Improving the metric in an open manifold with nonnegative curvature

Guijarro, Luis

doi:10.1090/S0002-9939-98-04287-7

Improving the metric in an open manifold with nonnegative curvature
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by Luis Guijarro PDF

Proc. Amer. Math. Soc. 126 (1998), 1541-1545 Request permission

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:

$g’$ has nonnegative sectional curvature and soul $S$.
The normal exponential map of $S$ is a diffeomorphism.
$(M,g’)$ 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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