Nonnegatively curved quotient spaces with boundary

TL;DR

This paper investigates the structure of nonnegatively curved quotient spaces with boundary, constructing specific submanifolds and applying the results to show rational ellipticity of certain torus manifolds with nonnegative curvature.

Contribution

It introduces a new soul construction for quotient spaces with boundary and applies it to prove rational ellipticity for simply connected torus manifolds with nonnegative curvature.

Findings

Constructed a smooth submanifold via soul construction

Diffeomorphic relation between manifold minus boundary preimage and normal bundle

Proved rational ellipticity for certain torus manifolds

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.