Improving the metric in an open manifold with nonnegative curvature

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.