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 π : M → 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 π is in fact C 2 , thus allowing us the use of O'Neill formulas in the study of M . For the proof, we study

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... efully how the conjugate locus of S behaves in regard to the fibers of π. As applications, we study souls with totally geodesic Bieberbach submanifolds, and also obtain some rigidity results concerning the distribution of the rays of M . Proof. We just need to check that if E and F are smooth vector fields, thenĀ E H F V andT F V E H are continuous. So assume we have some sequence p i → p. For J i (t), J(t) the holonomy Jacobi fields corresponding to F V (p i ), E H (p i ) and F V (p), E H (p), we know already that J i (t) → J(t). Thus J i (0) → J (0), and hence Lemma 8.2. H and V are C 1 .