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 $$\pi : M \rightarrow M/\mathsf {G}$$ 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.