Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition

Matthias Hammerl ; Katja Sagerschnig

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

Given a maximally non-integrable $2$-distribution $\mathcal D$ on a $5$-manifold $M$, it was discovered by P. Nurowski that one can naturally associate a conformal structure $[g]_{\mathcal D}$ of signature $(2,3)$ on $M$. We show that those conformal structures $[g]_{\mathcal D}$ which come about by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of $[g]_{\mathcal D}$ can be decomposed into a symmetry of $\mathcal D$ and an almost Einstein scale of $[g]_{\mathcal D}$.

Détail

Keywords: generic distributions; conformal geometry; tractor calculus; Fefferman construction; conformal Killing fields; almost Einstein scales.

Bibliographie

[1] Alt J., Fefferman constructions in conformal holonomy, Thesis, Humboldt-Universität zu Berlin, 2008

[2] Armstrong S., “Definite signature conformal holonomy: a complete classification”, J. Geom. Phys., 57 (2007), 2024–2048 ; math.DG/0503388 | DOI | MR | Zbl

[3] Bryant R. L., “Metrics with exceptional holonomy”, Ann. of Math. (2), 126 (1987), 525–576 | DOI | MR | Zbl

[4] Burns D. Jr., Diederich K., Shnider S., “Distinguished curves in pseudoconvex boundaries”, Duke Math. J., 44 (1977), 407–431 | DOI | MR | Zbl

[5] Calderbank D. M. J., Diemer T., “Differential invariants and curved Bernstein–Gelfand–Gelfand sequences”, J. Reine Angew. Math., 537 (2001), 67–103 ; math.DG/0001158 | MR | Zbl

[6] Čap A., “Parabolic geometries, CR-tractors, and the Fefferman construction”, Differential Geom. Appl., 17 (2002), 123–138 | DOI | MR | Zbl

[7] Čap A., “Correspondence spaces and twistor spaces for parabolic geometries”, J. Reine Angew. Math., 582 (2005), 143–172 ; math.DG/0102097 | MR | Zbl

[8] Čap A., “Two constructions with parabolic geometries”, Rend. Circ. Mat. Palermo (2) Suppl., 79 (2006), 11–37 ; math.DG/0504389 | MR | Zbl

[9] Čap A., “Infinitesimal automorphisms and deformations of parabolic geometries”, J. Eur. Math. Soc. (JEMS), 10 (2008), 415–437 ; math.DG/0508535 | MR

[10] Čap A., Gover A. R., “Tractor bundles for irreducible parabolic geometries”, Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999), Sémin. Congr., 4, Soc. Math. France, Paris, 2000, 129–154 | MR | Zbl

[11] Čap A., Gover A. R., “Tractor calculi for parabolic geometries”, Trans. Amer. Math. Soc., 354 (2002), 1511–1548 | DOI | MR | Zbl

[12] Čap A., Gover A. R., A holonomy characterisation of Fefferman spaces, ESI Preprint 1875, 2006; math.DG/0611939

[13] Čap A., Gover A. R., “CR-tractors and the Fefferman space”, Indiana Univ. Math. J., 57 (2008), 2519–2570 ; math.DG/0611938 | DOI | MR | Zbl

[14] Čap A., Sagerschnig K., “On Nurowski's conformal structure associated to a generic rank 2 distributions in dimension 5”, J. Geom. Phys., 59:7 (2009), 901–912 ; arXiv:0710.2208 | DOI | MR | Zbl

[15] Čap A., Slovák J., Parabolic geometries. I: Background and general theory, Mathematical Surveys and Monographs, 1, American Mathematical Society, Providence, RI, 2009 | MR

[16] Čap A., Slovák J., Souček V., “Bernstein–Gelfand–Gelfand sequences”, Ann. of Math. (2), 154 (2001), 97–113 ; math.DG/0001164 | DOI | MR | Zbl

[17] Čap A., Žádník V., “On the geometry of chains”, J. Differential Geom., 82 (2009), 1–33 ; math.DG/0504469 | MR

[18] Cartan É., “Les espaces á connexion conforme”, Ann. Soc. Pol. Math., 2 (1923), 172–202

[19] Cartan É., “Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre”, Ann. Sci. École Norm. Sup. (3), 27 (1910), 109–192 | MR | Zbl

[20] Doubrov B., Slovák J., Inclusions between parabolic geometries, arXiv:0807.3360

[21] Eastwood M. G., Michor P. W., “Some remarks on the Plücker relations”, The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999), Rend. Circ. Mat. Palermo (2) Suppl., 63, 2000, 85–88 ; math.AG/9905090 | MR | Zbl

[22] Fefferman C. L., “Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains”, Ann. of Math. (2), 103 (1976), 395–416 | DOI | MR | Zbl

[23] Gover A. R., “Laplacian operators and $Q$-curvature on conformally Einstein manifolds”, Math. Ann., 336 (2006), 311–334 ; math.DG/0506037 | DOI | MR | Zbl

[24] Gover A. R., Almost Einstein and Poincaré–Einstein manifolds in Riemannian signature, arXiv:0803.3510

[25] Gover A. R., Šilhan J., “The conformal Killing equation on forms – prolongations and applications”, Differential Geom. Appl., 26 (2008), 244–266 ; math.DG/0601751 | DOI | MR | Zbl

[26] Graham C. R., “On Sparling's characterization of Fefferman metrics”, Amer. J. Math., 109 (1987), 853–874 | DOI | MR | Zbl

[27] Hammerl M., Natural prolongations of BGG-operators, Thesis, University of Vienna, submitted | Zbl

[28] Hammerl M., “Homogeneous Cartan geometries”, Arch. Math. (Brno), 43, suppl. (2007), 431–442 ; math.DG/0703627 | MR | Zbl

[29] Hammerl M., “Invariant prolongation of BGG-operators in conformal geometry”, Arch. Math. (Brno), 44 (2008), 367–384 ; arXiv:0811.4122 | MR

[30] Hitchin N., “Stable forms and special metrics”, Global Differential Geometry: the Mathematical Legacy of Alfred Gray (Bilbao, 2000), Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2001, 70–89 ; math.DG/0107101 | MR | Zbl

[31] Kostant B., “Lie algebra cohomology and the generalized Borel–Weil theorem”, Ann. of Math. (2), 74 (1961), 329–387 | DOI | MR | Zbl

[32] Leistner T., Nurowski P., Conformal structures with $G_{2(2)}$-ambient metrics, arXiv:0904.0186

[33] Leitner F., “Conformal Killing forms with normalisation condition”, Rend. Circ. Mat. Palermo (2) Suppl., 75 (2005), 279–292 | MR | Zbl

[34] Leitner F., A remark on conformal $\mathrm{su}(p,q)$-holonomy, math.DG/0604393

[35] Leitner F., “A remark on unitary conformal holonomy”, Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., 144, Springer, New York, 2008, 445–460 | MR | Zbl

[36] Nurowski P., “Differential equations and conformal structures”, J. Geom. Phys., 55 (2005), 19–49 ; math.DG/0406400 | DOI | MR | Zbl

[37] Nurowski P., “Conformal structures with explicit ambient metrics and conformal $G_2$ holonomy”, Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., 144, Springer, New York, 2008, 515–526 ; math.DG/0701891 | MR | Zbl

[38] Sagerschnig K., “Split octonions and generic rank two distributions in dimension five”, Arch. Math. (Brno), 42, suppl. (2006), 329–339 | MR | Zbl

[39] Sagerschnig K., Weyl structures for generic rank two distributions in dimension five, Thesis, University of Vienna, 2008

[40] Šilhan J., Cohomologies of real Lie algebras, available at http://bart.math.muni.cz/~silhan/lie/lac/formR.php

[41] Šilhan J., “A real analog of Kostant's version of the Bott–Borel–Weil theorem”, J. Lie Theory, 14 (2004), 481–499 | MR