To a Lie group G and a closed subgroup H of G, one associates a differential geometric structure, called a Cartan geometry of type (G,H). Manifolds carrying such a geometry can then be thought of as "curved analogs" of the homogeneous space G/H. The most classical example of this concept is that G is the group of rigid motions and H is the orthogonal group, so G/H is Euclidean space of some fixed dimension n. A torsion free Cartan geometry of type (G,H) on an n-dimensional manifold M is then equivalent to a Riemannian metric on M.
Parabolic geometries are Cartan geometries of type (G,P), where G is a (real or complex) semisimple Lie group and P is a parabolic subgroup of G. The homogeneous models G/P are the so-called generalized flag manifolds. On the one hand, this allows one to use tools from representation theory to deal with the algebraic problems that usually come up in the study of Cartan geometries. On the other hand, a variety of (seemingly very different) geometric structures can be equivalently described as parabolic geometries. Usually, this equivalence is not obvious be the result of a difficult theorem, a typical example being the canonical Cartan connection for CR structures, which has been found by Tanaka in [T1] and Chern-Moser in [CM].
Although the name "parabolic geometries" is of rather recent origin, many examples of these structures were intensively studies between 1900 and 1940. Elie Cartan showed that classical projective structures, conformal structures, and three-dimensional CR structures can be equivalently described as parabolic geometries. Also Cartan's famous "five variables paper" [C], which related generic rank two distributions on five-dimensional manifolds to the exceptional Lie group G_{2}, actually shows that such a distribution is equivalent to a parabolic geometry. Also path geometries were a very active field of research at that time.
In the process of working up Cartan's ideas in the 1950's, the main interest was not in Cartan connections (which are the basic ingredient for a Cartan geometry) but rather in principal connections. This slowed down the development of the theory of Cartan geometry, but then new impact came from several directions: C. Fefferman's study of the Bergmann kernel of strictly pseudoconvex domains initiated research on invariants of CR structures and, more generally, parabolic invariant theory, see [F2]. Second, R. Penrose's twistor theory lead to renewed interest in conformal structures. These ideas were also generalized to almost quaternionic structures, which are another interesting example of a parabolic geometry. On the level of the homogeneous model, many of Penrose's ideas apply to arbitrary generalized flag manifolds (see [BE]), so the question of curved analogs naturally arose in this context. Finally, in his pioneering work [T2], N. Tanaka showed (in the setting of differential systems) that parabolic geometries are always determined by underlying geometric structures. Unfortunately, this work did not receive much attention.
My own work on parabolic geometries started with a series of joint papers with J. Slovak (Masaryk University, Brno) and V. Soucek (Charles University, Prague). This series was devoted to the simplest class of examples of parabolic geometries, called AHS (almost Hermitian symmetric) structures. These structures correspond to parabolic subgroups with Abelian nilradical, and they include the examples of conformal, projective, almost Grassmannian, and almost quaternionic structures. AHS structures are equivalent to certain classical first order G-structures, so the description of the underlying structure is very easy in this case. In [15], the first paper in the series, we developed a differential calculus based on Cartan connections and applied it to the case of AHS-structures. In particular, we introduced semiholonomic jet modules, which allow the construction of invariant differential operators in a purely algebraic way. The second paper in the series, [16], gives a simple construction of the normal Cartan connection on an AHS-manifold from the underlying geometric structure. The general machinery immediately leads to a complete set of obstructions to local flatness for AHS structures. These are analyzed for the main examples in [13]. In the third and last part of the series, [18], we proved existence of a large class of invariant differential operators on manifolds with AHS structures and deduced universal formulae for these operators in terms of underlying linear connections.
For the case of the homogeneous model G/P of a parabolic geometry, invariant differential operators can be equivalently described by homomorphisms between generalized Verma modules, which allows the application of deep tools from representation theory. On of these tools is the Jantzen-Zuckermann translation principle. In [ER], M. Eastwood and J. Rice developed a curved version of this principle for 4-dimensional conformal structures, which motivated a lot of further research in that direction. In my article [14], I developed a technique to translate invariant differential operators on AHS-manifolds. Some algebraic background is provided in [11].
In the joint article [17] with H. Schichl (University of Vienna), we gave a uniform description of the geometric structure underlying a general parabolic geometry, as well as a new procedure for constructing the canonical Cartan connection from this underlying structure. The construction is independent of the earlier procedures developed by N. Tanaka in [T2] and T. Morimoto in [M]. In the second part of the paper we analyzed basic properties of parabolic geometries. An improved version of the description of the underlying structure can be found in the first part of the overview article [31].
The explicit description of the structures underlying parabolic geometries led to new structures being identified as (equivalent to) parabolic geometries. A surprising example is provided by partially integrable almost CR manifolds of CR dimension and codimension two (of elliptic or hyperbolic type). This is studied in the joint article [23] with G. Schmalz (University of Bonn), which builds on earlier work of G. Schmalz and J. Slovak. In the elliptic case, these results show that certain types of generic rank four distributions on six-dimensional manifolds admit a canonical almost complex structure. This is the subject of the joint paper [26] with M. Eastwood (University of Adelaide).
There is an alternative description of Cartan geometries in terms of vector bundles and linear connections, which, for some examples, goes back to the work of Tracy Thomas in the 1930's. In the case of parabolic geometries, this leads to a new type of geometric objects, which are called tractors. For general parabolic geometries, the description in terms of tractor bundles and connections was developed in two joint articles, [22] and [19], with Rod Gover (University of Auckland). The advantage of this approach is that it leads to an invariant differential calculus, which was successfully applied to the study of invariants and the construction of invariant differential operators in many examples.
With the basic setting for general parabolic geometries at hand, the joint work with J. Slovak and V. Soucek was continued in this generality. A breakthrough was the article [21]. This gives a construction of a very large class of invariant differential operators generalizing the Bernstein-Gelfand-Gelfand resolutions from representation theory. The operators are obtained by compressing the twisted de-Rham sequence associated to the canonical connection on a tractor bundle to a pattern of higher order operators acting between certain subquotients. To obtain these subquotients and to compress the operators, one uses ideas from Lie algebra cohomology, which are related to Kostant's version of the Bott-Borel-Weil theorem. The resulting pattern of invariant differential operators is called a BGG sequence. The method not only provides the BGG sequence itself, but also a calculus relating it to the twisted de-Rham sequence.
More recently, the theory of BGG sequences has been developed further. In the joint article [34] with V. Soucek, we show that for certain classes of geometries, one obtains subcomplexes in all BGG sequences under appropriate torsion freeness and/or semi-flatness assumptions. In the special case of quaternionic structures, we could prove that a large family of complexes obtained in that way is elliptic. A particularly important example of a BGG sequence is the one associated to the adjoint representation. For the case of the homogeneous model, it was known that this BGG sequence can be interpreted as a deformation complex. The paper [35] studies the relation between infinitesimal automorphisms and deformations on one hand, and the adjoint BGG sequence on the other hand. Basically, infinitesimal automorphisms and deformations on the level of the Cartan geometry are described in terms of the twisted de-Rham sequence and the BGG machinery provides the relation to the underlying geometric structure. One obtains an explicit description of the formal tangent space to the moduli space of normal parabolic geometries. Moreover, for some structures subcomplexes of the type described above admit an interpretation as deformation complexes in the subcategory of torsion free and/or semi-flat geometries. In particular, one obtains an elliptic deformation complex for quaternionic structures. Another nice feature of the description of infinitesimal automorphisms on the level of Cartan geometries is that simple algebraic ideas lead to strong restrictions on the size of the automorphism group. This is studied in my article [30].
The first operator in each BGG gives rise to an overdetermined system of PDEs. The systems obtained in that way contain important examples, like infinitesimal automorphism equations and the conformal to Einstein equation. Ignoring the issue of invariance, a simplified version of the BGG machinery can be used to study general semi-linear PDEs with the same symbol. This is described in the joint article [32] with T. Branson (University of Iowa), M. Eastwood, and R. Gover.
It was mentioned above that twistor theory was developed for arbitrary generalized flag manifolds. The question of curved version of this is taken up in my article [29]. Given two parabolic subgroups P and Q of G, such that Q is contained in P, one easily obtains a functor which from a (normal) geometry of type (G,P) on M constructs a (normal) geometry of type (G,Q) on the total space of a certain natural fiber bundle over M. Principal bundle geometry leads to a characterization of the geometries obtained in that way in terms of the Cartan curvature. Using the BGG machinery, one obtains equivalent conditions in terms of the harmonic curvature, which are much more efficient. If these are satisfied, then the geometry actually comes from a smaller space, the twistor space. Twistor correspondences are obtained by combining the two constructions for different parabolics.
Using a mixture between general ideas for Cartan geometries and representation theory arguments, several elements of the general theory of parabolic geometries were developed. Similarly to the case of conformal structures, any parabolic geometry has a family of distinguished underlying connections. By analogy, they are called Weyl structures, and their general theory is developed in the joint article [25] with J. Slovak. There is also a general notion of distinguished curves in parabolic geometries, which generalizes the notion of conformal circles. The basic general theory of these distinguished curves is developed in the paper [28] with J. Slovak and V. Zadnik (Masaryk University, Brno). For parabolic contact structures, these distinguished curves in particular provide a concept of chains, which generalizes Chern-Moser chains on CR manifolds. In the joint article [33] with V. Zadnik, we give a direct construction of the Cartan connection associated to the path geometry defined by the chains. This leads to the surprising result that CR structures and integrable Lagrangean contact structures can be essentially recovered from their chains.
A direction of research which currently is particularly active is the ambient metric construction. Originally this was first done by Fefferman in [F1] for boundaries of strictly pseudoconvex domains and then by Fefferman-Graham in [FG] for conformal structures. There is also the closely related Poincaré-metric, which has gained a lot of interest during the last years because of its relation to theoretical physics via the AdS/CFT correspondence, to scattering theory, and deep analytical work on conformally compact Einstein manifolds. The relation of the ambient metric associated to a conformal structure to tractor bundles was completely described in the joint paper [27] with R. Gover. Joint work with him on the CR version is in progress, an outline can be found in [24]. The ambient metric for CR structures led Fefferman to a construction of a natural conformal structure on the total space of a circle bundle over a CR manifold. This construction admits a nice interpretation and several generalizations which are reported on in [24] and [31], more work in this direction is in progress.
[BE] | R.J. Baston, M.G. Eastwood: "The Penrose Transform" Its Interaction with Representation Theory. Oxford Science Publications, Clarendon Press, 1989 |
[C] | E. Cartan, Les systèmes de Pfaff a cinq variables et les équations aux derivées partielles du second ordre, Ann. Ec. Normale 27 (1910), 109-192 |
[CM] | S.S. Chern, J. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. |
[ER] | M.G. Eastwood, J.W. Rice, Conformally invariant differential operators on Minkowski space and their curved analogues. Comm. Math. Phys. 109 (1987), 207-228.; Erratum: 144 (1992) 213. |
[F1] | C. Fefferman, Monge-Ampère equations, the Bergman kernel and geometry of pseudoconvex domains, Ann. of Math. 103 (1976), 395-416; Erratum 104 (1976) 393-394 |
[F2] | C. Fefferman, Parabolic invariant theory in complex analysis, Adv. Math. 31 (1979), 131-262. |
[FG] | C. Fefferman, C.R. Graham, Conformal invariants, in "Élie Cartan et les Mathématiques d'Adjourd'hui", Astérisque, hors serie (1985) 95-116 |
[M] | T. Morimoto, Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993), 263-347. |
[T1] | N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math. 2 (1976), 131-190. |
[T2] | N. Tanaka, On the equivalence problem associated with simple graded Lie algebras, Hokkaido Math. J., 8 (1979), 23-84. |