The smoothness of Riemannian submersions with nonnegative sectional curvature

Abstract

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 $Ψ: M^n \to \Cal S$ must give rise to a $C^{\infty }$-smooth Riemannian submersion.