We give a classification of many closed Riemannian manifolds M whose universal cover M possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that Isom( 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

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... for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.