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Jacobi fields and geodesic spheres

Published online by Cambridge University Press:  14 November 2011

L. Vanhecke  and
T. J. Willmore
L. Vanhecke
Affiliation:
Department Wiskunde, Faculteit der Wetenschappen, Katholieke Universiteit te Leuven, Belgium
T. J. Willmore
Affiliation:
Department of Mathematics, University of Durham
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Synopsis

This 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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 82 , Issue 3-4 , 1979 , pp. 233 - 240
Copyright
Copyright © Royal Society of Edinburgh 1979

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