On the Homogeneity Conjecture

Joseph A. Wolf

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arXiv · 2303.16365

On the Homogeneity Conjecture

Joseph A. Wolf

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Abstract

Consider a connected homogeneous Riemannian manifold $(M,ds^2)$ and a Riemannian covering $(M,ds^2) \to \Gamma \backslash (M,ds^2)$. If $\Gamma \backslash (M,ds^2)$ is homogeneous then every $\gamma \in \Gamma$ is an isometry of constant displacement. The Homogeneity Conjecture suggests the converse: if every $\gamma \in \Gamma$ is an isometry of constant displacement on $(M,ds^2)$ then $\Gamma \backslash (M,ds^2)$ is homogeneous. We survey the cases in which the Homogeneity Conjecture has been verified, including some new results, and suggest some related open problems.

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Joseph A. Wolf. 2023-03-29. On the Homogeneity Conjecture. https://arxiv.org/abs/2303.16365

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