Free CR distributions

• Autorzy: Gerd Schmalz , Jan Slovák
• Języki publikacji: EN

Abstrakty

EN

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n 2 are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n 2)-dimensional submanifolds in $\mathbb{C}^{n + n^2 } $ for all n > 1. In particular, we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.

Słowa kluczowe

EN

Cartan connection; Cartan curvature; Parabolic geometry; Fefferman construction

Kategorie tematyczne

• 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)
• 32V20: Analysis on CR manifolds

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 5
• Strony: 1896-1913

Daty

wydano

2012-10-01

online

2012-07-24

Twórcy

autor

Gerd Schmalz

• University of New England

autor

Jan Slovák

• Masaryk University

Bibliografia

• [1] Armstrong S., Free 3-distributions: holonomy, Fefferman constructions and dual distributions, preprint available at http://arxiv.org/abs/0708.3027
• [2] Čap A., Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math., 2005, 582, 143–172
• [3] Čap A., Slovák J., Parabolic Geometries I, Math. Surveys Monogr., 154, American Mathematical Society, Providence, 2009
• [4] Doubrov B., Slovák J., Inclusions between parabolic geometries, Pure Appl. Math. Q., 2010, 6(3), Special Issue: In Honor of Joseph J.Kohn, Part 1, 755–780
• [5] Ežov V.V., Schmalz G., Poincaré automorphisms for nondegenerate CR quadrics, Math. Ann., 1994, 298(1), 79–87 http://dx.doi.org/10.1007/BF01459726
• [6] Schmalz G., Slovák J., The geometry of hyperbolic and elliptic CR-manifolds of codimension two, Asian J. Math., 2000, 4(3), 565–598
• [7] Schmalz G., Slovák J., Addendum to ”The geometry of hyperbolic and elliptic CR-manifolds of codimension two”, Asian J. Math., 4, 565–598, 2000, Asian J. Math., 2003, 7(3), 303–306
• [8] Šilhan J., A real analog of Kostant’s version of the Bott-Borel-Weil theorem, J. Lie Theory, 2004, 14(2), 481–499
• [9] Tanaka N., On the equivalence problem associated with simple graded Lie algebras, Hokkaido Math. J., 1979, 8(1), 23–84
• [10] Yamaguchi K., Differential systems associated with simple graded Lie algebras, In: Progress in Differential Geometry, Adv. Stud. Pure Math., 22, Mathematical Society of Japan, Tokyo, 1993, 413–494

Identyfikatory

DOI

10.2478/s11533-012-0090-y