Local and Global Homogeneity for Manifolds of Positive Curvature

Joseph A. Wolf

Published 2019-06-15Version 1

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 of positive curvature and (ii) showing that in most cases the normality condition can be dropped.