Riemannian manifolds with local symmetry

TL;DR

This paper classifies closed Riemannian manifolds whose universal covers have significant symmetry, showing they often form fiber bundles over locally homogeneous spaces, extending previous work on symmetric and aspherical manifolds.

Contribution

It provides a new classification of manifolds with highly symmetric universal covers, generalizing earlier results and characterizing manifolds with both compact and finite volume quotients.

Findings

Manifolds with noncompact isometry components are often fiber bundles over homogeneous spaces.

The work extends Eberlein and Farb-Weinberger's results to broader classes of manifolds.

Characterization of simply-connected manifolds with both compact and finite volume quotients.

Abstract

We give a classification of many closed Riemannian manifolds M whose universal cover possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds $M$ such that Isom$(\widetilde{M})$ has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for nonpositively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.