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(To appear in The Journal of Differential Geometry)

CONNECTION PRESERVING ACTIONS OF CONNECTED ANDDISCRETE LIE GROUPS

EDWARD R. GOETZE

Abstract. This paper examines connection preserving actions of a noncom-pact semisimple Lie group G on a compact fiber bundle and connection pre-serving actions of a lattice Γ ⊂ G on a compact manifold. The results rely ona new technique that increases the regularity of sections of bundles naturallyassociated to the actions under consideration.

1. Introduction

Let M be a connected smooth n-dimensional manifold, and H a subgroup ofGL(n,R). An H-structure on M is a reduction of the full frame bundle overM to H. If we allow H to be a subgroup of GL(n,R)(k), the subgroup of k-jets at 0 of diffeomorphisms of Rn fixing 0, we can extend the notion of an H-structure to include reductions of higher order frame bundles to H. Given anH-structure P → M , the automorphism group of P , Aut(P ), is the subgroup ofDiff(M) consisting of the diffeomorphisms of M whose induced action on the framebundle preserves P . We wish to examine relationships between a Lie group G andmanifolds M with H-structures such that G ⊂ Aut(P ). Also, we are interestedin the situation where, instead of a G action, we have only Γ ⊂ Aut(P ) whereΓ ⊂ G is a lattice subgroup. This case deals with the issue of the rigidity ofthe action of a higher rank lattice, an area of much recent research. The use ofhyperbolic dynamical systems by Hurder in [7], and Katok and Lewis in [9] and[10] has produced recent results.

If we assume M is a compact manifold and G preserves a volume form on M ,then the study of the ergodic theory of the action has been a successful technique inanswering some of these questions. In particular, we mention Zimmer’s work in [15]and [16] as examples of this technique. One drawback of this approach, however, isthat the use of ergodicity provides measurable information which is often difficultto translate into meaningful information of a higher regularity. This information,which typically is represented as a section of an associated bundle to a principalbundle over M , is often defined only on an open dense subset of M . Hence, anadditional difficulty in applying these methods is the problem of demonstrating thatsuch a section has completeness properties rich enough to imply useful geometricinformation, for example, the section is defined on all of M . See [17] for a discussion.

In Section 2 of this paper, we develop a technique that under suitable hypothesesallows one to improve the regularity of this information, e.g., to pass from themeasurable to the smooth. The approach we take is to combine elements of thetheory of hyperbolic dynamical systems and the ergodic theory of the action, inthe form of Zimmer’s Measurable Superrigidity, with geometric considerations toconstruct smooth sections of bundles naturally associated to M . We prove

1

2 EDWARD R. GOETZE

Theorem 2.19. Let X be a compact fiber bundle over Y with fibers F . Let G bea connected semisimple Lie group of higher rank without compact factors. SupposeG acts ergodically on X via bundle automorphisms preserving a volume densityand a Cr connection such that the fibers are autoparallel. If the algebraic hull ofα′ : G × X → SL(f), the derivative cocycle in the fiber direction, equals SL(f),then, by possibly having to pass to a finite algebraic cover of G, there exists anabelian subgroup A ⊂ G such that the Oseledec decomposition of TF correspondingto A is parallel, Cr regular, and everywhere defined on Fx for almost every x ∈ X.In particular, there exists a Cr section of the full flag bundle over T (Fx).

Although this result assumes the algebraic hull is SL(f), this assumption is muchstronger than necessary, and the proof can easily be adapted to other situations.Theorem 2.20 describes a similar result for actions of lattices on manifolds.

The existence of these sections is often too weak to provide meaningful geometricresults. To strengthen this information we can use either of two methods describedin this paper. The first of these methods, developed in Section 3, is to employC(r,s) Superrigidity which is a generalization of Zimmer’s Topological Superrigidity,[14]. Where applicable, use of Topological or C(r,s) Superrigidity allows us toconclude that sections of an associated bundle over M come from sections of thecorresponding principal bundle over M . More specifically, if P →M is a principalbundle H bundle, and EV = (P ×V )/H is an associated bundle with Φ a section ofEV →M , then Topological or C(r,s) Superrigidity ensures the existence of a sectionφ of P → M and an element v0 ∈ V such that Φ(m) = [φ(m), v0]. The essentialpoint to note is that φ possesses the same regularity and completeness propertiesas Φ. C(r,s) Superrigidity extends Topological Superrigidity in that it holds in thecase where one makes more delicate regularity or completeness assumptions on Φ.See the definition of C(r,s) regularity below.

By exploiting the algebraic properties of H, this section of the principal bundleP → M can be used to provide much stronger information about M , and we usesuch a section below to classify the possibilities for M under certain conditions.

The second technique, developed with Renato Feres, is discussed in Section 4.2.Here we analyze the local holonomy of the connection, and, if conditions are appro-priate, we conclude directly from this information that our original sections haveoriginated from sections of principal bundles. In [5], Feres uses this technique tocome to similar conclusions under more general situations, illustrating the abilityto generalize the techniques to broader situations.

The motivation for the development of these techniques was to analyze the fol-lowing geometric problems. Suppose G is a higher rank noncompact semisimpleLie group with Γ ⊂ G a lattice. Let Γ act ergodically on a compact manifold Mpreserving a volume density and a connection. Does this action place any restric-tion on the possibilities for M? More generally, let X be a fiber bundle over Y withcompact fibers F . Suppose G acts ergodically on X via bundle automorphismspreserving a volume density and a smooth connection. Does the G action restrictthe choices for F? If Γ is cocompact, by inducing the action of Γ to G, we find thatdetermining the possibilities for F restricts the possibilities for M .

We use the methods described above to obtain the following results, which arepresented in Section 4.

CONNECTION PRESERVING ACTIONS OF LIE GROUPS 3

Theorem 4.1. Let X,Y, F , and G be as described above, and assume the fibers inX are autoparallel. Let L be the algebraic hull for the cocycle α′.

(1) If L is compact, then there exists a smooth Riemannian metric g on F withrespect to which F has a transitive group of isometries, i.e.,

F ∼= Isom(F, g)Isom(F, g)x

,

(2) If L = SL(f), where f = dim(F ), then F is a torus.

Theorem 4.4. Let Γ ⊂ G be an irreducible lattice in a higher rank semisimpleLie group without compact factors. Suppose Γ acts ergodically on a compact n-dimensional manifold M preserving a volume and a smooth connection. Let α :Γ×M → SL(n) be the derivative cocycle with measurable algebraic hull L.

(1) If L is compact, Γ acts isometrically on M preserving a smooth Riemannianmetric.

(2) If L = SL(n) and π(Γ) ⊂ SL(n) contains a lattice, where π is the Super-rigidity homomorphism, then M admits a torus as a finite affine cover.

This last theorem is a generalization of results presented in [6] where we nolonger require the rather restrictive assumption that the connection has boundedparallel transport.

As a simple corollary to Theorem 4.4, we mention

Corollary 4.8. Suppose SL(n,Z), n ≥ 3, acts ergodically on an n-dimensionalmanifold M preserving a connection and a volume density. Then M admits a torusas a finite affine cover.

Since the requirement that L = SL(f) is stronger than necessary in Theorem2.19, the same holds true for these results. There should be little problem inadapting these results to hold when L is any noncompact semisimple Lie group,regardless of its type, as long as the dimension of L is comparable to the dimensionof F . This point will be addressed in future work.

We wish to thank Robert Zimmer and Renato Feres for their helpful conversa-tions and suggestions.

2. Improving the Regularity of Sections

Throughout this section and those that follow, we assume the reader is familiarwith the elements of Zimmer’s Measurable Superrigidity as presented in [15] and[16].

2.1. Measurable Superrigidity and the Multiplicative Ergodic Theorem.Let X be a compact fiber bundle over Y with fibers F , dim(X) = x, dim(F ) = f .Let G be a connected noncompact semisimple Lie group of higher rank. SupposeG acts ergodically on X via bundle automorphisms preserving a connection and avolume density. A bundle automorphism is a diffeomorphism of X which factors toa diffeomorphism of Y . Naturally associated to this situation are two cocycles:

β : G× Y → Diff(F )

α : G×X → GL(x).

4 EDWARD R. GOETZE

β describes the lift of the G action on Y to X, and α is the derivative cocycle(x = dim(X)). Since the G action maps fibers to fibers, α induces another naturalcocycle

α′ : G×X → GL(f)

which describes how the G action maps tangent vectors in one fiber to tangentvectors in another fiber. We let L be the algebraic hull of α′.

We shall make extensive use of the following result.

Theorem 2.1 (Superrigidity for cocycles, [15]). Suppose G is a connected semisim-ple Lie group, R-rank(G) ≥ 2, with no compact factors. Let X be an irreducibleergodic G-space, H and algebraic R-group, and α : G × X → H a cocycle withalgebraic hull L. If LR is noncompact and center free, then α ∼ π : G→ L.

Remark 2.1. If LR has a finite center Z, we apply the theorem to LR/Z, and bylifting this homomorphism we obtain a homomorphism from a finite algebraic coverof G into LR itself, Remark 6.2, [14]. This point necessitates the inclusion of the“up to finite cover” phrases in most of our results.

Remark 2.2. We may dispense with the assumption of irreducibility if, instead,we assume every simple factor of G has higher rank.

Remark 2.3. If Γ ⊂ G is a lattice acting on X as above, then the conclusion stillholds, i.e., if α : Γ×X → H is a cocycle, there exists π : G→ H such that α ∼ π.See Theorem 9.4.14 in [15].

We will use this theorem in conjunction with some classical results from ergodictheory.

Theorem 2.2. Let M be a compact smooth manifold, f : M → M a C1 diffeo-morphism, and ‖ · ‖ the norm of some Riemannian metric on M . Let B be theσ-algebra of Borel subsets of M and let M be the set of all f -invariant probabilitymeasures on B. Then there exists an f -invariant set B ∈ B such that

(1) µ(B) = 1 for every µ ∈M,(2) there exists a measurable f -invariant function s : B → Z+,(3) there exist measurable f -invariant functions χi : B → R, for i = 1, 2, . . . , s,(4) there exists a measurable decomposition TM |B = E1 ⊕ E2 ⊕ · · · ⊕ Es into

f -invariant subbundles, and(5) if x ∈ B, V ∈ Ei(x)r {0}, 1 ≤ i ≤ s(x), then

limn→±∞

1n

log ‖Tfnx (V )‖ = χi(x).

The objects are unique and independent of the choice of Riemannian metric.

Proof. This is Oseledec’s Multiplicative Ergodic Theorem, Theorem 10.4 in [13].


The decomposition is called the Oseledec decomposition, the functions {χi} arecalled the Lyapunov exponents, and B is called the set of regular points.

If A consists of a family of commuting C1 diffeomorphisms, we can choose theOseledec decomposition to be common to all elements of A on some conull set Λ.The Lyapunov exponents then become functions χi : A→ Func(Λ,R).

We wish to combine Superrigidity and the Multiplicative Ergodic Theorem toyield information concerning the Lyapunov exponents in the direction of the fibersin X. Let TF ⊂ TX consist of the subbundle of vectors tangent to the fibers inX. Assuming L is not compact, Superrigidity yields a measurable trivialization ofTF ∼= X × Rf such that the G action becomes

g(x, V ) = (gx, π(g)V )

where α′ ∼ π : G→ SL(f). For an abelian subgroup A ⊂ G, let {χi(a)} be the setof Lyapunov exponents corresponding to the Oseledec decomposition

TFx ∼=⊕

j

Fj ,

the tangent space through the fiber at x ∈ X.

Proposition 2.3. The Lyapunov exponents for a ∈ A are {log |ai|}, where ai arethe eigenvalues for π(a) ∈ SL(f).

Proof. [6].

We now assume the fibers in X are autoparallel with respect to the given con-nection [11], i.e., we need to assume the restriction of the connection to a fiberyields a connection. For a vector V ∈ TFx, let PV denote parallel translation alongthe geodesic exp(tV ), t ∈ [0, 1], provided that the geodesic is defined for all such t.Also, recall that χ+(a, Z) = limn→∞ 1

n log ‖Tanx(Z)‖.Lemma 2.4. There exist constants C and K such that if V ∈ TFx and ‖V ‖ < K,then ‖PV ‖ < C.

Proof. Let l(t) be a geodesic in F starting at x. By continuity of the connection,for any V ∈ TFx, Pl(t)(V ) is a continuous function in t. Hence, by continuity of thenorm on F , f(t) = ‖Pl(t)(V )‖ is also continuous.

For a fixed C > 1, continuity of f implies for every x ∈ F , there exists aneighborhood Ux of x such that for any geodesic generated by V ∈ TFx, startingat x and staying in Ux, we have ‖PV ‖ < C. Using the local diffeomorphic propertyof exp, there exists Kx such that for every V ∈ TFx with ‖V ‖ < Kx, exp(tV ) liesin Ux for t ∈ [0, 1]. For such V , we thus have ‖PV ‖ < C. Note that Kx variescontinuously with x. Using compactness of F we can choose a K > 0 such thatK ≤ Kx for every x ∈ F . The lemma now follows.

By the usual arguments, the previous result is independent of our choice of norm.

Proposition 2.5. Fix a ∈ A and suppose x, y ∈ Λ and l(t) = exp(tZ) for t ∈ [0, 1]is a geodesic from x to y with χ+(a, Z) < 0.

(1) If V ∈ Fj(x), then χj(a, Pl(V )) ≤ χj(a, V ).(2) If V ∈ Fj(x) where χj(a, V ) is minimal, then Pl(V ) ∈ Fj(y), i.e., Pl

preserves the maximal contracting direction.

6 EDWARD R. GOETZE

(3) If χ1(a) < χ2(a) < · · · < χf (a), then Pl preserves the flag F1 ⊂ F1 ⊕ F2 ⊂· · · ⊂ F1 ⊕ · · · ⊕ Ff .

Proof. Since the G action preserves the connection, we have

Tay ◦ Pl = Pa◦l ◦ Tax.Note that an ◦ l(t) = an ◦ exp(tZ) is a geodesic at an(x) in the direction Tanx(Z).Since χ+(a, Z) < 0, we have {‖Tanx(Z)‖}n≥0 is bounded, and, in fact, converges to0. Thus, there exists N such that for every n ≥ N , ‖Tanx(Z)‖ < K, where K is asin the previous lemma. Therefore, there exists C > 0 such that

‖Tany ◦ Pl(V )‖ ≤ C‖Tanx(V )‖ ∀n ≥ 0.

Thus,

limn→∞

1n

log ‖Tany ◦ Pl(V )‖ ≤ limn→∞

1n

log(C‖Tanx(V )‖) = χj(a).

This proves the first claim. The second statement follows as minimality of χjassures us that equality is achieved, and that Pl(V ) ∈ Fj(y). The third claimsfollows immediately from the first.

Remark 2.4. Without mention, we have made substantial use of our assumptionthat the fibers are autoparallel. This assumption assures us that parallel translationalong a path in the fiber takes tangent vectors to the fiber to tangent vectors to thefiber.

If l(t) = exp tZ is a geodesic and there exists a ∈ A such that χ+(a, Z) < 0 thencall l a contracting geodesic for a.

Using Fubini’s theorem, there exists x ∈ Λ such that Λ ∩ Fx is conull in F . Wenow fix such an x.

Proposition 2.6. Let R and T be the curvature and torsion tensors of the con-nection on Fx, and let Xi ∈ Fi(x). Then

(1) (a) R(X1, X2)X3 = 0, or(b) R(X1, X2)X3 has Lyapunov exponent χ1 + χ2 + χ3.

(2) (a) T (X1, X2) = 0, or(b) T (X1, X2) has Lyapunov exponent χ1 + χ2.

Proof. Let W = R(X1, X2)X3. We have

‖W‖ ≤ ‖R‖ ·∏

i

‖Xi‖.

Hence,

log(‖Tanx(W )‖) = log(‖(R(TanxX1, TanxX2)(TanxX3)‖)

≤ log(‖R‖∏

i

‖TanxXi‖)

= log ‖R‖+∑

i

log ‖TanxXi‖.


Thus,

limn→∞

1n

log ‖Tanx(W )‖ ≤∑

i

χi(a).(1)

Replacing a with b = a−1, we obtain

limn→∞

1n

log ‖Tbnx(W )‖ ≤∑

i

χi(b) = −∑

i

χi(a),(2)

and if W 6= 0, then

limn→−∞

1n

log ‖Tanx(W )‖ ≥∑

i

χi(a).(3)

Combining 1 and 3, and using regularity of x ∈ Λ,

limn→±∞

1n

log ‖Tanx(W )‖ =∑

i

χi(a).

A similar argument works for T as well.

2.2. Construction of Cr Sections. Throughout this subsection we assume thefibers F in the bundle X over Y are f dimensional (f ≥ 3), and that the algebraichull L = SL(f,R). Using Superrigidity, by passing to a finite (algebraic) cover ofG, we may assume α′ ∼ π : G → L is a surjection. If D is the set of diagonalmatrices in SL(f,R), then we may choose A ⊂ G such that π(A) = D. Thus theeigenvalues for π(a) are just the elements along the diagonal. Using Lemma 2.3,we have

Lemma 2.7. Let {χi} be the set of Lyapunov exponents corresponding to the Aaction. Then,

(1) If x ∈ Λ and TFx =⊕Fi(x) is the Oseledec decomposition corresponding

to the A action, then dim(Fi(x)) = 1 for all i.(2) If σ is a permutation on the set of f elements, then there exists a ∈ A such

thatχσ(1)(a) < χσ(2)(a) < · · · < 0 < χσ(f)(a).

Recall we have x ∈ X such that Fx ∩ Λ is conull in Fx. Since the fibers areautoparallel, the connection on X yields one on Fx. Let R and T be the curvatureand torsion of this connection.

Proposition 2.8. R ≡ 0 and T ≡ 0.

Proof. Let i, j, k ∈ {1, 2, . . . , f}. If R(Xi, Xj)Xk 6= 0 then we have χi + χj + χkis a Lyapunov exponent, and hence, must be χl for some l ∈ {1, 2, . . . , f}. UsingLemma 2.3, this corresponds to an algebraic relation among the ai’s for every π(a) ∈D. In other words, we have that aiajak = al for all diagonal matrices in SL(f).However, no such relation exists for all matrices in D. Thus, R(Xi, Xj)Xk = 0 forall i, j, k. Hence, R ≡ 0. A similar argument shows T ≡ 0.

8 EDWARD R. GOETZE

If x ∈ Λ and TFx =⊕f

i=1 Fi(x), let Wj(x) =⊕

i 6=j Fi(x) be a hyperplane inTFx (so Wj(x) has constant j-coordinate), and let Vj(x) = expBj(x) (where expis defined). Since R, T ≡ 0, in some neighborhood U of x we may assume U is flataffine space, and that exp is the identity map. For x ∈ Λ, let {Xi(x)} be a basis ofTFx with Xi(x) ∈ Fi(x).

Lemma 2.9. Let x ∈ Λ∩U and suppose z ∈ Vi(x)∩Λ∩U . Then Fj(x) is parallelto Fj(z) for j 6= i.

Proof. Pick a ∈ A such that ai > 1 and aj is minimal. Then any vector in Wi(x) iscontracting for a, and since aj is minimal, parallel translation along any geodesicfrom x to z in Vi(x) maps Fj(x) to Fj(z). However, the parallel translation in U isjust ordinary translation, and so Fj(x) is parallel to Fj(z).

Lemma 2.10. Suppose l(t) = exp tXi(x) is a geodesic in U from x through z ∈ Λ.Then Fj(x) is parallel to Fj(z) for all j = 1, 2, . . . , f .

Proof. Pick k 6= i. Then l(t) lies in Vk(x), so by Lemma 2.9, Fj(x) is parallel toFj(z) for all j 6= k. It remains only to see Fk(x) is parallel to Fk(z). However, werepeat the same argument using l 6= i, k (possible since F is at least 3 dimensional),and conclude Fk(x) is parallel to Fk(z).

We define a geodesic of the form l(t) = exp tXi(x) for x ∈ Λ to be a primarygeodesic.

Remark 2.5. The upshot of this lemma is that if we can join two regular points bya primary geodesic, then the entire decomposition is preserved. If we can connectenough of the regular points together in this fashion (enough, of course, meaninga conull set), we will have established that the Oseledec decomposition is preservedby parallel translation. From there, we simply define, via parallel translation, adecomposition on all of Fx, which is consistent with the Oseledec decomposition.Of course, we need to demonstrate that we can join enough regular points via theseprimary geodesics.

2.2.1. Ultraregular Points. We now wish to establish the existence of a conull setof regular points with some special properties.

Proposition 2.11. There exists a conull set Λ0 ⊂ Λ such that if x ∈ Λ0 thenalmost every point of exp tXi(x) lies in Λ0 for all i = 1, 2, . . . , f and all t ∈ R.

We begin the proof by noting that we need only show the proposition holds in aneighborhood of every point. Throughout this section, then, we fix an x ∈ Fx, anda neighborhood U of x which we may assume is flat affine space.

Lemma 2.12. Suppose x, y ∈ U ∩ Λ. If Vi(x) ∩ Vi(y) ∩ U 6= ∅, then Vi(x) = Vi(y)for any i = 1, 2, . . . , f .

Proof. Pick a ∈ A such that ai < 1. Let z ∈ Vi(x) ∩ Vi(y) ∩ U . Then there existsgeodesic l1(t) = exp(tZ1) with Z1 ∈Wi(x) joining x to z, a geodesic l2(t) = exp tZ2

with Z2 ∈ Wi(y) joining y to z, and there exists l3(t) = exp tZ3 joining x to y.Since Z1 lies in Wi(x), and Z2 lies in Wi(y), both li’s are contracting for a. Hence‖Tanx(Z1)‖, ‖Tany (Z2)‖ → 0 exponentially as n → ∞. By the triangle inequality,


‖Tanx(Z3)‖ → 0 exponentially as n → ∞. Since Wi(x) consists of vectors withexactly this property, Z3 ∈Wi(x), and therefore, Vi(x) = Vi(y).

A Lipschitz Function. We now wish to view V1 : U ∩ Λ → {Hyperplanes in U}.Choose a fixed x0 ∈ U ∩Λ and a basis for U at x0. Use this basis to define a norm,so that for r > 0, Br(x) = {y | ‖x − y‖ < r}. Then there exists a > 0 such thatBa(x0) ⊂ U . Let U0 = Ba/4(x0).

Let L = exp tX1(x0). Then L intersects V1(x) transversally for all x ∈ Ba(x0) (ifnot, then L ⊂ V1(x) and therefore x0 ∈ V1(x) ∩ V1(x0) contradicting the previouslemma). For all x ∈ Ba(x0), define G(x) to be the intersection of V1(x) with L,and let Θ(x) be the angle between L and the plane V1(x). Since L is transversal toV1(x), Θ(x) > 0.

Lemma 2.13. There exists θ > 0 such that Θ(x) ≥ θ for all x ∈ U0 ∩ Λ.

Proof. Choose xn, x ∈ U0 ∩ Λ such that xn → x. If Θ(xn) 6→ Θ(x), then thereexists ε > 0 and a subsequence xnk such that xnk → x, but |Θ(xnk) − Θ(x)| ≥ ε.But then for large nk, V1(xnk) ∩ V1(x) ∩ U 6= ∅, contradicting the previous lemma.So we have Θ(x) is continuous on U0 ∩ Λ, and therefore, for θ > 0,Θ−1(θ, π/2] isan open set in U0 ∩ Λ. By shrinking U0 if necessary, the lemma follows.

Proposition 2.14. G : U0 ∩ Λ→ L is Lipschitz. More specifically,

|G(x)−G(y)| ≤ 7sin θ

‖x− y‖.

Proof. Let x, y ∈ U0∩Λ so that ‖x−y‖ = ε < a/2. If x, y ∈ L, then |G(x)−G(y)| =‖x−y‖. So, we assume that x /∈ L. Let P be the plane containing L and x, and forevery z, let l(z) be the line of intersection of V1(z) and P . Note, then, that Θ(x) isless than the angle between L and l(x) in P . Let B0 = U0 ∩P , a circle with radiusa/4 (since L ⊂ P , and L is a diameter of U0).

Since V1(x) and V1(y) do not intersect in U , l(x) and l(y) do not intersect inB0. Let R be the intersection of the line perpendicular to l(y) through x, and letr = |x−R|. So r < ε.

Assume that all lines are defined on all of Rn. If l(x) and l(y) never intersect,then

|G(x)−G(y)| ≤ r

sin Θ(x)<

ε

sin θ=‖x− y‖

sin θ.

So assume l(x) and l(y) intersect at the point S. Let C be the point of intersec-tion of l(x) and the line perpendicular to l(y) through G(y), and let γ = |C−G(y)|.Also, let D be the point on l(x) closest to G(y) and let δ = |D −G(y)|.

Exploiting the similarity of the triangles SCG(y) and SxR we have

δ < γ =r|S −G(y)||S −R| ≤ ε(|S −R|+ |R−G(y)|)

|S −R| = ε

(1 +|R−G(y)||S −R|

).

Now,

|R−G(y)| ≤ |R− x|+ ‖x− y‖+ ‖y −G(y)‖ ≤ a/2 + a/2 + 2a = 3a

10 EDWARD R. GOETZE

by Lemma 2.15 below. Since R ∈ Ba/4(x0) and S /∈ Ba(x0), |S −R| > a/2. Thus,

δ < ε

(1 +|R−G(y)||S −R|

)< ε

(1 +

3a|S −R|

)< ε

(1 +

3aa/2

)= 7ε.

Finally, using the right triangle G(x)DG(y),

|G(x)−G(y)| ≤ δ

sin Θ(x)≤ 7ε

sin θ=

7sin θ

‖x− y‖,

as claimed.

The proof shows that G is uniformly continuous on U0∩Λ. Hence, we can extendG to a continuous and Lipschitz function on all of U0.

Lemma 2.15. Using the notation from Lemma 2.14, |x − G(x)| < 2a for all x ∈U0 ∩ Λ.

Proof. If l(x) and l(x0) do not intersect in P , then l(x) and l(x0) are parallel, andsince L is perpendicular to V1(x0), the closest point on l(x) to l(x0) is G(x). But|x−x0| < a/4 so |x0−G(x)| < a/4 and therefore |x−G(x)| ≤ |x−x0|+|x0−G(x)| ≤a/2.

Suppose now that l(x) and l(x0) intersect at S. Then |x0−S| > a. Let k be theline tangent to B0 through S such that if K is the intersection of k and L, thenG(x) lies on L between x0 and K. Let J be the intersection of k and B0. Since x0

is the center of B0 and k is tangent to B0, the triangles x0JK and x0JS are bothright triangles. Note |x0 − J | = a/4. Hence,

|J − x0||x0 − S| =

a/4|x0 − S| <

a/4a

= 1/4.

Thus,

|J − S||x0 − J | =

√1−

( |x0 − J ||X0 − S|

)2

>

√154

> 3/4.

By similar triangles,

|J −K||x0 − J | =

|x0 − J ||J − S| ≤

1/43/4

= 1/3.

But,

|x0 −K| ≤ |x0 − J |+ |J −K| ≤ |x0 − J |+ |x0 − J |3

< a,

and as G(x) lies on L between x0 and K, this implies |x0−G(x)| < a. From whichwe conclude

|x−G(x)| ≤ |x− x0|+ |x0 −G(x)| < a/4 + a < 2a,

which proves the lemma.


A Type of Fubini’s Theorem.

Theorem 2.16. Consider a Lipschitz function f : Rm → Rn with m > n. If U isa Lebesgue measurable set, then

∫

U

Jnf(x)dLmx =∫

RnHm−n(U ∩ f−1{y})dLny.

Here, Ln is n-dimensional Lebesgue measure, and Hm is m-dimensional Hausdorffmeasure. Jnf(x) = ‖ ∧n Df(x)‖ is the n-dimensional Jacobian of f at x. (Thisis well defined as Lipschitz functions are differentiable almost everywhere, 3.1.6 in[4].)

Proof. This is 3.2.11 from [4].

By applying this theorem, we see that a type of Fubini’s Theorem result holds, andwe may conclude that almost every point on almost every V1(y) ∩ U0 lies in Λ.

Lemma 2.17. Let S ⊂ {1, 2, . . . , f}, let WS(x) =⋂i∈SWi(x), and let VS(x) =

expWS(x).

(1) There exists a conull dense Λi ⊂ Λ such that if x ∈ Λi then almost everypoint in Vi(x) lies in Λi.

(2) There exists a conull set ΛS ⊂ Λ such that if x ∈ ΛS, then almost everypoint of VS(x) lies in ΛS.

Proof. The first statement has been proven above. To see the second statement, as-sume that S = {i, j}. Note that each Vi(x) is a hyperplane, and that by Lemma 2.9,Vi(x) ∩ Vj(x) are parallel hyperplanes of codimension 2 for all x. Hence, the usualFubini’s Theorem applies. Similarly, for general S, the FS(x) are also hyperplanesof higher codimension, and so the usual Fubini’s theorem still holds.

Proof of Proposition 2.11. Let Λ0 =⋂Sj

ΛSj where Sj = {1, 2, . . . , f} \ {j}.Proposition 2.18. Fix x ∈ Λ0. Then there exists a conull set Λx ⊂ Λ0 such thatfor every y ∈ Λx there exists a path consisting of broken primary geodesics from xto y.

Proof. Pick a neighborhood U of x which we assume is flat affine space. Sincex ∈ Λ0, the set Γ1 = {exp tX1(x)}t∈R ∩ Λ0 is conull in {exp tX1(x)}t∈R. ByLemma 2.10, if y ∈ Γ1, then Xi(x) is parallel to Xi(y) for all i, and therefore,exp tX2(y) lies in the plane

⋂fj=3 Vj(x) for all y ∈ Γ1, t ∈ R. By Fubini’s Theorem,

the set Γ2 = {{exp tX2(y)}t∈R ∩ U}y∈Γ1 is conull in⋂fj=3 Vj(x) ∩ U . Further,

for every z ∈ Γ2, Xi(x) is parallel to Xi(z) for all i. By Fubini again, Γ3 ={{exp tX3(z)}t∈R∩U}z∈Γ2 is conull in

⋂fj=4 Vj(x)∩U . We can continue constructing

these Γi’s for all i = 1, 2, . . . , f . Let Λx = Γf , then Λx is conull in U , and byconstruction, there exists a broken primary geodesic from x to y for any y ∈ Λx.

If V is another neighborhood of flat affine space which intersects U , then thereexists y ∈ Λx ∩ V . Repeating the above argument for y, we may conclude that Λxis conull in U ∪ V . The proposition now follows since we can cover Fx with a finitenumber of such neighborhoods.

12 EDWARD R. GOETZE

2.2.2. Regularity of the Oseledec Decomposition.

Theorem 2.19. Let X be a compact fiber bundle over Y with fibers F . Let G bea connected semisimple Lie group of higher rank without compact factors. SupposeG acts ergodically on X via bundle automorphisms preserving a volume densityand a Cr connection such that the fibers are autoparallel. If the algebraic hull ofα′ : G × X → SL(f), the derivative cocycle in the fiber direction, equals SL(f),then, by possibly having to pass to a finite algebraic cover of G, there exists anabelian subgroup A ⊂ G such that the Oseledec decomposition of TF correspondingto A is parallel, Cr regular, and everywhere defined on Fx for almost every x ∈ X.In particular, there exists a Cr section of the full flag bundle over T (Fx).

Proof. We follow the outline described in Remark 2.2.1. By Proposition 2.18, thereexists a conull set of regular points such that any two may be connected by a seriesof primary geodesics. By Lemma 2.10, the Oseledec decomposition is preservedby parallel translation along this path. Note that any path is homotopic to apath consisting of a series of primary geodesics. (This is certainly true locally, sowe need only cover our (compact) path with a finite number of neighborhoods.)Since R ≡ 0, parallel translation along homotopic paths is identical. Hence, theOseledec decomposition is preserved by parallel translation along any path. Definethe decomposition to be parallel translation to points in the complement of Λx. Byconstruction, the decomposition is parallel.

A similar result holds for certain actions of lattices on manifolds.

Theorem 2.20. Let G be as in Theorem 2.19. Suppose Γ ⊂ G is a lattice actingon an f -dimensional manifold M preserving a volume density and a Cr connection.If the algebraic hull of the derivative cocycle α : Γ ×M → SL(f) equals SL(f),and if π(Γ) ⊂ SL(f) contains a lattice, where π : G → SL(f) is the Superrigidityhomomorphism, then, by possibly having to pass to a finte cover of Γ, the Oseledecdecomposition on M corresponding to an abelian subgroup A ⊂ Γ is parallel, Cr

regular, and everywhere defined. So, there exists a Cr section of the full flag bundleover M .

Proof. The proof follows exactly as in Theorem 2.19 once we have establishedthe existence of a suitable abelian subgroup, i.e., we need a result comparableto Lemma 2.7. Since π(Γ) contains a lattice in SL(f), using Lemma 3.2 in [6] andits preceding remarks, the proof follows.

3. C(r,s) Superrigidity

We will need to prove a generalization of Zimmer’s Topological Superrigidityfor the sequel. The difference between the results here and those presented in [14]lie in the regularity of the sections under consideration. In [14], Zimmer proves aresult analogous to Theorem 3.4 for Cr sections. The point here is to strengthenthe geometric meaning of the previously constructed sections, and these sectionsare not Cr regular.


3.1. Preliminaries. Let G be a semisimple Lie group acting on set S. Thens ∈ S is called a parabolic invariant if there exists a parabolic subgroup Q ⊂ Gsuch that Q fixes s. In particular, if G acts by automorphisms of a principal H-bundle P →M , V is an H-space with EV →M the associated bundle, a parabolicinvariant section is a section of EV invariant under a parabolic subgroup of G.

Let P →M be a principal H-bundle on which G acts via bundle automorphisms.If π : G → H is a homomorphism, then a section s : M → P is called totally π-simple if for g ∈ G and m ∈M ,

s(gm) = g.s(m).π(g)−1.

Here, of course, M and P are right G-spaces, and P is a left H-space.Let X be a fiber bundle over Y with fibers F . For any function φ : X → M ,

and any neighborhood U ⊂ Y trivializing X, we have φ |p−1(U): F × U →M . Callφ : X → M a C(r,s) function if for any such U , φ |p−1(U) is Cr as a function of Fand Cs as a function of U .

Similarly, if P → X is a principal H-bundle over X, V an H-space, and Φ :P → V is an H-equivariant map, call Φ an H-equivariant C(r,s) function if locallyΦ is a Cr function of F and a Cs function of Y . Note that by H-equivariance, Φis necessarily C∞ as a function of H.

Although most of what follows will be valid for most (r, s), the most interestingresults follow when either r or s is measurable. Of particular interest to us is thecase where s is measurable. Let C(r,s)(X;EV ) be the set of C(r,s) sections of theassociated bundle EV over X.

Proposition 3.1. There exists a natural bijective correspondence between C(r,s)(X;EV )and H-equivariant C(r,s) maps P → V , and G-invariant elements of C(r,s)(X;EV )correspond to H-equivariant G-invariant C(r,s) maps P → V .

Proof. The proof is standard noting the appropriate changes in regularity. See [1],for example.

Let r ∈ [0,∞] and s be measurable. Call a set U ⊂ X a C(r,s) generic set if Uis conull in X and for almost every x ∈ U,Fx ∩U is open, dense in Fx, where Fx isthe fiber in X through the point x.

Now suppose G acts by automorphisms of a principal H-bundle P → X whereH is an algebraic group. Assume that the G action on X is ergodic. An algebraicsubgroup L ⊂ H is called a C(r,s) algebraic hull for the G action on P if

(1) there exists a G-invariant C(r,s) generic set U ⊂ X and a G-invariant C(r,s)

section of EH/L |U→ U , and(2) the first assertion is false for any proper algebraic subgroup of L.

Proposition 3.2. Assume the situation described in the previous definition. Then

(1) C(r,s) algebraic hulls always exist.(2) Any two C(r,s) algebraic hulls are conjugate in H.(3) If there exists a G-invariant C(r,s)-generic U ⊂ X and a C(r,s) section of

EH/L1 |U→ U , then L1 contains a C(r,s) hull.

The first and last statements follow from the descending chain condition on alge-braic subgroups. The second statement will require the following

14 EDWARD R. GOETZE

Lemma 3.3. Let G be ergodic on X, P → X a principal H-bundle on whichG acts by bundle automorphisms. Let H be a real algebraic group and let V bea quasi-algebraic H-space. Let φ be a G-invariant C(r,s) section of EV → X,with Φ : P → V the corresponding H-equivariant G-invariant C(r,s) map. Thenthere exists a G-invariant C(r,s) generic U ⊂ X and an H-orbit O ⊂ V such thatφ(P |U ) ⊂ O.

Proof. Since Φ : P → V is H-equivariant, factoring by the H action, we obtainΦ̃ : X → H\V . The H action on V is quasi-algebraic and therefore tame. As the Gaction on X is ergodic, Φ̃ is constant on U ⊂ X a conull set. By Fubini, for almostevery x ∈ U , Fx∩U is conull in Fx. But then Φ̃ |Fx is a Cr function constant on theconull set Fx ∩U ⊂ Fx. Hence Fx ∩U = Fx for such x, and U is C(r,s) generic.

Proof of Theorem 3.2. Let Ui ⊂ X be two C(r,s) generic G-invariant sets withLi ⊂ H algebraic subgroups and Φi : P |Ui→ H |Li G-invariant H-equivariantC(r,s) maps. Let Φ = (Φ1,Φ2) : P |U1∩U2→ H/L1 ×H/L2. (Note that U1 ∩ U2 isa C(r,s) generic set.) Apply Lemma 3.3 to U1 ∩ U2, to obtain a C(r,s) generic setU ⊂ U1 ∩ U2 and a single H orbit O ⊂ H/L1 × H/L2 such that Φ(P |U ) ⊂ O.Note that the stabilizer of a point in H/L1 × H/L2 under the H action takesthe form h1L1h

−11 ∩ h2L2h

−12 for some h1, h2 ∈ H. Thus, we have Φ(P |U ) →

H |h1L1h−11 ∩h2L2h

−12

. By definition of C(r,s) hull, h1L1h−11 ∩h2L2h

−12 contains L1 so

that L1 is contained in a conjugate of L2. Similarly, L2 is contained in a conjugateof L1. Since the Li’s are algebraic subgroups, L1 and L2 are actually conjugate.

Let G act via principal bundle automorphisms on P (X,H) with C(r,s) algebraichull L. The G action is C(r,s) complete if

(1) there exists a C(r,s) generic set U such that for almost every x ∈ U , Fx ⊂ U ,and

(2) there exists a C(r,s) section EH/L |U→ U .

Given a principal H-bundle P → X, and V an H-space, a section φ of EVis called effective for P if H acts effectively on Φ(P ) where Φ : P → V is thecorresponding H-map to φ. Suppose G acts ergodically and C(r,s) completely onP where H is an algebraic group and V is an algebraic variety. Then a C(r,s)

section φ : X → EV is C(r,s)-G-effective if it is effective for P1 ⊂ P where P1 is aG-invariant reduction to L ⊂ H, the C(r,s) algebraic hull.

We are now ready to state the main result of this section.

Theorem 3.4 (C(r,s) Superrigidity). Let G be a connected semisimple Lie group,R-rank(G) ≥ 2, with G acting via bundle automorphisms on P (X,H), H an alge-braic R-group, V an R-variety on which H acts algebraically, and G acting ergod-ically on X with respect to a probability measure µ where supp(µ) = X. Assumethe action is C(r,s) complete. If there exists a G effective C(r,s) parabolic invariantsection ψ of EV → X, then, by possibly passing to a finite cover of G, there exists

(1) a homomorphism π : G→ H,(2) v0 ∈ V , and(3) a totally π-simple C(r,s) section s of P → X

such that ψ is the associated section (s, v0), i.e. ψ(x) = [s(x), v0].


Remark 3.1. (1) We can dispense with the completeness assumption, how-ever, the totally π-simple C(r,s) section s will be defined only where ψ isdefined. So, in essence, the necessity of the completeness assumption issimply to ensure that the section ψ exists.

(2) Effectiveness of ψ will ensure that we can see enough of the C(r,s) algebraichull by looking at V . This is to avoid the obvious degeneracy problems,e.g., if the C(r,s) algebraic hull were in the kernel of the H action on V .By passing to a suitable subquotient, we can still obtain results without theeffectiveness assumption. Let P1 ⊂ P be the G-invariant reduction to theC(r,s) algebraic hull L, and let N ⊂ L be the maximal normal subgroup ofL pointwise fixing Φ(P1). We can then obtain a C(r,s)-G-effective sectionof the V associated bundle to the principal bundle P1/N , and apply thetheorem to this new situation.

(3) The hypothesis concerning the parabolic invariant section is the key pointin extending Measurable Superrigidity to Topological and C(r,s) Superrigid-ity. The proof of all versions of Superrigidity depend on the existence ofparabolic invariance, however, in the measurable case, it is always possi-ble to deduce the existence of relevant parabolic invariant sections. Thisis not the case in the situations of higher regularity, thus the need for thisadditional assumption.

3.2. Proof of C(r,s) Superrigidity. By C(r,s) completeness, we may assume His the C(r,s) algebraic hull. Let Q be the parabolic subgroup such that ψ is Q-invariant. Let B ⊂ Q be a minimal parabolic subgroup. By Lemma 3.1, ψ corre-sponds to a Q-invariant, H-equivariant C(r,s) map Ψ : P → V . If Ψ0 : X → H\V isthe induced map on H orbits, then Ψ0 is B-invariant. Moore’s Theorem implies Bis ergodic on X, therefore, by Lemma 3.3, there exists a C(r,s) generic set X0 ⊂ Xand an H orbit V0 ⊂ V such that Ψ0(x) ∈ V0 for every x ∈ X0. We may assumeV0 = V .

Let L be the measurable algebraic hull of the G action on P . Let R be themeasurable G-invariant subbundle which is a measurable principal L bundle. Thenfor almost every q ∈ R, Ψ0(q) lies in a single L orbit, say W0 ⊂ V . Note thatW0 ⊂ V0 and H.W0 = V0.

Now define F : P×G/B → V so that F (p, g) = Ψ(g−1p). Since Ψ is B-invariant,F is indeed well-defined. From the proof of Measurable Superrigidity [15], we havethe following

Proposition 3.5. By passing to a finite (algebraic) cover of G,

(1) there exists a regular homomorphism σ : G→ L, and(2) for almost every r ∈ R, there exists a conjugate σr of σ in L,

such that for almost all r ∈ R, F (r, g) = σr(g).Ψ(r). Further, if L1 is the stabilizerin L of W0, then σ : G→ L/L1 is a surjection.

Next define Φ : P → Func(G/B, V ) as Φ(p)(g) = F (p, g) = Ψ(g−1p). Propo-sition 3.5 implies that for almost every q ∈ R, Φ(q) : G/B → W0 is a regularsurjective map. As B is parabolic, G/B is a complete variety, and therefore W0 isZariski closed in V . We also have the following

16 EDWARD R. GOETZE

Proposition 3.6. There exists Φ0 ∈Reg(G/B,W0) such that for almost every q ∈R, Φ(q) ∈ L.Φ0.

Proof. Since Ψ is L-equivariant, so is the map Φ : R → Reg(G/B,W0). Let Φ̃ :L\R : X → L\ Reg(G/B,W0) be the map on L orbits. Note that Φ̃ is B-invariantsince Ψ is B-invariant. Since B is ergodic on X, Φ̃ must be constant on a conullset.

Let R1 = {q ∈ R | Ψ(q) ∈ W0 and Φ(q) ∈ L.Φ0}. Then R1 is conull in R, andif P1 is the H saturation in P of R1, then P1 is conull in P and for every p ∈ P1,Φ(p) is regular and Φ(p) ∈ H.Φ0 ⊂ Reg(G/B, V0).

By Theorem 3.3.1 in [15], the H action on Reg(G/B, V ) is tame, so H.Φ0 is openin its closure in C(r,s)(G/B, V ). If P2 = {p ∈ P | Φ(p) ∈ H.Φ0}, then since Φ isH-equivariant C(r,s) regular, P2 is C(r,s) generic.

Let S(V ) be the space of closed subvarieties of V . Define X : P2 → S(V )so that X (p) = Φ(p)(G/B). Then X is H-equivariant and partially G-invariant(i.e., if p, gp ∈ P2, then X (p) = X (gp)). Also, X (P2) ⊂ H.W0 ⊂ S(V ). If H1

is the stabilizer of W0 in S(V ), then we obtain an H-equivariant, partially G-invariant C(r,s) map P2 → H/H1. This corresponds to a G-invariant C(r,s) sectionof EH/H1 |U→ U , where U = G.p(P2) (p : P → X is the standard projection).Since H is the C(r,s) algebraic hull for the G action on P , we have H1 = H. So, Hstabilizes W0 and as W0 is closed, we have

W0 = V0 = V0 = V.

Thus, both H and L are transitive on V , and for every p ∈ P2, Φ(p) : G/B → V isa regular surjection.

To summarize, we have anH-equivariant C(r,s) map Φ : P2 → H.Φ0 ⊂ Reg(G/B, V ).However, we will need to know that p(P2) contains entire fibers, not just open densesubsets of fibers.

Lemma 3.7. Let U = p(P2). If Fx ∩ U is conull in Fx, then Fx ⊂ U .

Proof. Note that V is complete and a transitive H-space, and can therefore beembedded as a closed orbit in a projective space via a representation of H. Letf0 ∈ Fx. Choose {pn}n≥1 ∈ P2 such that pn → p0 and p(pi) = fi. Since Φ iscontinuous in the direction of F and H, and G/B is compact, Φ(pn) → Φ(p0)uniformly. As Φ(pn) ∈ H.Φ0, we can write Φ(pn) = hn.Φ0 for hn ∈ H. ByLemma 6.3 in [14], hn is bounded in PGL(n + 1), and therefore converges toh0 ∈ PGL(n + 1). Since H is algebraic, h0 lies in the image of H in PGL(n + l).We conclude that Φ(p0) = h0.Φ0.

Since Φ0 is surjective, and H is effective on V , the stabilizer of Φ0 in H is trivial,implying that Φ defines an H-equivariant C(r,s) map P2 → H, and therefore definesa C(r,s) section s of P2 → U ⊂ X(U = p(P2)) as follows: for m ∈ U , s(m) isthe element of Pm such that Φ(s(m)) = Φ0. We have Φ(p)(ag) = Ψ(g−1a−1p) =Φ(a−1p)(g). Thus, for every p ∈ P2, theG orbit of Φ(p) in Reg(G/B, V ) is containedin Φ(P ), and therefore in H.Φ0.


Using the proof of Measurable Superrigidity (or Lemma 3.5.2 from [15]), weconclude that for each p ∈ P2, there exists a homomorphism πp : G→ H such that

Φ(p)(g) = πp(g).Φ(p)(e).

Define πm = πs(m). Hence, for all m ∈ U, g ∈ G,

Φ0(g) = πm(g).Φ0(e).

For any a ∈ G,

Φ0(ga) = πm(ga).Φ0(e) = πm(g)πm(a).Φ0(e) = πm(g).Φ0(a).

So, for any g, a ∈ G,πm1Φ0(a) = πm2Φ0(a),

for any m1,m2 ∈ U . Again, Φ0 is surjective and H is effective on V , so we haveπm1(g) = πm2(g) for all such m1,m2. Hence, there exists π : G → H such thatπ = πm for all m ∈ U .

As Φ : P2 → H.Φ0 ⊂ Reg(G/B, V ) is injective on the fibers of P , we need onlyshow Φ(g.s(m).π(g)−1) = Φ(s(gm)) to see that s is totally π-simple. But,

Φ(g.s(m).π(g)−1)(a) = Ψ(a−1g.s(m).π(g)−1)= π(g).Ψ(a−1g.s(m))= π(g).Φ(s(m))(g−1a)= π(g).Φ0(g−1a)= Φ0(a) = Φ(s(gm))(a).

Finally, it is routine to verify that ψ is the section associated to s and Φ0(e) ∈ V .

4. Geometric Implications

The purpose of this section is to explore some of the geometric consequences ofour previous work.

4.1. Connection Preserving Actions on Fiber Bundles. Let X be a compactfiber bundle over Y with fibers F . Let G act ergodically on X via fiber bundleautomorphisms preserving a smooth connection and a volume density, where G isa connected semisimple Lie group of higher rank. Also, assume that the fibers areautoparallel. Let

β : G× Y → Diff(F ),

α : G×X → GL(x), and

α′ : G×X → GL(f)

be the cocycles described above in Section 2.1. Let L be the algebraic hull of α′.

Theorem 4.1. (1) If L is compact, then there exists a smooth Riemannianmetric g on F with respect to which F has a transitive group of isometries,i.e.,

F ∼= Isom(F, g)Isom(F, g)x

,

(2) If L = SL(f), where f =dim(F ), then F is a torus.

18 EDWARD R. GOETZE

Remark 4.1. Since isometry groups of compact manifolds are rather special (par-ticularly when they are large in comparison to the manifold), as are their closedsubgroups, we have special restrictions on the possibilities for F . In low dimen-sions, these possibilities are few and easy to calculate. Examples not precluded bythese results are also easy to construct.

4.1.1. Proof of Theorem 4.1.L Compact. Compactness of L implies the existence of a measurable invariant met-ric in the direction of the fibers. Equivalently, there exists a β-invariant functionΦ : Y → {measurable Riemannian metrics on F}.

Theorem 4.2. There exists a smooth invariant metric on F .

Proof. This is a main result of [2]. Note that this requires the fibers to be autopar-allel.

Thus, we have a cocycle β : G× Y → Diff(G) and a β-invariant function Φ : Y →Met(F ), the space of Riemannian metrics on F . Next, we employ

Theorem 4.3. The Diff(F ) action on Met(F ) is tame.

Proof. Theorem 7.4 of [3] constructs a slice for the action of Diff(F ) on Met(F ). Inparticular, for every m ∈ Met(F ), there exists a submanifold Sm of Met(F ) suchthat there exists a local cross section χ : Diff(F )/Diff(F )m → Diff(F ) defined ona neighborhood U of the identity coset such that the map U × Sm → Met(F ) is ahomeomorphism onto a neighborhood of m (Diff(F )m being the stabilizer of m inDiff(F )). This is enough to ensure the action of Diff(F ) on Met(F ) is tame. See,for instance, 2.1.12 in [15].

Applying the Cocycle Reduction Lemma, 5.2.11 in [15], there exists g ∈ Met(F )such that β is equivalent to β′ : G × Y → Diff(F )g = Isom(F, g). As G actsergodically on X, it must act ergodically on F (under the appropriate measurabletrivialization of X ∼= Y × F ), therefore F has an ergodic group of isometries.Compactness of F allows us to conclude that F therefore has a transitive group ofisometries.L Noncompact. Let P → X be the principal SL(f) bundle over X. Let V be thespace of full flags on Rf . Theorem 2.19 establishes the existence of a section Φ ofthe associated bundle EV → X. Note that this section varies measurably in theY direction on X, but is Cr regular in the direction of F . Also, note that this isa parabolic invariant section, with the parabolic subgroup being the stabilizer ofthe appropriate flag. We apply C(r,s) Superrigidity to conclude that there exists asection φ of P → X with the same regularity as Φ, and v ∈ V fixed by our parabolicsubgroup, such that Φ is associated to (φ, v). We now restrict ourselves to somefiber Fx with a conull set of regular points. Since Φ is parallel, Cr regular, anddefined on all of Fx, so is φ, i.e. there exists linearly independent parallel vectorfields on Fx. Since T ≡ 0 (for the connection on Fx), parallel vector fields arecommuting. The commuting vector fields now yield a locally free, transitive Rfaction on Fx, allowing us to conclude F is homeomorphic to a torus.


4.1.2. Examples. It is possible to construct examples of bundles whose fibers F areany of those allowed by Theorem 4.1.

Example 4.1. Let i : G ↪→ H be the inclusion of G into H. It is possible to chooseH = SL(n) with n large enough so that Tf ⊂ H as a closed subgroup and G and Tfcommute. Let Γ be a cocompact torsion free irreducible lattice in H. If X = H/Γ,then X admits a G action and a Tf action that commute. If Y = X/Tf , we havea G action on Y that lifts to a G action on X. Ergodicity of G on X follows byirreducibility of Γ (Moore’s Ergodicity Theorem), and the existence of a connectionpreserved by the G-action follows by reductiveness of X, [11].

Example 4.2. The construction above can generalized to show that F can be thehomogeneous space of any compact Lie group K. Repeat the construction above,replacing Tf with any compact K ⊂ H commuting with G. Thus, X = H/Γ andY = X/K. Now choose F such that K acts by diffeomorphisms on F , and formthe associated fiber bundle EF to X with fibers F , i.e.,

EF =X × FK

.

The G-action on E will be connection preserving, inheriting this property from theaction of G on X. Also, if K acts transitively on F then the G-action on E willbe ergodic. Hence, by setting F = K/K0, we obtain any homogeneous space ofany compact Lie group. Of course, the fiber bundle we obtain will generally be of avery large dimension, since H itself is quite large. It is an interesting problem todetermine, for a given G, the possible types of fibers for a fiber bundle X of a givendimension.

We remark that this construction illustrates a fundamental difference betweenactions of a connected Lie group on a fiber bundle considered in Theorem 4.1and actions of a lattice subgroup on a manifold considered in Theorem 4.4. In theexample just constructed, we were able to exploit properties of compact principalK-bundles to obtain a fiber bundle whose fiber admit a transitive group of isometries.The analogous construction for the action of a lattice Γ on a manifold M wouldrequire a homomorphism

Γ→ Isom(M),

which by ergodicity, must have non-finite image. Using the rigidity and arithmetic-ity theory of lattices in higher rank semisimple Lie groups, such a homomorphismis possible only if the complexifications of G and Isom(M) have simple componentswhich are isomorphic, see [8] and [12]. Taking M = K, where K is any compactLie group, we see that this rarely possible.

4.2. Connection Preserving Actions of Lattices. The following generalizesresults in [6].

Theorem 4.4. Let Γ ⊂ G be an irreducible lattice in a higher rank semisimpleLie group without compact factors. Suppose Γ acts ergodically on a compact n-dimensional manifold M preserving a volume and a smooth connection. Let α :Γ×M → SL(n) be the derivative cocycle with measurable algebraic hull L.

(1) If L is compact, Γ acts isometrically on M preserving a smooth Riemannianmetric.

20 EDWARD R. GOETZE

(2) If L = SL(n) and π(Γ) ⊂ SL(n) contains a lattice, where π is the Super-rigidity homomorphism, then M admits a torus as a finite affine cover.

4.2.1. Proof of Theorem 4.4. In the first case, where L is compact, the proof followsas in Theorem 4.1. In the second case we employ Theorem 2.20 to deduce theexistence of a Cr section φ of the flag bundle EV associated to the principal SL(n)bundle P →M . We wish to show that modulo a finite subgroup, this decompositioncomes from a smooth framing. Let {Xi} be the measurable framing associated withthis Cr decomposition. If x is a regular point, and C is a loop at x, then let PC beparallel translation along the loop C. Then

PC(Xi(x)) =∑

j

Hij(C)Xj(x).(4)

for some matrix H(C) = (Hij(C)). Since the measurable framing is associated tothe Cr decomposition, we have Hij(C) is a diagonal matrix for all C, i.e.,

PC(Xi(x)) = Hii(C)Xi(x).(5)

Lemma 4.5. If γ ∈ Γ such that x and γx are regular points, then

π(γ)H(γ ◦ C) = H(C)π(γ),

for any loop C at x, where π is the Superrigidity homomorphism.

Proof. Using Equation 5 above and the fact that Γ preserves the connection, andhence commutes with parallel translation, we have

γPC(Xi(x)) = PC(γXi(x))

= PC(ε(γ, x)∑

k

πik(γ)Xk(γx))

=∑

k

ε(γ, x)πik(γ)Pγ◦C(Xk(γx))

=∑

k,l

ε(γ, x)πik(γ)Hkl(γ ◦ C)Xl(γx),

where ε(γ, x) is an element in the (compact) center of SL(n), and π : Γ→ SL(n) isthe homomorphism from Superrigidity, and πij(γ) is the (i, j) entry of the matrixπ(γ).

On the other hand, we have,

γPC(Xi(x)) = γ∑

k

Hik(C)Xk(x)

=∑

k

Hik(C)(γXk(x))

=∑

k

Hik(C)(ε(γ, x)∑

l

πkl(γ)Xl(γx))

= ε(γ, x)∑

k,l

Hik(C)πkl(γ)Xl(γx).

By equating and summing these two equations, we obtain the desired conclu-sion.


Proposition 4.6. H(C) = ±I.

Proof. Since Γ action is ergodic and preserves a volume form, we haveH(C) must liein SL(n). Additionally, from Equation 5 we know that H(C) is a diagonal matrix.Hence, the proposition follows once we demonstrate H(C) is a scalar matrix.

Using Lemma 4.5 and diagonality of H(C), we have

πij(γ)Hjj(γ ◦ C) = Hii(C)πij(γ).(6)

for all γ ∈ Γ such that x and γx are regular points. Since we can choose a finiteset {γi} of generators for Γ and an x such that {γix} is a regular for all i, we havethat for a fixed x, Equation 6 holds for all γ. By Zariski density of π(Γ), we haveEquation 6 holds for any matrix in SL(n). By fixing j and varying i with anymatrix with a nonzero (i, j) term, we see that all entries Hii(C) are equal.

So, modulo a finite subgroup, we have established the decomposition from aframing. Arguing as in [6], on some suitable covering, we actually have a framing,and as in Theorem 4.1, we conclude M must be a torus, thereby completing theproof of Theorem 4.4.

4.2.2. A Few Corollaries.

Corollary 4.7. Let Γ be a lattice in SL(n,R) acting ergodically on an n-dimensionalmanifold M preserving a volume density and a connection. Then the conclusion toTheorem 4.4 holds.

Proof. If L is not compact, then from Superrigidity, we know that π must be asurjection with finite kernel, hence π(Γ) is a lattice. The result now follows fromTheorem 4.4.

Corollary 4.8. Suppose SL(n,Z), n ≥ 3, acts ergodically on an n-dimensionalmanifold M preserving a connection and a volume density. Then M admits a torusas a finite affine cover.

Proof. Since SL(n,Z) is a noncocompact lattice in SL(n,R), the first case in The-orem 4.4 is not possible. That the second case holds follows from Corollary 4.7.

References

1. A. Borel and N. Wallach. Continuous Cohomology, Discrete Subgroups, and Representationsof Reductive Groups. Princeton University Press, 1980.

2. Anthony Capozzoli. Bundle automorphisms for fiber G-structures of finite type. GeometriaeDedicata. To appear.

3. David G. Ebin. The manifold of riemannian metrics. In Global Analysis (Proc. Sympos. PureMath., Vol XV, Berkeley, 1968), pages 11–40. Amer. Math. Soc., Providence, R.I., 1970.

4. Herbert Federer. Geometric Measure Theory. Springer, New York, 1969.5. Renato Feres. Actions of discrete linear groups and invariant continuous connections. Preprint.6. Renato Feres. Connections preserving actions of lattices in SL(n,R). Israel Journal of Math-

ematics, 79:1–21, 1992.7. S. Hurder. Rigidity for Anosov actions of higher rank lattices. Annals of Mathematics,

135(2):361–410, 1992.8. F.E.A. Johnson. On the existence of irreducible discrete subgroups in isotypic Lie groups of

classical type. Proceedings of the London Mathematical Society, 56:51–77, 1988.9. A. Katok and J. Lewis. Global rigidity results for lattice actions on tori and new examples of

volume preserving actions. Preprint.

22 EDWARD R. GOETZE

10. A. Katok and J. Lewis. Local rigidity for certain groups of toral automorphisms. Israel Journalof Mathematics, 75(2–3):203–241, 1991.

11. Shoshichi Kobayashi and Katsumi Nomizu. Foundations of Differential Geometry. John Wiley& Sons, New York, 1963.

12. G. A. Margulis. Discrete Subgroups of Lie Groups. Springer-Verlag, New York, 1991.13. Peter Walters. An Introduction to Ergodic Theory. Springer-Verlag, New York, 1982.14. Robert J. Zimmer. Topological superrigidity. Preprint.15. Robert J. Zimmer. Ergodic Theory and Semisimple Groups. Birkhauser, Boston, 1984.16. Robert J. Zimmer. Ergodic theory and the Automorphism Group of a G-structure. In C. C.

Moore, editor, Group representations, ergodic theory, operator algebras, and mathematicalphysics, pages 247–278. Springer, New York, 1987.

17. Robert J. Zimmer. Actions of simple Lie groups. In Workshop on Lie Groups, Ergodic Theoryand Geometry & Problems in Geometric Rigidity. M.S.R.I., Berkeley, CA, 1992.

University of Michigan, Department of Mathematics, 317 West Engineering Building,Ann Arbor, MI 48109-1092

E-mail address: [email protected]

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