Highly curved orbit spaces

TL;DR

This paper investigates the sectional curvature bounds of orbit spaces from isometric actions on spheres, showing that for most actions the infimum is 1 and identifying cases with higher bounds.

Contribution

It establishes that the infimum of sectional curvatures is 1 for most actions and characterizes when it exceeds 1, refining previous curvature bounds.

Findings

Infimum of sectional curvatures is 1 for most actions.

Identifies cases where the curvature bound exceeds 1.

Refines understanding of curvature bounds in orbit spaces.

Abstract

It is known that the infimum of the sectional curvatures (on the regular part) of orbit spaces of isometric actions on unit spheres in bounded above by $4$. We show that the infimum is $1$ for "most" actions, and determine the cases in which it is bigger than $1$.