Equivariant geometry and Kervaire spheres

Author:

Back Allen,Hsiang Wu-Yi

Abstract

The intrinsic geometry of metrics on the Kervaire sphere which are invariant under a large transformation group (cohomogeneity one) is studied. Invariant theory is used to describe the behavior of these metrics near the singular orbits. Nice expressions for the Ricci and sectional curvatures are obtained. The nonexistence of invariant metrics of positive sectional curvature is proven, and Cheeger’s construction of metrics of positive Ricci curvature is discussed.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference12 articles.

1. A. Back, M. do Carmo, and W. Y. Hsiang, On some fundamental equations of equivariant riemannian geometry, preprint.

2. Les variétés riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive;Berard-Bergery, L.;J. Math. Pures Appl. (9),1976

3. General position of equivariant maps;Bierstone, Edward;Trans. Amer. Math. Soc.,1977

4. Some examples of manifolds of nonnegative curvature;Cheeger, Jeff;J. Differential Geometry,1973

5. W. T. Hsiang and W. Y. Hsiang, On the construction of exotic and/or knotted minimal spheres in the standard Riemannian sphere by means of equivariant differential geometry, preprint.

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