Local and Global Homogeneity for Three Obstinate Spheres

TL;DR

This paper completes the proof of the Homogeneity Conjecture for positively curved homogeneous spaces, including three challenging spheres, establishing conditions under which quotients remain homogeneous.

Contribution

It verifies the Homogeneity Conjecture for all cases in positive curvature, including three previously unresolved spheres, using new methods.

Findings

The Homogeneity Conjecture holds for all positively curved homogeneous spaces.

The conjecture is verified for three specific spheres that resisted earlier approaches.

The results confirm that quotients are homogeneous if and only if the original space is homogeneous and elements have constant displacement.

Abstract

In this note we complete a study of globally homogeneous Riemannian quotients $\Gamma\backslash (M,ds^2)$ in positive curvature. Specifically, $M$ is a homogeneous space $G/H$ that admits a $G$-invariant Riemannian metric of strictly positive sectional curvature, and $ds^2$ is a $G$--invariant Riemannian metric on $M$, not necessarily normal and not necessarily positively curved. The Homogeneity Conjecture is that $\Gamma\backslash (M,ds^2)$ is (globally) homogeneous if and only if $(M,ds^2)$ is homogeneous and every $\gamma \in \Gamma$ is of constant displacement on $(M,ds^2)$. In an earlier paper we verified that conjecture for all homogeneous spaces that admit an invariant Riemannian metric of positive curvature -- with three exceptions, all odd dimensional spheres, which surprisingly did not yield to the earlier approaches. Here we develop some methods that let us verify the Homogeneity Conjecture for those three obstinate spheres. That completes verification of the Homogeneity Conjecture in positive curvature.