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Rigidity of Transformation Groups in Differential Geometry

Karin Melnick

Communicated by Notices Associate Editor Chikako Mese

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Introduction and Framework

A generic Riemannian manifold has no isometries at all. But mathematicians and physicists do not really study such manifolds. The first Riemannian manifolds we learn about, and the most important ones, are Euclidean space, hyperbolic space, and the sphere, followed by quotients of these spaces, and by other symmetric spaces. These spaces go hand in hand with Lie groups and their discrete subgroups.

In this survey, symmetry provides a framework for classification of manifolds with differential-geometric structures. We highlight pseudo-Riemannian metrics, conformal structures, and projective structures. A range of techniques has been developed and successfully deployed in this subject, some of them based on algebra and dynamics and some based on analysis. We aim to illustrate this variety below.

Historical currents

As with much of modern geometry, this subject has intellectual foundations in the 1872 Erlangen Program of F. Klein. He proposed to study geometries via their transformation groups. Euclidean geometry thus comprises the properties and quantities invariant by translations, rotations, and reflections, while conformal geometry is further enriched with dilations and circle inversions. Klein’s classical geometries are homogeneous spaces—their transformation groups act transitively—and the groups are Lie groups.

The concept of Lie group arose from the search for a theory of symmetries of differential equations. In the 1930s, É. Cartan laid several foundations of differential geometry with Lie groups: he found a complete set of local differential invariants for many important and interesting geometric structures via his method of moving frames. He defined the notion of -structure of finite type and his beautiful theory of Cartan connections, in which a manifold is “infinitesimally modeled” on one of Klein’s homogeneous spaces.

Several decades later, geometers such as S. Sternberg, V. Guillemin, and S. Kobayashi proved that the local symmetries of any -structure of finite type form a finite-dimensional Lie pseudogroup. This notion was then greatly extended by Gromov in the 1980s with his rigid geometric structures, which arose from his theory of partial differential relations.

Meanwhile, in the study of discrete subgroups of Lie groups, A. Weil, G. Mostow, and, ultimately, G. Margulis discovered remarkable rigidity of lattices in semisimple Lie groups. Among the breakthroughs in these monumental results was the surprisingly powerful application of ergodic theory.

Notions of rigid geometric structures will be explained and partially defined below. The reader could opt to read that section now, to read it later, or to skip it. The article can be well appreciated by just considering the examples of geometric structures given. After this introductory section, the three main sections on isometries, conformal, and projective transformations could be read separately without much extra effort, as each is relatively self-contained. The number of references is limited; readers seeking additional references may refer to other surveys or contact the author.

Zimmer’s Program

In the 1970s and 80s, R. Zimmer developed rigidity theory of Lie groups and their lattices in the context of their actions on manifolds (see Zim84). A lattice in a Lie group is a discrete subgroup such that the quotient has finite Haar measure.

Let be a lattice in a simple Lie group ; we assume for this subsection and the next that is noncompact, has finite center, and is not locally isomorphic to or , meaning the Lie algebra is not isomorphic to that of either of these groups. Superrigidity in this case says that homomorphisms of to linear Lie groups are the restrictions of homomorphisms , up to some “precompact noise,” and possibly passing to a cover of . The linear manifestations of these complicated and mysterious objects are thus reduced to those of —an elementary algebraic matter, solved by Schur’s Lemma. (Superrigidity and the remaining results and conjectures of this section are valid for semisimple, but this entails additional definitions and assumptions.)

Zimmer’s Program asserts that, in many cases, homomorphisms from , or , as above to the group of volume-preserving diffeomorphisms of a compact manifold arise from a short list of algebraic constructions. We illustrate two such actions, along with invariant differential-geometric structures for each. Let for a connected Lie group and a cocompact lattice.

(1)

A homomorphism gives an action of on , preserving the volume determined by the Haar measure.

For semisimple, the Killing form on the Lie algebra is a nondegenerate bilinear form, invariant by conjugation via the adjoint representation, . It gives rise to a bi-invariant metric on , of indefinite signature when is noncompact. This Cartan-Killing metric descends to a -invariant pseudo-Riemannian metric on .

(2)

Any subgroup , the automorphisms of normalizing , acts on . The standard example is , and .

Any Lie group has a bi-invariant affine connection , defined by declaring left-invariant vector fields to be parallel. This connection descends to and is -invariant. Although is torsion-free only if is abelian, the existence of a -invariant affine connection on implies existence of a torsion-free affine connection that is also -invariant.

Pseudo-Riemannian metrics and affine connections are examples of -structures of finite type. For a lattice in a simple Lie group as above, an affine action of is an action on via a homomorphism to . We can refine the classification claim from Zimmer’s Program to say, infinite actions of or on a compact manifold preserving a -structure of finite type and a volume are affine actions—up to some additional “precompact noise,” for which we refer to Fis11 for details.

A famous conjecture of Zimmer is a dimension bound, with no invariant rigid geometric structure assumed: Let be the minimal dimension of a nontrivial representation of . For any compact manifold of dimension , any action of on by volume-preserving diffeomorphisms preserves a smooth Riemannian metric. Depending on , the conclusion can often be strengthened to the action being finite. Affine actions satisfy this conjecture, which can be proved with Zimmer’s Cocycle Superrigidity Theorem. Given a volume-preserving action of on any -dimensional manifold, for any , cocycle superrigidity gives a nontrivial homomorphism or a measurable, -invariant Riemannian metric on . In the case also preserves a -structure of finite type, Zimmer improved the measurable Riemannian metric to a smooth one.

For decades, researchers struggled to improve the invariant, measurable Riemannian metric in the general case. The breakthrough came in 2016, when Brown–Fisher–Hurtado proved the dimension conjecture for cocompact and any classical simple Lie group, aside from a few exceptional dimensions for types and . They also proved a dimension bound for smooth actions not necessarily preserving a volume: in this case, , or the action is finite. A sharp example to have in mind is the action of , or one of its lattices, on . Their proof opens up new avenues for the classification conjecture of Zimmer’s Program—see below.

Zimmer-Gromov Program

Gromov generalized the -structures of finite type in Zimmer’s earlier work to the richer rigid geometric structures. Inspired in large part by Zimmer’s work, he proved results for a wider class of Lie groups with actions preserving these structures. His vision to some extent parts ways with another, large branch of Zimmer’s Program, focused on rigidity of lattices in semisimple Lie groups, which has been demonstrated for a breathtaking range of actions.

Gromov attempted to formulate Zimmer’s conjecture about affine actions more broadly, in particular, without assuming a finite, invariant volume. In DG91, he and D’Ambra state the following.

Vague General Conjecture (VGC).

All triples , where is a compact manifold with rigid geometric structure and H is a “sufficiently large” transformation group, are almost classifiable.

“Large” here often means noncompact. It could also be a transitivity condition, such as having a dense orbit or an ergodic invariant measure. It could be a geometric condition of being essential, generally meaning it does not preserve a finer geometric structure subsidiary to . They note, “we are still far from proving (or even starting) this conjecture, but there are many concrete results which confirm it.” Some such results will be presented below.

Here are examples of actions that join the “almost classification” for large actions of arbitrary groups not necessarily preserving a volume. For a more complete list, see Gro88, 0.5.

3.

with closed—that is, an arbitrary homogeneous space. Assuming semisimple, an important class are the quotients by parabolic subgroups, algebraic subgroups of for which is a compact projective variety. These -actions never preserve a finite volume.

The basic example is projective space , with , and the stabilizer of a line in . This is a Klein geometry, in which the invariant notions are lines and intersection.

Another important parabolic Klein geometry is the round sphere , a homogeneous space of . It is the projectivization of the light cone in Minkowski space . It carries an invariant conformal structure, for which the invariant notion is angle.

4.

, where acts freely, properly discontinuously, and cocompactly on .

An example is the unit tangent bundle of a compact hyperbolic manifold, where and is the stabilizer of a unit vector in . The image of the holonomy representation of is . The geodesic flow on is furnished by a one-parameter subgroup of centralizing ; it is isometric for a metric obtained by pseudo-Riemannian submersion from the Cartan-Killing metric on .

5.

, where is open, and acts freely, properly discontinuously, and cocompactly on . The example of the conformal Lorentzian Hopf manifold is presented later.

The spaces in examples 4 and 5 both carry -structures, which are defined in the next section.

Gromov’s first key contribution, based on his theory of partial differential relations, is known as the Frobenius Theorem. At points satisfying a regularity condition, it can produce local transformations of from pointwise, infinitesimal input. The infinitesimal input corresponds to points in an algebraic variety. The Rosenlicht Stratification for algebraic actions on varieties yields a corresponding stratification by local transformation orbits in . A consequence is the Open-Dense Theorem: if an orbit for local transformations in is dense, then an open, dense subset is locally homogeneous, meaning that the pseudogroup of local transformations is transitive on .

The Open-Dense Theorem lends support to the VGC. In specific cases, researchers can show that and that the locally homogeneous structure is a -structure; some have obtained significant theorems in the Zimmer-Gromov Program by this route. But may not, in general, equal . Classifying compact -manifolds compatible with a given geometric structure is a challenging subject unto itself.

A beautiful alternative relating examples 1–2 with example 3 is given by results of Nevo–Zimmer on smooth projective factors. They use Gromov’s Stratification and his Representation Theorem to prove: given a connected, simple preserving a real-analytic rigid geometric structure on a compact manifold , together with a stationary measure, there is (i) a smooth, -equivariant projection of an open, dense to a parabolic homogeneous space ; or (ii) the Gromov representation of contains in its Zariski closure. A stationary measure is a finite measure, here assumed of full support, invariant by convolution with a certain finite measure on , but not necessarily under the -action.

Notions of rigid geometric structure

The infinitesimal data of a geometric structure on inhabits the frame bundle of , of the appropriate order. The frames at a point are the linear isomorphisms , where . These form a principal -bundle, denoted . For , a -structure is a reduction of to —that is, a smooth -prinicipal subbundle. The frames in determine the structure. For example, a -semi-Riemannian metric is a -structure for . The frames of are the bases of in which the inner product has the form .

The fiber comprises the first derivatives of coordinate charts at . The order- frame bundle has fibers comprising -jets of coordinate charts at with . Here and have the same -jet at if for all . The -jets at of diffeomorphisms of fixing form the group . A -structure of order is a reduction of to . For example, a linear connection on is a reduction to . Elements of are -jets of geodesic coordinate charts.

A -structure is of finite type if a certain prolongation process stabilizes, which essentially means there is such that, at all , the -frames at adapted to the structure are determined by the adapted -frames at for all (see Kob95). Conformal Riemannian structures, for example, are -reductions of , where . The prolongation gives a reduction of to a bigger group, namely the parabolic subgroup fixing a point of . É. Cartan found, for , the prolongation stabilizes here, and thus any conformal transformation is determined by the 2-jet at a point.

A -reduction of is the same as an equivariant map (and similarly in higher order). Gromov’s geometric structures of algebraic type are equivariant maps to algebraic varieties. They are rigid if a similar prolongation process stabilizes (see Gro88). Some very useful additional flexibility is afforded by varieties which are not necessarily homogeneous. For example, one may add to a -structure of finite type some vector fields, which correspond to maps to , and the result is again a rigid geometric structure. All rigid geometric structures in this article are understood to be of algebraic type.

For a Lie group and a closed subgroup, a Cartan geometry on modeled on comprises a principal -bundle and a -valued -form on , called the Cartan connection. The Lie group carries a left-invariant -valued -form, the Maurer-Cartan form , which simply identifies the left-invariant vector fields on with . The Cartan connection is required to satisfy three axioms, mimicking properties of . One of them says is an isomorphism on each , so it determines a parallelization of (see Sha97).

Essentially all classical rigid geometric structures canonically determine a Cartan geometry, such that the transformations correspond to automorphisms of the Cartan geometry—diffeomorphisms of lifting to bundle automorphisms of preserving . A conformal Riemannian structure in dimension , for example, determines a Cartan geometry modeled on , with and as above. The bundle in this case is the -reduction of obtained by prolongation of the -structure.

Cartan geometries have proven very useful for the study of transformation groups. Fundamental results of the Zimmer-Gromov theory of rigid geometric structures have been translated to this setting. A key asset is the curvature 2-form

Vanishing of over an open is the obstruction to having a -structure with .

The notion of -structure was developed by Ehresmann and later by Thurston Thu97: on a manifold , it comprises an atlas of charts to with transitions equal to transformations in (one on each connected component of the chart overlap). When for some , it is called complete; quotients of open subsets are called Kleinian.

Isometries of Pseudo-Riemannian Manifolds

By the classical theorems of Myers and Steenrod, the isometry group of a compact Riemannian manifold is a compact Lie group. Isometries of pseudo-Riemannian manifolds need not act properly; in particular, when the manifold is compact, this group can be noncompact. In this section we describe some important results on isometry groups and illustrate a couple techniques which have been influential. We focus on compact Lorentzian manifolds, mostly because there are as yet few answers to the corresponding questions in higher signature. The term semi-Riemannian means Riemannian or pseudo-Riemannian.

Simple group actions: Gauss maps

Let be a noncompact, simple Lie group. For a lattice, the -action on is locally free, meaning stabilizers are discrete. We sketch an argument of Zimmer, using the Borel Density Theorem, that any isometric -action on a finite-volume pseudo-Riemannian manifold is locally free on an open, dense subset. First suppose that acts ergodically, meaning that any -invariant measurable function is constant almost everywhere. In fact, any -invariant measurable map to a countably separated Borel space is constant almost everywhere; countably separated means there is a countable collection of Borel subsets such that for any points , some contains one of and not the other. This property holds for quotients of algebraic varieties by algebraic actions. Then we can apply this fact to -equivariant maps from to -algebraic varieties, sometimes called Gauss maps.

Let assign to the Lie algebra of the stabilizer of in , which lies in the disjoint union of the Grassmannians , . The map satisfies the equivariance relation . We conclude that there is a single -orbit containing for almost every . The volume on pushes forward to an -invariant finite volume on . After passing to Zariski closures, this orbit can be identified with an algebraic homogeneous space. The Borel Density Theorem says that , hence also the -orbit , must be a single point. Thus is an ideal for almost all , namely, because is simple, it is or .

In general, the metric volume decomposes into ergodic components, and for almost all . The points with comprise the fixed set of . The fixed set of any nontrivial isometry has null volume (assuming connected). Thus , and the stabilizer of is discrete for all in a full-volume subset . It follows from upper semicontinuity of that is open and dense.

By comparable arguments involving ergodicity and the Borel Density Theorem, Zimmer obtained, for compact, a Lie algebra embedding and concluded that, in particular, the real-rank—the dimension of a maximal abelian, -diagonalizable subgroup of —satisfies

Such an embedding for of real-rank at least two follows from Zimmer’s Cocycle Superrigidity Theorem; here, with a pseudo-Riemannian metric, a stronger result is obtained by a more elementary proof.

A more general embedding theorem for connected , not necessarily simple, preserving any rigid geometric structure, not necessarily determining a volume, was proved by Gromov Gro88. A version of this generality was proved for compact Cartan geometries by Bader–Frances–Melnick in 2009.

Connected isometry groups of compact Lorentzian manifolds

The group with the Cartan-Killing metric is a three-dimensional Lorentzian manifold with isometry group . This space has constant negative sectional curvature and is known as anti-de Sitter space, . (Together with its higher-dimensional analogues, this space is important in string theory and M-theory.)

Let be a compact, connected, Lorentzian manifold and a noncompact simple group. Zimmer’s bound is ; he in fact proved that the identity component is locally isomorphic to , with compact. Gromov improved Zimmer’s result to conclude

where is a Riemannian manifold, is a warping function, and acts isometrically, freely, and properly discontinuously on the warped product.

The complete determination of connected isometry groups of compact Lorentzian manifolds was simultaneously achieved by Adams–Stuck AS97aAS97b and Zeghib Zeg98aZeg98b.

Theorem (Adams–Stuck/Zeghib 1997/8).

The identity component of the isometry group of a compact Lorentzian manifold is locally isomorphic to , with compact, , and one of

,

, a -dimensional Heisenberg group,

an oscillator group, a solvable extension of by .

Adams and Stuck proceed by studying the dynamics of a Gauss map , where is the Lie algebra of . The map sends to the pullback of the metric on restricted to the subspace tangent to the -orbit at . It is equivariant for the representation on . Zeghib works with the average over of these forms on .

The geometry of when is a Heisenberg or oscillator group has not been completely described.

Dynamical foliations from unbounded isometries

The anti-de Sitter space can be identified with the unit tangent bundle of the hyperbolic plane. For a hyperbolic surface , the quotient is identified with , and the geodesic and horocycle flows with the right-action of the diagonal and upper-triangular unipotent subgroups, respectively, of .

The geodesic flow is an Anosov, Lorentz-isometric flow on . The weak-stable and -unstable foliations are by totally geodesic surfaces on which the metric is degenerate. These foliations, and , are the orbits of the two-dimensional upper-triangular and lower-triangular subgroups, respectively (see Figure 1). The horocycle flow is not hyperbolic in the dynamical sense. Nonetheless, it also preserves a foliation by totally geodesic, degenerate surfaces, namely . Zeghib calls this the approximately stable foliation of the flow. He proved the following Zeg99aZeg99b.

Theorem (Zeghib 1999).

Let be a compact Lorentzian manifold, and suppose that is noncompact. Any unbounded sequence in determines, after passing to a subsequence, a foliation by totally geodesic, degenerate hypersurfaces.

The space of all these approximately stable foliations gives a compactification of and reflects algebraic properties of .

Figure 1.

Two-dimensional anti-de Sitter space with bifoliation by lightlike geodesics.

Graphic for Figure 1.  without alt text

The construction of such foliations was suggested by Gromov, who provided the following argument. The reader may note the significance of the index , a way in which Lorentzian geometry is “closest” to Riemannian. Let in . For any , we may pass to a subsequence so that converges to a point . Equip with the pseudo-Riemannian metric . The Levi-Civita connection on the product is . The graphs of are totally isotropic and totally geodesic in , and they converge to an -dimensional, isotropic, totally geodesic submanifold near .

Because does not converge, cannot be a graph near , so it has positive-dimensional intersection with . This intersection is isotropic and geodesic in , so coincides with an isotropic geodesic segment near . The projection of to must be orthogonal to , so it is contained in the degenerate hyperplane . Therefore, has nonzero intersection with , which again must be isotropic; the projection to is contained in a degenerate hyperplane . A dimension count implies that the projection equals . Because is totally geodesic, it projects to the totally geodesic, degenerate hypersurface obtained by exponentiation of .

A classical theorem of Haefliger says that a compact, simply connected, real-analytic manifold admits no real-analytic, codimension-one foliation. Zeghib used his foliations to give another proof of the following theorem of D’Ambra D’A88.

Theorem (D’Ambra 1988).

Let . For a compact, simply connected, real-analytic Lorentzian manifold, is compact.

D’Ambra’s proof was a tour de force of Gromov’s theory of rigid geometric structures and belongs to the underpinning of the VGC.

Current questions

After reading about these wonderful accomplishments of the 1980s and 90s, the reader is likely to have at least two obvious questions: what about nonconnected groups? and, what about higher signature?

Some progress has been made on compact Lorentzian manifolds for which has infinitely-many components when is stationary—that is, it admits a timelike Killing vector field. The latter property implies that has closed timelike geodesics, or “time machines.” Piccione–Zeghib have a nice structure theorem for such spaces: contains a torus , and there is a Lorentzian quadratic form on such that .

For smooth, three-dimensional, compact Lorentzian manifolds, Frances recently completely classified those with noncompact isometry group, topologically and geometrically. The topological classification says that such a manifold is, up to a cover of order at most four:

for a cocompact lattice,

the mapping torus of an automorphism of .

In the first case, has a left-invariant metric. Geometries of the second type can have infinite, discrete isometry group. A corollary is an improvement of D’Ambra’s Theorem to smooth metrics, in the three-dimensional case.

For , it is not known which homogeneous, compact, -pseudo-Riemannian manifolds have noncompact isometry group. Much current work focuses on classifying left-invariant metrics on Lie groups. To illustrate the complexity of higher signature, we indicate Kath–Olbrich’s survey on the classification problem for pseudo-Riemannian symmetric spaces KO08. While Riemannian symmetric spaces were essentially classified by É. Cartan, and reductive pseudo-Riemannian symmetric spaces by Berger in 1957, a complete classification is not really expected, without restricting to a low index , or assuming some additional invariant structure.

Conformal Transformations

Compact Lorentzian manifolds are marvelous for the mathematical study of transformation groups, but the possibility of “time machines” makes them less appealing to physicists. Conformal compactifications of Lorentzian manifolds are, on the other hand, quite natural in relativity.

We have seen above that conformal Riemannian transformations are those preserving angles between tangent vectors. For arbitrary signature, we define the conformal class of to be , with ranging over smooth functions on . The conformal transformations are those preserving . In higher signature, these are the transformations preserving the causal structure—that is, the null cones in each tangent space.

Stereographic projection from a point is a conformal equivalence of with Euclidean space. Thus is conformally flat—locally conformally equivalent to a flat Riemannian manifold. The higher signature analogue of is a parabolic homogeneous space for , called the Möbius space . It is a two-fold quotient of ; the conformal class can be realized by , where are the constant-curvature metrics on the respective spheres. These metrics are also conformally flat. The conformal structure of a -pseudo-Riemannian metric, , determines a Cartan geometry modeled on . It is also a -structure of finite type, with order 2. Throughout this section, is an integer greater than or equal to .

The Ferrand-Obata Theorem

Figure 2.

Iterates of a conformal transformation of the sphere with source-sink dynamics.

Graphic for Figure 2.  without alt text

While the isometry group of a compact Riemannian manifold is compact, the conformal group need not be. A dilation of Euclidean space corresponds via stereographic projection to a conformal transformation of having source-sink dynamics under iteration—see Figure 2. A sequence of transformations has source-sink dynamics if there are points and in such that converges uniformly on compact subsets of to the constant map , and similarly uniformly on compact subsets of . The full conformal group of is the noncompact simple Lie group . This isomorphism comes from the fact that is the visual boundary in the conformal compactification of . Being the visual boundary of hyperbolic space is quite a special property. Lichnerowicz conjectured the following striking fact.

Theorem (Ferrand/Obata 1971).

Let be a compact Riemannian manifold. If is noncompact, then is globally conformally equivalent to .

(This theorem is also true for .) Lichnerowicz’s original conjecture was actually under the stronger assumption noncompact, and this was proved by Obata Oba71, using Lie theory and differential geometry. Ferrand proved the theorem in full generality LF71, using quasiconformal analysis and no Lie theory.

In 1994, Ferrand extended her result to noncompact and gave a streamlined proof of her original result. She uses conformal capacities to define a conformally invariant function on pairs of points. Sometimes it gives a metric on ; in this case, acts isometrically for , and, therefore, properly.

When does not define a distance, then Ferrand uses capacities to construct conformally invariant functions on distinct quadruples or triples of points, no three of which are equal, depending on whether is compact or not. In the former case, this function is essentially a log-cross-ratio, which can detect whether distinct points are converging under a sequence of conformal transformations. In the latter case, the function partly extends to the Alexandrov compactification and detects divergence to infinity, as well. With these functions, she reconstructs source-sink dynamics for unbounded sequences in when it acts nonproperly, and ultimately concludes or .

PDEs proof, CR analogue

When deforming a metric to one with desirable properties, a natural choice is to restrict to conformal deformations. The Yamabe Problem is a famous case: on any compact Riemannian manifold , there is with constant scalar curvature.

In 1995, Schoen published a different proof of Ferrand’s theorem Sch95, in the wake of his work on the completion of the Yamabe Problem. Given a Riemannian metric with scalar curvature function and Laplace-Beltrami operator , if has constant scalar curvature , then satisfies

where the conformal Laplacian

is an elliptic operator satisfying a certain conformal invariance. This equation can be applied to if has constant scalar curvature and . Schoen’s arguments are based on the analytic properties of the elliptic operator .

CR, or Cauchy-Riemann, structures model real hypersurfaces in complex vector spaces. On the unit sphere in , for example, the tangent bundle carries a totally nonintegrable, or contact, hyperplane distribution, equal at each to the maximal complex subspace of . In general, a CR structure (nondegenerate, of hypersurface type) on an odd-dimensional manifold comprises a contact hyperplane distribution and an almost-complex structure on , satisfying a compatibility condition: for any 1-form on with , the Levi form on given by is required to obey . In this case, is the imaginary part of a Hermitian form on . The real part of this Hermitian form is a semi-Riemannian metric on

A different choice of gives a metric conformal to , that is, related by a positive function on . The CR structure is strictly pseudoconvex if any is positive definite. In this case, there is a subelliptic Laplace operator on , satisfying a certain CR invariance. Schoen proved, in analogy with his conformal results, that the automorphisms of a strictly pseudoconvex CR manifold act properly, unless is CR-equivalent to , or to a Heisenberg group carrying a certain left-invariant CR structure. S. Webster previously proved such a theorem, in 1977, for compact with noncompact, connected automorphism group, following some analogy with Obata’s conformal proof.

Cartan connections proof, rank-one analogues

The CR sphere can be identified as a conformal sub-Riemannian space with the visual boundary of complex hyperbolic space . The quaternionic hyperbolic space and the Cayley hyperbolic plane have visual boundaries diffeomorphic to spheres, with respective differential-geometric structures invariant by the isometries of the interior. These boundaries of hyperbolic spaces form a family; they are parabolic homogeneous spaces with simple of -rank 1—see Table 1. Cartan geometries modeled on one of these homogeneous spaces canonically correspond to certain differential-geometric structures. Contact quaternionic and contact octonionic structures were introduced by O. Biquard in 2000.

Table 1.

Rank-one parabolic geometries.

structure
conformal Riemannian
strictly pseudoconvex CR
contact quaternionic
contact octonionic

Let be a manifold carrying one of the above structures, and suppose that a group of transformations acts nonproperly—that is, there are in and points in such that . Let be the Cartan bundle. Choose lifts and of and , respectively, to , such that .

Recall that the Cartan connection determines a parallelization of , invariant by the -action on . It follows that acts properly on ; in particular, diverges. This divergence is necessarily “in the fiber direction”—that is, there exist in such that converges to some