Divergence-preserving geodesic transformations

1998 ◽  
Vol 128 (6) ◽  
pp. 1309-1323
Author(s):  
E. García-Río ◽  
L. Vanhecke
Keyword(s):  
Principal Curvatures ◽  
Isoparametric Hypersurfaces ◽  
Hopf Hypersurfaces ◽  
Geodesic Spheres ◽  
Constant Principal Curvatures ◽  
Volume Preserving ◽  
Special Classes

We discuss divergence- and volume-preserving geodesic transformations with respect to submanifolds and in particular, with respect to hypersurfaces. We use these transformations to derive characterisations of special classes of hypersurfaces such as isoparametric hypersurfaces and Hopf hypersurfaces with constant principal curvatures. Furthermore, we consider divergence-preserving geodesic transformations with respect to geodesic spheres.

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.

HYPERSURFACES IN P2 AND H2 WITH TWO DISTINCT PRINCIPAL CURVATURES

2015 ◽  
Vol 58 (1) ◽  
pp. 137-152 ◽  
Author(s):  
THOMAS A. IVEY ◽  
PATRICK J. RYAN
Keyword(s):  
Parameter Family ◽  
Hopf Hypersurface ◽  
Principal Curvatures ◽  
Hopf Hypersurfaces ◽  
Constant Principal Curvatures ◽  
Two Parameter ◽  
Standard List

AbstractIt is known that hypersurfaces in ${\mathbb C}$Pn or ${\mathbb C}$Hn for which the number g of distinct principal curvatures satisfies g ≤ 2, must belong to a standard list of Hopf hypersurfaces with constant principal curvatures, provided that n ≥ 3. In this paper, we construct a two-parameter family of non-Hopf hypersurfaces in ${\mathbb C}$P2 and ${\mathbb C}$H2 with g=2 and show that every non-Hopf hypersurface with g=2 is locally of this form.

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.

Jacobi fields and geodesic spheres

1979 ◽  
Vol 82 (3-4) ◽  
pp. 233-240 ◽  
Author(s):  
L. Vanhecke ◽  
T. J. Willmore
Keyword(s):  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Holomorphic Sectional Curvature ◽  
Principal Curvatures ◽  
Jacobi Fields ◽  
Geodesic Spheres ◽  
Jacobi Vector ◽  
Nearly Kähler ◽  
Nearly Kähler Manifolds

SynopsisThis is a contribution to the general problem of determining the extent to which the geometry of a riemannian manifold is determined by properties of its geodesic spheres. In particular we show that total umbilicity of geodesic spheres determines riemannian manifolds of constant sectional curvature; quasi-umbilicity of geodesic spheres determines Kähler and nearly-Kähler manifolds of constant holomorphic sectional curvature; and the condition that geodesic spheres have only two different principal curvatures, one having multiplicity 3, determines manifolds locally isometric to the quaternionic projective spaces. The use of Jacobi vector fields leads to a unified treatment of these different cases.

SUBMANIFOLDS WITH CONSTANT PRINCIPAL CURVATURES AND NORMAL HOLONOMY GROUPS

1991 ◽  
Vol 02 (02) ◽  
pp. 167-175 ◽  
Author(s):  
E. HEINTZE ◽  
C. OLMOS ◽  
G. THORBERGSSON
Keyword(s):  
Euclidean Space ◽  
Weyl Group ◽  
Principal Curvatures ◽  
Normal Field ◽  
Slice Theorem ◽  
Holonomy Groups ◽  
Shape Operators ◽  
Isoparametric Submanifold ◽  
Constant Principal Curvatures

A submanifold has by definition constant principal curvatures if the eigenvalues of the shape operators Aξ are constant for any parallel normal field ξ along any curve. It is shown that a submanifold of Euclidean space has constant principal curvatures if and only if it is an isoparametric or a focal manifold of an isoparametric submanifold. Furthermore a "homogeneous slice theorem" is proved which says that the fibres of the projection of an isoparametric submanifold onto a full focal manifold are homogeneous isoparametric. To this end it is shown that any two points of an isoparametric submanifold can be connected by a piecewise differentiable curve whose pieces are tangent to one of the simultaneous eigenspaces Ei, i ∈ I, of the shape operators provided that the corresponding reflections generate the Weyl group of the isoparametric submanifold.

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