Riemannian manifolds with local symmetry
We give a classification of many closed Riemannian manifolds whose universal cover possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds such that Isom 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.
A basic question in Riemannian geometry is to understand those manifolds whose metric is highly symmetric. As a measure of the degree of symmetry we can use the size of the isometry group. On the one hand manifolds with large isometry group, such as compact symmetric spaces, play important roles in many areas of mathematics, but on the other hand there are many closed manifolds equipped with very special geometric structures with small isometry groups. For example, if is closed and hyperbolic, Isom is finite. However, the symmetry of the hyperbolic metric becomes apparent when one considers covers of . Indeed, the universal cover of a hyperbolic manifold is so that acts transitively on , revealing the symmetry of the hyperbolic metric.
In general, any isometry between covers of lifts to an isometry of , so that is the most symmetric cover of . Therefore a natural problem is to describe manifolds for which is large in some meaningful way. More precisely, it is well-known that is a Lie group with possibly infinitely many components, and containing the deck group as a discrete subgroup. Further if is a compact connected group acting isometrically on , then there is a cover acting isometrically on . Since we cannot hope to usefully classify compact group actions on manifolds, we should at least assume that is noncompact if we aim for a classification. Therefore the most general classification problem that we can hope to answer is
Describe the closed Riemannian manifolds such that is noncompact.
A sufficiently explicit solution of this problem has strong implications in many areas of mathematics. For example, as an application of our results we characterize simply-connected Riemannian manifolds that admit both a compact and a finite volume noncompact quotient (see Theorem 1.5 below). This generalizes a result for contractible of Farb-Weinberger [FW08].
We review progress on Problem 1.1. An answer has been obtained by Eberlein if is nonpositively curved [Ebe80, Ebe82], and more generally by Farb-Weinberger [FW08] if is aspherical. Melnick has considered Problem 1.1 for real-analytic closed aspherical Lorentzian manifolds if Isom is semisimple [Mel09].
Roughly, the theorem of Farb-Weinberger states that if is as in Problem 1.1, and in addition aspherical, then a finite cover of is a fiber bundle such that the fibers are locally homogeneous spaces. For a precise statement some orbifold phenomena need to be accounted for, see Theorem 6.1 below. Previously this was known for nonpositively curved manifolds by work of Eberlein, who used tools coming from differentialy geometry in nonpositive curvature. In this case the orbifold phenomena are not present, and a genuine fiber bundle is obtained that is actually a Riemannian product.
Neither of these methods generalize to the nonaspherical case, and in fact, the classification result obtained by Farb-Weinberger is false without the asphericity assumption (see Remark 1.3 and Example 6.2 below). The first progress for not necessarily aspherical manifolds is due to Frankel [Fra94], who proved that under the assumption that is semisimple without compact factors and with finite center, has a finite cover that fibers over a locally symmetric space of noncompact type.
So all the progress on Problem 1.1 either relies on topological restrictions on or algebraic restrictions on . The goal of this paper is to solve Problem 1.1 when has finitely many components. In addition we reduce the general case to being solvable. In particular, our results make no assumptions on the topology of , other than that is closed, and we do not require to be semisimple. To state our results precisely, we recall that has a Levi decomposition.
(see [Rag72]) Let be a connected Lie group. Then there exist
a unique maximal connected closed normal solvable subgroup of (the solvable radical) and
a closed, connected semisimple subgroup of
such that . Further is unique up to conjugation. Any such subgroup is called a Levi subgroup of .
We also introduce the following terminology.
Let be a closed Riemannian manifold. We say fibers locally equivariantly over a locally homogeneous space , where is homogeneous and simply-connected, if there exist
a smooth fiber bundle ,
a closed subgroup containing , and
a smooth group homomorphism with image acting transitively on
such that there is a lift of that is -equivariant. When there is no danger of confusion, we will often omit the locally homogeneous space , and we will just say that fibers locally equivariantly. We say virtually fibers locally equivariantly over a locally homogeneous space if a finite cover of fibers locally equivariantly over .
The power of a locally equivariant fibering comes from the equivariance with respect to a morphism where acts transitively on . For example, this implies that the fibers of are isometric with respect to the induced metric. Our main result is the following.
Let be a closed Riemannian manifold. Set and let be the Levi decomposition of . Then either
is compact and has infinitely many components, or
virtually fibers locally equivariantly over a locally homogeneous space.
In particular the theorem applies when has finitely many components. We now give a much more detailed description in this case.
Let be a closed Riemannian manifold such that is noncompact and has finitely many components. Then
virtually fibers locally equivariantly over a locally symmetric space of noncompact type (possibly a point) with isometric fibers . Further, is a point if and only if is compact,
virtually fibers locally equivariantly over a compact torus (possibly ) with isometric fibers ,
virtually fibers locally equivariantly over a nilmanifold (possibly a point) with isometric fibers . Further and are both zero if and only if is compact, and
if is finite, then Isom is compact.
This result should be viewed as a decomposition of into homogeneous directions (the base spaces in statements (i)-(iii)) and into a residual part (the fibers ). The conditions for (non)triviality of certain base spaces guarantees that Lie theoretic properties of are reflected in the geometry of .
Note that there are examples in which has infinitely many components. For example,consider the lift of a very bumpy metric on an -torus to its universal cover . It is easy to see the deck transformations will be finite index in the isometry group of .
One may hope to generalize Theorem 1.3 to also give a description of that belong to cases (i) or (ii). However, the problem in this generality is hopeless: any action by a connected compact Lie group on admits an invariant metric and lifts to an action of some group on the universal cover . Therefore a complete solution of (i) or (ii) would classify compact group actions on manifolds.
Even though the flavor of Theorem 1.3 is similar to that of the result of Farb-Weinberger, in the nonaspherical case new phenomena appear. As a consequence, the result is different. The fiber bundle constructed in [FW08] has locally homogeneous fibers. Indeed the fibers are projections to of -orbits in . In contrast, the fiber bundle in Theorem 1.3 has locally homogeneous base. Indeed, in Section 6 we give an example of a nonaspherical manifold such that is not discrete (in fact has finitely many components), but it is not a Riemannian orbibundle such that the fibers are projections of orbits of any closed subgroup containing .
As an application of Theorem 1.3, we characterize Riemannian manifolds that admit both a compact and a finite volume noncompact quotient.
Let be a simply-connected Riemannian manifold that admits both a compact quotient and a finite volume noncompact quotient . Then virtually locally equivariantly fibers over a locally symmetric space of noncompact type.
So the only source of Riemannian manifolds with both compact and finite volume noncompact quotients are symmetric spaces. This result generalizes a result of Farb-Weinberger for contractible manifolds .
Since Problem 1.1 is situated in the Riemannian category, it is natural to ask whether the fiber bundle obtained in Theorem 1.7 is a Riemannian submersion. However, this need not be true (see Example 6.4). Further, while for any virtual locally equivariant fibering , the fibers of are isometric, the fibers of need not be isometric (see Example 6.3). Still, we have the following regularity result for in the Riemannian category.
Let be a closed Riemannian manifold that virtually fibers locally equivariantly over a locally homogeneous space. In the notation of definition 1.2, admits a Riemannian metric such that acts isometrically on and is a Riemannian submersion with totally geodesic fibers.
The claims of Theorem 1.3 follow from a more detailed result that explicitly constructs the locally homogeneous space, given the assumptions of case (iii).
Let be a closed Riemannian manifold, and let . Assume that is noncompact. Let be the Levi decomposition of . Then
if is noncompact then virtually fibers locally equivariantly over a nontrivial locally symmetric space of noncompact type.
if is compact and has finitely many components, then virtually fibers locally equivariantly over a nontrivial nilmanifold.
Outline of proof
In both cases, we produce a cocompact lattice in a Lie group , a finite index subgroup and a morphism . In both cases induces a map for a finite cover of , which is determined up to homotopy. Here is a locally symmetric space in Case (i) and a nilmanifold in Case (ii). Our goal is then to show can be chosen to be a locally equivariant fiber bundle.
In case (i) we then use the theory of harmonic maps and a barycenter construction that generalizes both Proposition 3.1 of [FW08] and the work of Frankel [Fra94]. This method does not apply in case (ii), but instead we find and repeat the construction above to find maps
Using the Arzelà-Ascoli theorem, we can show lifts of to converge uniformly to a map
In general is not necessarily smooth, but just continuous. To remedy this, we approximate by an equivariant smooth map and show this approximation is a fiber bundle. This will prove Theorem 1.7.
The paper is organized as follows: In Section 2 we recall some preliminaries needed for the rest of the paper. In Section 3 we prove the existence of the virtual locally equivariant fibering as claimed by Theorem 1.7, and the existence of the metric such that this fibering is a Riemannian submersion (Theorem 1.6). In Section 4 we prove the decomposition of Corollary 1.4 and in Section 5 we prove Corollary 1.5 characterizing manifolds with uniform and nonuniform quotients. In Section 6 we give examples that show that in the nonaspherical situation one cannot expect a result as proved in [FW08], and we show that the fiber bundle obtained in Theorem 1.7 need not be Riemannian, or have isometric fibers. Finally in Section 7 we prove the generalization of the Mostow-Palais equivariant embedding theorem needed in the proof of Theorem 1.7.
I am pleased to thank Daniel Studenmund, Bena Tshishiku, and Shmuel Weinberger for many helpful conversations and suggestions. Many thanks to Karin Melnick, Daniel Stundenmund and Bena Tshishiku for reading an earlier version of this paper and helpful comments. I am very grateful to my thesis advisor Benson Farb for his suggestion that a version of this project should be true, his invaluable advice during the project, and continuing unbounded enthusiasm in all matters mathematical. I would like to thank the University of Chicago for support during the work on this project.
2.1. Lie theory
In this section we review some Lie theoretic preliminaries needed for the proof of our results. Most of the results in this section and their proofs can be found in [Rag72] or [OV00]. In the rest of this section, will be a connected Lie group with Levi decomposition .
Let be a lattice in . Which morphisms map to a lattice in ? When is is discrete in ? Here we will restrict our attention to maps that arise as projections . To answer these questions, we introduce the following terminology.
Definition 2.1 ([Ov00]).
Let be a closed subgroup. We say is (uniformly) -hereditary if is a (cocompact) lattice in . We say is -closed if the set is closed in .
Then we have the following criteria for to project to a lattice.
-(non)heredity of radicals:
In general need not be -hereditary.
Set and let be an infinite order element of . Consider the representation
defined by . Then is a lattice in but is not a lattice in .
However, there is the following result due to Auslander.
Theorem 2.4 (Auslander, [Ov00, 4.1.7(i)]).
Write where is the maximal compact factor of and is the maximal noncompact factor of . Let be a lattice in . Then is a lattice in .
Because is not solvable, one cannot expect to be solvable. However, there is a solvable group in sight.
Theorem 2.5 (Auslander, [Rag72, 8.24]).
Consider the closed subgroup of . Its connected component of the identity is solvable.
In Example 2.3 of a group and a lattice such that is not -hereditary, any Levi subgroup of centralizes the radical . This is in fact the only obstruction.
Theorem 2.6 (Mostow, [Wu88, 1.3]).
Let be a connected Lie group with Levi decomposition . Write where is the maximal compact factor of and is the maximal noncompact factor of , and assume that the centralizer of in is finite. Then for a lattice, is -hereditary.
Besides , there is another convenient subgroup of that we can work with, namely the nilradical of . The nilradical of is the unique maximal, connected, normal, nilpotent subgroup of . Theorem 2.6 has an analogue for .
Theorem 2.7 (Mostow, [Wu88, 1.3]).
Let be a connected Lie group with Levi decomposition and let be the nilradical of . Write where is the maximal compact factor of and is the maximal noncompact factor of , and assume that the centralizer of in is finite. Then for a lattice, is -hereditary.
In particular the theorem applies if is solvable.
Lattices in nilpotent Lie groups.
When studying connected subgroups of Lie groups, the exponential map is an essential tool to transfer to the linear setting of Lie algebras. In nilpotent groups this strategy also applies when studying lattices.
Let be a simply-connected nilpotent group. Then the exponential map is a diffeomorphism.
In particular simply-connected nilpotent groups are contractible. The theorem suggests a strong correspondence between lattices in and , and this is in fact true.
Theorem 2.9 ([Rag72, 2.12]).
Let be a simply-connected nilpotent group. Then
For any lattice , the -span of is a lattice (i.e. additive subgroup of maximal rank) in . Further, the structural constants of with respect to a basis for are rational.
Let be the -span of a basis for with respect to which the structural constants are rational. Then for any lattice in , is a lattice in .
In general it is not true that itself is a lattice in .
Let be a simply-connected nilpotent Lie group. We say a lattice is an exponentiated lattice if is an additive subgroup of .
The correspondence provided by Theorem 2.9 is one-to-one up to finite index subgroups.
Theorem 2.11 ([Ov00, 2.2.13]).
Let be a lattice in a simply-connected Lie group . Then there are exponentiated lattices in such that . Further both inclusions are as finite index subgroups.
Of course Theorem 2.8 fails if is not simply-connected. However, this failure is very controlled.
Let be a connected nilpotent group. Then has a unique maximal compact torus . Further is central and is simply-connected.
Lattices in semisimple groups.
Let be a connected semisimple Lie group. Its center is a closed abelian normal subgroup, hence it must be discrete. Using this fact, it is easy to see that has trivial center. Another classical fact we will need is that is finite.
Finally, we note that the theme of a lattice resembling the ambient group is especially powerful in the context of semisimple groups (e.g. the Mostow rigidity theorem). Here we will just note that the center of a lattice resembles the center of the ambient group, which follows from a version of the Borel density theorem:
Theorem 2.13 (Borel, [Rag72, 5.17, 5.18]).
Let be a semisimple group without compact factors and a lattice. Then , and is a lattice in .
2.2. Harmonic maps
A second ingredient in the proofs below is the theory of harmonic maps. For a smooth map between Riemannian manifolds , we define the energy
We say is harmonic if it locally minimizes the energy. There is a very rich theory of these maps and their rigidity, but we only mention the following existence and uniqueness results:
Theorem 2.14 ([Sy79]).
Let be closed Riemannian manifolds.
(Eells-Sampson) If is nonpositively curved, then there exists a harmonic map in each homotopy class.
(Hartman, Schoen-Yau) Suppose is harmonic, surjective on and that . Then is the unique harmonic map in its homotopy class.
3. Existence of locally equivariant fiberings
We introduce some notation that will be used below. Let be a closed Riemannian manifold and set . Let . Let be the solvable radical of and . Set .
We give an outline of the proof of Theorem 1.7.(i). Assume that is noncompact. The first part of the proof is entirely Lie theoretic. We will construct a cocompact lattice in a semisimple Lie group with no compact factors and trivial center, a finite index subgroup , and a morphism (Lemmas 3.1 and 3.2).
The second part of the proof is geometric, and relates the morphism to the geometry of . By asphericity of the locally symmetric space associated to , there is a smooth map (where is the cover corresponding to ) inducing on fundamental groups.
The map is only determined up to homotopy. We use the theory of harmonic maps between Riemannian manifolds to select a unique harmonic representative of the homotopy class of . The harmonicity will imply that this map is a fiber bundle.
Proof of Theorem 1.7.(i)
Assume is noncompact.
Step 1 (Lie theory):
First we prove:
is a cocompact lattice.
Proof of Lemma 3.1.
Because is compact, by the Milnor-Schwarz lemma we have that for any the map
is a quasi-isometry. Since also acts on properly, isometrically, and contains , the map
is also a quasi-isometry. It follows that the inclusion is a quasi-isometry, so is a cocompact lattice in . Since is a connected component of , is closed in . Because is compact, it follows is compact, as desired. ∎
We can use to obtain a lattice in a semisimple group. Let be a Levi subgroup of , so that . Let be the unique maximal connected closed normal compact subgroup of . By Theorem 2.4, is a lattice in . It is necessarily cocompact by Proposition 2.2, and again by Proposition 2.2, projects to a cocompact lattice in .
is a semisimple group, but it may have infinite center. However, below we need to have a semisimple group without center, so consider , where . Then is a semisimple group without center, and since is a (necessarily cocompact) lattice in by Theorem 2.13, the image of in is a cocompact lattice.
We will extend the projection map to a map for a suitable finite index subgroup . Note that this is nontrivial because may have infinitely many components. In order to obtain the extension, set . Then is a finite index subgroup of . Consider the short exact sequence
This sequence is almost a direct product.
There is a finite index subgroup containing such that the induced sequence
splits as a direct product, where .
The sequence 3.1 is determined by
A representation , and
A cohomology class in , where the module structure on is induced by .
Note that since has no center, the cohomology class in (ii) vanishes. Since is finite, there is a finite index subgroup of containing such that is trivial on . Hence for this choice of , the sequence 3.2 splits as a direct product. ∎
Step 2 (Geometric):
Let be a maximal compact subgroup and consider the locally symmetric space . Because is aspherical, the homomorphism induces a map determined up to homotopy. To select a representative of this homotopy class that will be a fiber bundle, we use the theory of harmonic maps between Riemannian manifolds.
In our situation, is nonpositively curved, so that by the theorem of Eells-Sampson (Theorem 2.14.(i)), there is a harmonic representative in the homotopy class of . From now on, we will denote this harmonic representative by .
Note that is surjective and by the Borel density theorem,
So by Theorem 2.14.(ii) of Hartman and Schoen-Yau, is the unique harmonic map in its homotopy class. We will show that this rigidity in the choice of implies that it is a fiber bundle. First lift to . Then we have:
This will follow from an ‘averaging’ construction due to Frankel [Fra94]. Frankel carried out this construction only for and , but his proof applies verbatim here, so we will merely recall the construction.
The intuition for the construction is to compare the points and for and . Note that is -equivariant precisely when these points always coincide, i.e. for every and , we have
To measure how far is from being equivariant, for , set
Because induces the map on and is a lift of , we know that is -equivariant, so Equation 3.3 holds for and any . Because is a cocompact lattice in , it follows is compact. Frankel shows [Fra94, Thm 3.5] that compactness of together with the the nonpositive curvature of implies that there is a well-defined unique barycenter of the set . The map is then the ‘average’ of . More precisely, define for the map
This is well-defined because is a lift of , hence -equivariant. Because has a lattice, it is unimodular, so Haar measure on descends to . This gives a compactly supported measure on . For , define
Because is nonpositively curved and contractible, the function is strictly convex. Together with the fact that is compactly supported, this implies that there is a unique with . The barycenter of is defined to be this point , and is denoted .
It is not hard to see that the map is -equivariant. In particular, descends to a map
and induces the map on . Because is aspherical, and both and induce the map on , it follows that and are homotopic, but Frankel also proves [Fra94, Thm 3.3] that the averaging construction decreases the energy. But because is the unique harmonic map in its homotopy class, it uniquely minimizes the energy functional in its homotopy class. Therefore . Hence is -equivariant, as claimed.∎
is a fiber bundle.
First we show is a fibration. Equivalently, admits a path lifting function, i.e. it is possible to lift paths from to continuously. Since this is a local property, it suffices to prove is a fibration. Since the composition
is a smooth fiber bundle, there is a path lifting function from to that lifts smooth paths to smooth paths. This naturally induces a path lifting for . If is a curve in starting at and lifts to a curve in , then for a point in over , the path
lifts to . It is clear that the constructed path depends continuously on and , so that we have a path-lifting function. It follows that is a fibration. Further we note that in this construction, smooth paths lift to smooth paths.
Finally, by a theorem of Ehresmann, a proper submersion is a fiber bundle. Since is compact, is clearly proper. It remains to show is a submersion. It is surjective since acts transitively on , and given a tangent vector , we can choose a path with . Since is a fibration, we can lift to a smooth path in , and it follows that . Therefore is a submersion.∎
This proves Theorem 1.7.(i). Now we turn to the proof of Theorem 1.7.(ii). Assume that has finitely many components and that is compact. As in the proof of Theorem 1.7.(i), the first step is completely Lie theoretic and aims to find a map for a finite index subgroup and a lattice in an appropriate nilpotent Lie group . This map naturally extends to a map for a closed connected subgroup .
As before, the second step is geometric, and uses to relate the geometry of to an appropriate locally homogeneous space: The map is induced by a homotopy class of maps . Again, we will select a representative that will be a fiber bundle by an averaging procedure using the structure of nilpotent groups.
Proof of Theorem 1.7.(ii)
Assume that has finitely many components and that is compact. Since has finitely many components, we may assume .
Consider the Levi decomposition . Let
be the natural projection. Set . This will be the Lie group alluded to above.
has finite index in .
is closed inside the compact group , so that is compact. Therefore has finitely many components. So has finite index in . Therefore has finite index in . ∎
Replace by the finite index subgroup . Note that is an extension
Theorem 2.5 implies that is solvable (see also [OV00, 4.1.7(ii)]). Hence is solvable-by-solvable, so is itself solvable. Let be the nilradical of . We consider two cases according to whether or not is cocompact in . Note that these cases are not disjoint: If is abelian, both methods below work. In fact the obtained maps are essentially the same.
Case 1: Noncocompact case. Suppose is not cocompact in . Since is abelian and noncompact, there is a torus and such that
Set and let be the natural projection. By Theorem 2.7, we have that is a lattice in . Hence is a lattice in . There is a homotopy class of maps
inducing on . Let be any representative in this homotopy class and let be defined by
where is induced by Haar measure on . Using invariance of Haar measure, it is easy to see that is -equivariant. This finishes the proof of Theorem 1.7.(ii) in this case.
Case 2: Cocompact case. Suppose is cocompact in . As above, we know that is a lattice in . By Proposition 2.2 the image of is a lattice in the compact group . Hence has finite index in , and we can replace by .
Since is a lattice in , we could now again split up into cases according to whether is cocompact in , and repeat the procedure of Case 1 in case it is not, and pass to if it is, etc. Some quotient of successive terms of the lower central series of is guaranteed to be noncompact since itself is noncompact. This argument shows that virtually fibers locally equivariantly over a torus. However, below we will show that a stronger conclusion holds, namely that actually fibers over a nilmanifold very closely related to .
Let be the unique maximal compact torus of . By Theorem 2.12, is central and the group is simply-connected and nilpotent. We claim that virtually fibers locally equivariantly over . Let
be the natural projection, and let be the image of in . It follows from Theorem 2.8 that is contractible. Therefore there is a homotopy class of maps
inducing on . Pick a representative and lift it to a map
In general need not be -equivariant, but only -equivariant. To construct a -equivariant map, set for :
and . We claim that is discrete. By Theorem 2.11, there exists an exponentiated lattice of that contains with finite index. Then
is discrete. Further, the condition is linear in since is an exponentiated lattice. It follows that is a lattice in , so that
is a lattice in . Since , it follows that is also a lattice in . By carrying out the same argument for on the universal cover of , it follows that is discrete.
Further the sequence
is increasing with dense union in . Now we would like to say that contractibility of implies that there is a homotopy class of maps
that induces on . However, may have torsion, so may not be a covering map. Consider instead and let
be the natural projection. Because is connected and is simply-connected, it follows is simply-connected (see e.g. [Bre72, II.6.3]). Further acts on . Because is compact and acts on properly, the action of on is also proper. Since is a lattice in a simply-connected nilpotent Lie group, it is torsion-free, so covers for every . Then we can choose continuous maps
Note that since is not necessarily a manifold, we can only require to be continuous, not necessarily smooth. We can lift to a -equivariant map
Choose a basepoint and fundamental domains for the action of on . Then we can arrange the lifts so that for all .
converges uniformly to some .
Note that is a fundamental domain for the action of on containing . For every , we know that is -equivariant, so it suffices to check the uniform convergence on . We note that is naturally a metric space because is compact, and acts isometrically on . Therefore is a compact metric space, so it suffices to check is a Cauchy sequence in the uniform topology.
Since form an increasing sequence of lattices in with dense union, we have monotonically as . Now let and choose such that diam. Let and . Since are both -equivariant, we can without loss of generality assume that belongs to . Further we can choose such that . Hence we have
Since as , it follows that converge uniformly.∎
It is clear that is -equivariant for every . Since the union of is dense in , is actually -equivariant. Now set
Then is -equivariant. However, note that we cannot expect to be smooth, for the maps were only continuous. Note that is linear, because it is a simply-connected nilpotent group, so that by Theorem 7.4 we can equivariantly homotope to a smooth equivariant map.
Finally, to show that is a fiber bundle, we note that the proof of Lemma 3.4 still applies in the current situation. The only change that needs to be made is to consider the fibration instead of . Otherwise the proof applies verbatim. This proves Theorem 1.7.(ii).
We will now prove Theorem 1.6.
Proof of Theorem 1.6.
Suppose is a locally equivariant fibering with lift that is equivariant with respect to a homomorphism , and acts transitively on . We want to show that is a Riemannian submersion with totally geodesic fibers in some Riemannian metric on .
Let be the stabilizer of some point . Then is -equivariantly diffeomorphic to . We claim first that the structure group of can be reduced to . Let be the natural projection. Then is a fiber bundle with fibers . Therefore we have an open covering of with local trivializations
of . Any such local trivialization naturally induces a local trivialization
of over defined by
It is easy to check that is actually a local trivialization and is precisely multiplication by when restricted to the fiber over . Therefore the structure group induced by this collection of trivializations is contained in and preserves a given fiber, say . It follows the structure group is contained in .
Let denote the Riemannian metric on . The tangent spaces to the fibers of give a -invariant vertical distribution , and restricting to induces a Riemannian metric on each fiber of . Further, since acts isometrically on , it follows that for any and , the restriction of to the fiber over is an isometry
In particular, the structure group of acts isometrically with respect to . To extend to a Riemannian metric on , consider the orthogonal complement of . Then is a -invariant distribution on , and maps isomorphically onto the tangent bundle of . Let denote the Riemannian metric on . Since is an isomorphism on , we can pull back to a Riemannian metric on . Note that since acts isometrically on , it follows that preserves .
Therefore is a -invariant Riemannian metric on . In this metric, is a Riemannian submersion. Vilms proves [Vil70] that has totally geodesic fibers. Finally, being a Riemannian submersion with totally geodesic fibers is a local condition, so the result follows for . ∎
4. Decomposition of manifolds with many local symmetries
Proof of Corollary 1.4.
Let be a closed Riemannian manifold and assume that has finitely many components. Because has finitely many components, we can replace by . As usual, let be the Levi decomposition. The proof consists of three steps. In the first step, we construct a virtually locally equivariant fibering of over a locally symmetric space of noncompact type if is noncompact. A fiber of will almost satisfy the hypotheses of Theorem 1.7.(ii). In fact, precisely satisfies the hypotheses if is finite. In the second step we construct a virtually locally equivariant fibering of over a torus. If is infinite, some care needs to be taken to apply the proof of Theorem 1.7.(ii). A fiber of will again almost satisfy the hypotheses of Theorem 1.7.(ii), and we construct a virtually locally equivariant fibering of over a nilmanifold.
Step 1 (fibering over symmetric space): If is noncompact, by Theorem 1.7.(i) there exists a locally symmetric space of noncompact type associated to the Lie group and a fiber bundle that is equivariant with respect to the natural projection , where is a finite cover of .
If is compact, fix the following notation: Set , let be a point, the constant map, and be the trivial morphism.
Step 2 (fibering over Euclidean space): Let be a fiber of . Note that regardless of (non)compactness of , we have a natural Riemannian metric on by restricting the Riemannian metric on to the tangent bundle of . Further, by applying the long exact sequence of homotopy groups for the fibration and using that is contractible, we see that a fiber of is simply-connected. Therefore for any , the fiber over is isometric to .
Under this identification, is a closed subgroup of Isom containing . If is infinite then has infinitely many components. In fact, as we will see below, the components of are indexed by a finite index subgroup of . Further, has compact Levi subgroups isogenous to , so unfortunately we cannot apply Theorem 1.7 directly to . However, we will show that the extension
almost splits as a direct product, and the ideas of the proof of Theorem 1.7 will then apply:
There is a finite index subgroup of containing such that the extension
splits as a direct product, where is the image of in .
Proof of claim.
Note that is finitely generated and abelian, so we can pass to a finite index torsion-free subgroup . First we lift a finite index subgroup of to a subgroup of . As before, write , where is the maximal noncompact factor. Then there is an exact sequence
for a finite index subgroup of and the restriction of the natural projection . Note that is a finite abelian group. Now consider the restriction of this sequence to as follows
This extension is trivial because is abelian and is torsion-free. Therefore lifts to a subgroup of , which we also denote by . It remains to show that a finite index subgroup of centralizes . To see this, consider the action of on . It induces an action on the Lie algebra of , giving a map
Since semisimple groups with infinite center are not linear (see e.g. [Kna02, 7.9]), we find that a finite index subgroup of acts trivially on . Since is connected, any automorphism of that is trivial on is trivial on . It follows that there is a finite index subgroup of such that
The proof of Theorem 1.7.(ii) applies verbatim in this situation and yields a closed connected subgroup of . Assume first that the has noncompact abelianization. Then the proof of Case 1 in Theorem 1.7.(ii) shows that there is a fiber bundle for a finite cover of and a torus . Further, is equivariant with respect to the composition
Here the first map is projection onto the first factor.
If has compact abelianization, let be the map to a point and be the trivial morphism. Then regardless of (non)compactness of the abelianization of , we see that is naturally a Riemannian manifold and acts isometrically on . Every fiber of is simply-connected by the long exact sequence on homotopy groups applied to the fibration , so that we can identify with a fiber of . Under this identification, is a closed subgroup of Isom containing . Further is a nilpotent group.
Step 3 (fibering over nilmanifold): The proof of Case 2 of Theorem 1.7.(ii) constructs a fiber bundle for a finite cover of and a nilmanifold , and is equivariant with respect to the composition
where is the unique maximal closed normal subgroup of , is the universal cover of , and the first map is projection onto the first factor. This construction proves Assertions (i)-(iii) of Corollary 1.4. It remains to prove Assertion (iv), that Isom is compact if is finite.
To see this, note that by the long exact sequence on homotopy groups and using that is contractible, we have . In particular, contains as a finite index subgroup. Therefore if is finite, the fibers of have finite fundamental group, so that the universal cover is compact. Hence in this case Isom is compact. ∎
5. Manifolds with uniform and nonuniform quotients
In this section, we prove Corollary 1.5, namely that Riemannian symmetric spaces are the only source of manifolds with uniform and nonuniform quotients.
Write and . Consider the group acting on . Let be the closure of in