Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Calderbank, David M. J.; Diemer, Tammo

doi:10.1515/crll.2001.059

Cited by 102 publications

(268 citation statements)

References 36 publications

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“…Then, the conformal symmetries of S n are induced by the action of SO(n + 1, 1) on R n+2 realised as those linear transformations preserving (8) and of unit determinant.…”

Section: Results In the Flat Casementioning

confidence: 99%

“…defines a smooth function f on a conical neighbourhood of (1, 0, 0) in the null cone N of the quadratic form (8). This is a homogeneous function of degree w, namely f (λx A ) = λ w f (x A ), for λ > 0.…”

Section: Results In the Flat Casementioning

confidence: 99%

“…This is to be expected since the equation (1) is conformally invariant. In fact, the differential operator that is the left-hand side of (1) is the first operator in a conformally invariant complex of operators known as the Bernstein-GelfandGelfand complex [3], [5], [8], [13], [37]. This implies that the conformal Killing tensors on R n form an irreducible representation of the conformal Lie algebra so(n + 1, 1), namely · · · · · · trace-free part s boxes in each row as a Young tableau.…”

Section: Results In the Flat Casementioning

confidence: 99%

“…Invariant bilinear differential pairings also appear as the cup product of Calderbank and Diemer [8]. The pairing (V, f ) → D V f of Theorem 4 is evidently in the same vein but only when w = s is it a special case (from the so(n + 1, 1)-invariant pairing K n,s ⊗ s R n+2 → s R n+2 ).…”

Section: Explicit Formulae and The Curved Casementioning

confidence: 88%

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Higher symmetries of the Laplacian

2005

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“…Then, the conformal symmetries of S n are induced by the action of SO(n + 1, 1) on R n+2 realised as those linear transformations preserving (8) and of unit determinant.…”

Section: Results In the Flat Casementioning

confidence: 99%

Section: Results In the Flat Casementioning

confidence: 99%

Section: Results In the Flat Casementioning

confidence: 99%

Section: Explicit Formulae and The Curved Casementioning

confidence: 88%

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Higher symmetries of the Laplacian

2005

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“…Finally, there are strong general results on invariant differential operators, i.e. differential operators intrinsic to a parabolic geometry (see [11] and [5]). …”

Section: Introductionmentioning

confidence: 99%

Correspondence spaces and twistor spaces for parabolic geometries

Čap¹

2005

Journal für die reine und angewandte Mathematik (Crelles Journa

155

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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.

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Overdetermined Systems, Conformal Differential Geometry, and the BGG Complex

Čap

2008

Symmetries and Overdetermined Systems of Partial Differential Equations

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This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work with T.P. Branson, M.G. Eastwood, and A.R. Gover. First a simple special case is discussed, then the (Riemannian) procedure is described in general.The prolongation procedure was derived from a simplification of the construction of Bernstein-Gelfand-Gelfand (BGG) sequences of invariant differential operators for certain geometric structures. The version of this construction for conformal structures is described next. Finally, we discuss generalizations of both the prolongation procedure and the construction of invariant operators to other geometric structures.

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Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Cited by 102 publications

References 36 publications

Higher symmetries of the Laplacian

Higher symmetries of the Laplacian

Correspondence spaces and twistor spaces for parabolic geometries

Overdetermined Systems, Conformal Differential Geometry, and the BGG Complex

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