Parabolic geometries and canonical Cartan connections

Čap, Andreas; Schichl, Hermann

doi:10.14492/hokmj/1350912986

Cited by 118 publications

(188 citation statements)

References 9 publications

Supporting

Mentioning

180

Contrasting

Unclassified

Order By: Relevance

“…Even in the general context, the observation that the curvature is uniquely determined by its (co)homology class is an old one: see [15,43]. Our approach reveals that the proofs in these references appear technical because they amount to the construction of Π 2 • repr in this special case.…”

Section: Theorem Let (G η) Be a Normal Regular Parabolic Geometry Omentioning

confidence: 96%

“…In order to keep the paper as self-contained as possible, we give proofs for all the basic properties of Lie algebra homology, although we only indicate briefly how Kostant's Hodge decomposition may be established. Additionally, there are some non-standard aspects to our treatment: firstly we concentrate on Lie algebra homology, rather than cohomology, since it is the Lie algebra homology that is p-equivariant, and secondly, we eschew the lamentably inverted notation ∂, ∂ * for the Lie algebra coboundary and boundary operators (for some reason, ∂, although a boundary operator in [32], is the coboundary operator in [1,15,18,43]). Instead, following Kostant in part [32], and by analogy with the deRham complex, we use d and δ, with subscripts to indicate that it is the Lie algebraic rather than differential operators we are considering.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Calderbank¹,

Diemer²

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

102

260

View full text Add to dashboard Cite

Abstract. We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential "cup product" on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A∞-algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a wide-reaching generalization of helicity raising and lowering for conformally invariant field equations.

show abstract

Section: Theorem Let (G η) Be a Normal Regular Parabolic Geometry Omentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Calderbank¹,

Diemer²

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

102

260

View full text Add to dashboard Cite

show abstract

“…This consists of a filtration {T i M} of T M and a principal G 0 -bundle G 0 → M endowed with certain partially defined differential forms. Under a cohomological condition, which is satisfied for all the structures considered in this paper, one can then apply involved prolongation procedures (see [27,21,13]). These extend G 0 to a principal P -bundle p : G → M endowed with a Cartan connection ω.…”

Section: 2mentioning

confidence: 99%

Subcomplexes in curved BGG-sequences

Čap

Souček

2011

Math. Ann.

Self Cite

View full text Add to dashboard Cite

Abstract. BGG-sequences offer a uniform construction for invariant differential operators for a large class of geometric structures called parabolic geometries. For locally flat geometries, the resulting sequences are complexes, but in general the compositions of the operators in such a sequence are nonzero. In this paper, we show that under appropriate torsion freeness and/or semi-flatness assumptions certain parts of all BGG sequences are complexes.Several examples of structures, including quaternionic structures, hypersurface type CR structures and quaternionic contact structures are discussed in detail. In the case of quaternionic structures we show that several families of complexes obtained in this way are elliptic.

show abstract

“…We start by briefly reviewing some general facts about parabolic geometries, see [7], [11], and [9] for more information.…”

Section: Correspondence Spaces and Twistor Spacesmentioning

confidence: 99%

“…We sketch a construction using the procedure from [7], which uses a simpler description of the underlying structures. We have noted above that the Lie bracket [ , ] : q −1 × q −1 → q −2 is non-degenerate, and one observes that the subspaces q L −1 and q R −1 are isotropic.…”

Section: 1mentioning

confidence: 99%

Correspondence spaces and twistor spaces for parabolic geometries

Čap¹

2005

Journal für die reine und angewandte Mathematik (Crelles Journa

Self Cite

155

View full text Add to dashboard Cite

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.

show abstract

Parabolic geometries and canonical Cartan connections

Cited by 118 publications

References 9 publications

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Subcomplexes in curved BGG-sequences

Correspondence spaces and twistor spaces for parabolic geometries

Contact Info

Product

Resources

About