The Smoothness of Riemannian Submersions with Nonnegative Sectional Curvature

In this article, we study the smoothness of Riemannian submersions for open manifolds with non-negative sectional curvature. Suppose thatM is a C-smooth, complete and non-compact Riemannian manifold with nonnegative sectional curvature. Cheeger-Gromoll [ChG] established a fundamental theory for such a manifold. Among other things, they showed that M admits a totally convex exhaustion {Ωu}u≥0 of M, where Ω0 = S is a totally geodesic and compact submanifold without boundary. Furthermore, M is diffeomorphic to the normal vector bundle of the soul S. Sharafutdinov found that there exists a distance non-increasing retraction Ψ : M → S from the open manifold M of non-negative sectional curvature to its soul, (cf. [Sh], [Y2]). Perelman [Per] further showed that such a map Ψ is indeed a C-smooth Riemannian submersion. Furthermore, Ψ[Expq(t~v)] = q for any q ∈ S and ~v⊥Tq(S). Therefore, the fiber Fq = Ψ(q) is a k-dimensional submanifold, which is C-smooth almost everywhere, where k = dim(Mn)− dim(S) > 0. Guijarro [Gu] proved that the fiber Fq is indeed a C -smooth submanifold for each q ∈ S. In this paper, we prove that the fibres are C-smooth. Theorem 1. Let M be a complete, non-compact and C-smooth Riemannian manifold with nonnegative sectional curvature. Suppose S is a soul of M. Then any distance non-increasing retraction Ψ : M → S must give rise to a C-smooth Riemannian submersion with the property that