We study geometric properties of the metric projection r: M S of an open manifold M with nonnegative sectional curvature onto a soul S. ir is shown to be C? up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of S also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of M, and stu(dy the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of M. The resolution of the Soul conjecture of Cheeger and Gromoll by Perelinan showed that the structure of open manifolds with nonnegative sectional curvature is more rigid than expected. Orne of the key results in [14] is that the metric projection wF: M -> S which maps a point p in M to the point 7r(p) in S that is closest to p is a Riemannian submersioni. Perelman also observed that this map is at least of class C1, and later it was shown in [9] that 7w is at least C2, and C? at almnost every point. The existence of such a Riemannian submersion combined with the restriction on the sign of the curvature suggests that the geometry of M inust be special. The purpose of this paper is to illustrate this in several different directions. In fact, we show that most natural geometric objects classically used in the study of these spaces are intrinsically related to the structure of the map -w. The paper is essentially structured as follows: After introducing notation and recalling some basic results in sectioni 1, we establish in section 2 the existence of maps similar to -w for any submanifold of M homologous and isometric to the soul, and use this to prove the existence of convex tubular neighborhoods for them. In particular, this also proves: Theorem 2.5. The unit normal bundle vti(S) admits a metric with nonnegative sectional curvature. As a consequence, open manifolds with nonnegative curvature provide examples of compact manifolds with the same lower curvatuire bound (see also [81). Theorem 2.5 also implies that nontrivial plane bundles over a Bieberbach maanifold do not admit nonnegatively curved metrics, thereby providing a shorter proof of the main result in [13]. We are grateful to the referee for pointing out this fact to us. Properties such as these are established via stanidard submersion techniques, but the fact that 7r is not known to be smooth everywhere prevents us fromi using these techniques to study the geometry of M far away from the soul. One way around Received by the editors August 18, 1997. 1991 Mathematics Subject Classification. Primary 53C20.