Let $(M,g)$ be a complete, connected, noncompact smooth Riemannian manifold whose sectional curvature satisfies $\operatorname{sec}_g(\sigma) \geq 0$ for every two-dimensional subspace $\sigma \subset T_pM$ and every $p \in M$. Then there exists a compact, connected, totally geodesic, totally convex embedded submanifold $S \subset M$, called a soul of $M$, such that the normal exponential map $\exp^\perp:\nu(S) \to M$ is a diffeomorphism. In particular, $M$ is diffeomorphic to the total space of the normal bundle $\nu(S)$.