Isometries, rigidity and universal covers By Benson Farb and Shmuel Weinberger* 1. Introduction The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers f M have a nontrivial amount of symmetry. By this we mean that Isom( f M ) is not discrete. By the well-known theorem of Myers-Steenrod [MS], this condition is equivalent to [Isom( f M ) : 1 (M )] = 1. Also note that if any cover of M has a nondiscrete isometry group, then so does its universal cover f M . Our description of such M is given in Theorem 1.2 below. The proof of this theorem uses methods from Lie theory, harmonic maps, large-scale geometry, and the homological theory of transformation groups. The condition that f M have nondiscrete isometry group appears in a wide variety of problems in geometry. Since Theorem 1.2 provides a taxonomy of such M , it can be used to reduce many general problems to veri cations of speci c examples. Actually, it is not always Theorem 1.2 which is applied di-rectly, but the main subresults from its proof. After explaining in Section 1.1 the statement of Theorem 1.2, we give in Section 1.2 a number of such applica-tions. These range from new characterizations of locally symmetric manifolds, to the classi cation of contractible manifolds covering both compact and nite volume manifolds, to a new proof of the Nadel-Frankel Theorem in complex geometry. 1.1. Statement of the general theorem. The basic examples of closed, as-pherical, Riemannian manifolds whose universal covers have nondiscrete isom-etry groups are the locally homogeneous (Riemannian ) manifolds M , i.e. those M whose universal cover admits a transitive Lie group action whose isotropy subgroups are maximal compact. Of course one might also take a product of such a manifold with an arbitrary manifold. To nd nonhomogeneous examples which are not products, one can perform the following construction. *Both authors are supported in part by the NSF. 916 BENSON FARB AND SHMUEL WEINBERGER Example 1.1. Let F ! M ! B be any Riemannian ber bundle with the induced path metric on F locally homogeneous. Let f : B ! R + be any smooth function. Now at each point of M lying over b, rescale the metric in the tangent space TMb = TFb TBb by rescaling TFb by f (b). Almost any f gives a metric on M with dim(Isom( f M )) > 0 but with f M not homogeneous, indeed with each Isom( f M )-orbit a ber. This construction can be further extended by scaling bers using any smooth map from B to the moduli space of locally homogeneous metrics on F ; this moduli space is large for example when F is an n-dimensional torus. Hence we see that there are many closed, aspherical, Riemannian mani-folds whose universal covers admit a nontransitive action of a positive-dimensional Lie group. The following general result says that the examples described above exhaust all the possibilities for such manifolds. Before stating the general result, we need some terminology. A Rieman-nian orbifold B is a smooth orbifold where the local charts are modelled on quotients V=G, where G is a nite group and V is a linear G-representation endowed with some G-invariant Riemannian metric. The orbifold B is good if it is the quotient of V by a properly discontinuous group action. A Riemannian orbibundle is a smooth map M ! B from a Riemannian manifold to a Riemannian orbifold locally modelled on the quotient map p : V G F ! V=G, where F is a xed smooth manifold with smooth G-action, and where V F has a G-invariant Riemannian metric such that projection to V is an orthogonal projection on each tangent space. Note that in this de nition, the induced metric on the bers of a Riemannian orbibundle may vary, and so a Riemannian orbibundle is not a ber bundle structure in the Riemannian category. Theorem 1.2. Let M be a closed, aspherical Riemannian manifold. Then either Isom( f M ) is discrete, or M is isometric to an orbibundle (1.1) F ! M ! B where: B is a good Riemannian orbifold, and Isom( e B) is discrete. Each ber F , endowed with the induced metric, is isometric to a closed, aspherical, locally homogeneous Riemannian n-manifold, n > 0. 1 Note that B is allowed to be a single point. 1 Recall that a manifold F is locally homogeneous if its universal cover is isometric to G=K, where G is a Lie group, K is a maximal compact subgroup, and G=K is endowed with a left G-invariant, K bi-invariant metric.