Stability of geodesic spheres in $$\varvec{\mathbb {S}^{n+1}}$$ S n + 1 under constrained curvature flows

2016 ◽  
Vol 17 (4) ◽  
pp. 1209-1225
Author(s):  
David Hartley
Keyword(s):  
Geodesic Spheres

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.

Isometric reflections with respect to submanifolds and the Ricci operator of geodesic spheres

1989 ◽  
Vol 108 (2-3) ◽  
pp. 211-217 ◽  
Author(s):  
Philippe Tondeur ◽  
Lieven Vanhecke
Keyword(s):  
Ricci Operator ◽  
Geodesic Spheres

scholarly journals An integral formula for compact hypersurfaces in space forms and its applications

2003 ◽  
Vol 74 (2) ◽  
pp. 239-248 ◽  
Author(s):  
Luis J. Alías
Keyword(s):  
Scalar Curvature ◽  
Integral Formula ◽  
Flat Space ◽  
Gauss Map ◽  
Ambient Space ◽  
Geodesic Ball ◽  
Space Forms ◽  
Geodesic Spheres

AbstractIn this paper we establish an integral formula for compact hypersurfaces in non-flat space forms, and apply it to derive some interesting applications. In particular, we obtain a characterization of geodesic spheres in terms of a relationship between the scalar curvature of the hypersurface and the size of its Gauss map image. We also derive an inequality involving the average scalar curvature of the hypersurface and the radius of a geodesic ball in the ambient space containing the hypersurface, characterizing the geodesic spheres as those for which equality holds.

Symmetric-like Riemannian manifolds and geodesic symmetries

1995 ◽  
Vol 125 (2) ◽  
pp. 265-282 ◽  
Author(s):  
Jürgen Berndt ◽  
Friedbert Prüfer ◽  
Lieven Vanhecke
Keyword(s):  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Jacobi Operators ◽  
Geodesic Spheres ◽  
Shape Operators

We treat several classes of Riemannian manifolds whose shape operators of geodesic spheres or Jacobi operators share some properties with the ones on symmetric spaces.

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.

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