THE SMOOTHNESS OF RIEMANNIAN SUBMERSIONS WITH NON-NEGATIVE SECTIONAL CURVATURE

Abstract

Let Mn be a complete, non-compact and C∞-smooth Riemannian manifold with non-negative sectional curvature. Suppose that [Formula: see text] is a soul of Mn given by the fundamental theory of Cheeger and Gromoll, and suppose that [Formula: see text] is a distance non-increasing retraction from the whole manifold to the soul (e.g. the retraction given by Sharafutdinov). Then we show that the retraction Ψ above must give rise to a C∞-smooth Riemannian submersion from Mn to the soul [Formula: see text]. Moreover, we derive a new flat strip theorem associated with the Cheeger–Gromoll convex exhaustion for the manifold above.