Improving the metric in an open manifold with nonnegative curvature

Abstract

The soul theorem states that any open Riemannian manifold (Mg) 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.