Local and Global Homogeneity for Manifolds that admit a Positive Curvature Metric

TL;DR

This paper investigates the conditions under which quotients of homogeneous Riemannian manifolds remain homogeneous, providing evidence for the Homogeneity Conjecture especially in cases with positive curvature and relaxing normality assumptions.

Contribution

It verifies the Homogeneity Conjecture for certain positively curved normal homogeneous manifolds and shows that normality can often be omitted in low-dimensional cases.

Findings

Confirmed the conjecture for positively curved normal homogeneous manifolds.

Demonstrated that normality condition can be dropped in most low-dimensional cases.

Provided new insights into the structure of homogeneous Riemannian quotients.

Abstract

In this note we study globally homogeneous Riemannian quotients $\Gamma\backslash (M,ds^2)$ of homogeneous Riemannian manifolds $(M,ds^2)$. 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 $(M,ds^2)$. We provide further evidence for that conjecture by (i) verifying it for normal homogeneous Riemannian manifolds that also admit an invariant Riemannian metric of strictly positive sectional curvature and (ii) showing that in most (three or less) cases the normality condition can be dropped.