  Let $M^n$ be a complete, non-compact and $C^\infty$-smooth Riemannian
manifold with nonnegative sectional curvature. Suppose $\Cal S$ is a soul of
$M^n$. Then any distance non-increasing retraction $\Psi: M^n \to \Cal S$ must
give rise to a $C^\infty$-smooth Riemannian submersion.
