Actions on positively curved manifolds and boundary in the orbit space

TL;DR

This paper classifies actions of compact Lie groups on positively curved manifolds with boundary in the orbit space, introduces a new geometric invariant, and explores reductions of group representations.

Contribution

It provides a classification of quotients of spheres by simple Lie group actions with boundary and introduces a novel invariant for symmetric spaces.

Findings

Classified quotients of spheres with boundary under simple Lie group actions.

Identified representations of simple Lie groups with non-trivial reductions.

Introduced a new geometric invariant for symmetric spaces.

Abstract

We study isometric actions of compact Lie groups on complete orientable positively curved $n$-manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere by actions of compact simple Lie groups with non-empty boundary. We deduce from this the list of representations of compact simple Lie groups that admit non-trivial reductions. As a tool of special interest, we introduce a new geometric invariant of a compact symmetric space, namely, the minimal number of points in a "spanning set" of the space.