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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Improving the metric in an open manifold
with nonnegative curvature

Author: Luis Guijarro
Journal: Proc. Amer. Math. Soc. 126 (1998), 1541-1545
MSC (1991): Primary 53C20
MathSciNet review: 1443388
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Abstract: The soul theorem states that any open Riemannian manifold $(\!M\!,\!g\!)$ with nonnegative sectional curvature contains a totally geodesic compact submanifold $S$ such that $M$ is diffeomorphic to the normal bundle of $S$. In this paper we show how to modify $g$ into a new metric $g'$ so that:

  1. $g'$ has nonnegative sectional curvature and soul $S$.
  2. The normal exponential map of $S$ is a diffeomorphism.
  3. $(M,g')$ splits as a product outside of a compact set.
As a corollary we obtain that any such $M$ is diffeomorphic to the interior of a convex set in a compact manifold with nonnegative sectional curvature.

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Additional Information

Luis Guijarro
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104

Received by editor(s): October 25, 1996
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society