  
Vol 2021 (779) ◽  
pp. 189-222
Author(s):  
José Carlos Díaz-Ramos ◽  
Miguel Domínguez-Vázquez ◽  
Alberto Rodríguez-Vázquez
Keyword(s):  
Symmetric Spaces ◽  
Hyperbolic Spaces ◽  
Orbit Equivalence ◽  
Principal Curvatures ◽  
Cohomogeneity One ◽  
Isoparametric Hypersurfaces ◽  
Rank One ◽  
Constant Principal Curvatures

Abstract We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces.

scholarly journals A new characterization of complete linear Weingarten hypersurfaces in real space forms

2013 ◽  
Vol 261 (1) ◽  
pp. 33-43 ◽  
Author(s):  
Cícero Aquino ◽  
Henrique de Lima ◽  
Marco Velásquez
Keyword(s):  
Real Space ◽  
Space Forms ◽  
Linear Weingarten Hypersurfaces ◽  
Real Space Forms
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