Correspondence spaces and twistor spaces for parabolic geometries

Abstract

Abstract. For a semisimple Lie group G with parabolic subgroups Q ⊂ P ⊂ G, we associate to a parabolic geometry of type (G, P ) on a smooth manifold N the correspondence space CN , which is the total space of a fiber bundle over N with fiber a generalized flag manifold, and construct a canonical parabolic geometry of type (G, Q) on CN .Conversely, for a parabolic geometry of type (G, Q) on a smooth manifold M , we construct a distribution corresponding to P , and find the exact conditions for its integrability. If these conditions are satisfied, then we define the twistor space N as a local leaf space of the corresponding foliation. We find equivalent conditions for the existence of a parabolic geometry of type (G, P ) on the twistor space N such that M is locally isomorphic to the correspondence space CN , thus obtaining a complete local characterization of correspondence spaces.We show that all these constructions preserve the subclass of normal parabolic geometries (which are determined by some underlying geometric structure) and that in the regular normal case, all characterizations can be expressed in terms of the harmonic curvature of the Cartan connection, which is easier to handle. Several examples and applications are discussed. IntroductionThis paper is devoted to the study of relations between different geometric structures via the construction of correspondence spaces and twistor spaces. The structures we deal with are the so-called parabolic geometries, which may be viewed as curved analogs of homogeneous spaces of the form G/P , where G is a semisimple Lie group and P ⊂ G is a parabolic subgroup. Parabolic geometries form a rather large class of structures, including for example projective, conformal and non-degenerate CR-structures of hypersurface type, as well as certain higher codimension CR structures.The starting point of twistor theory was R. Penrose's idea to associate to the Grassmannian Gr 2 (C 4 ) of planes in C 4 , which is viewed as compactified complexified Minkowski space, the twistor space CP 3 , and to study the conformal geometry of Gr 2 (C 4 ) via the complex geometry of the twistor space. The connection between these two manifolds is the correspondence space F 1,2 (C 4 ), the flag manifold of lines in planes in C 4 , which canonically fibers over Gr 2 (C 4 ) and over CP 3 and defines a correspondence between the 1991 Mathematics Subject Classification. primary: 53B15, 53C15, 53C28, secondary: 32L25, 53A20, 53A30, 53C10, 53D10.

Inclusions between Parabolic Geometries

“…For example, in the case of l = 3 which was considered by R. Bryant, we get the quadratic non-degenerate cone. So, in this smallest possible dimension the almost spinorial structure coincides with the conformal structure of signature (3,3).…”

Subcomplexes in curved BGG-sequences

“…There are various real forms of the situation discussed in 3.6 which are of interest in geometry. Putting G := SL(n + 2, R), one obtains Lagrangean (or Legendrean) contact structures, see [26] or [11]. Such a structure on a manifold M of dimension 2n + 1 is given by a codimension one subbundle H ⊂ T M which defines a contact structure, and a fixed decomposition of H = E ⊕F as the direct sum of two Legendrean subbundles.…”

Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction

“…Baston and M. Eastwood [2] have developed a general theory of the Penrose transform for correspondences G/R η ← − G/Q τ − → G/P between generalized flag varieties, where Q = P ∩ R, and their theory makes sense in the more general context of parabolic geometries studied by A.Čap [6]. In this context, G/P is replaced by a parabolic geometry N = G/P .Čap shows that M = G/Q is then also a parabolic geometry, and characterizes when it fibres over a generalized twistor space Y such that G is a local principal R-bundle over Y .…”