An open manifold $M$ with nonnegative sectional curvature contains a compact totally geodesic submanifold $S$ called the soul. In his solution of the Cheeger-Gromoll conjecture, G. Perelman showed that the metric projection $\pi:M\to S$ was a $C^1$ Riemannian submersion which coincided with a map previously constructed by V. Sharafutdinov.

In this paper we improve Perelman's result to show that $\pi$ is in fact $C^2$, thus allowing us the use of O'Neill formulas in the study of $M$. For the proof, we study carefully how the conjugate locus of $S$ behaves in regard to the fibers of $\pi$. As applications, we study souls with totally geodesic Bieberbach submanifolds, and also obtain some rigidity results concerning the distribution of the rays of $M$.