Nonnegatively curved quotient spaces with boundary

Abstract

Let $M$ be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group $\mathsf{G}$ in a way that the quotient space $M/\mathsf{G}$ has nonempty boundary. Let $\pi :M\to M/\mathsf{G}$ denote the quotient map and $B$ be any boundary stratum of $M/\mathsf{G}$. Via a specific soul construction for $M/\mathsf{G}$ we construct a smooth closed submanifold $N$ of $M$ such that $M\setminus {\pi }^{-1}(B)$ is diffeomorphic to the normal bundle of $N$. As an application we show that a simply connected torus manifold admitting an invariant metric of nonnegative curvature is rationally elliptic.

25 pages