Free CR distributions

Gerd Schmalz; Jan Slovák

Open Mathematics (2012)

  • Volume: 10, Issue: 5, page 1896-1913
  • ISSN: 2391-5455

Abstract

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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 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.

How to cite

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Gerd Schmalz, and Jan Slovák. "Free CR distributions." Open Mathematics 10.5 (2012): 1896-1913. <http://eudml.org/doc/269158>.

@article{GerdSchmalz2012,
abstract = {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.},
author = {Gerd Schmalz, Jan Slovák},
journal = {Open Mathematics},
keywords = {Cartan connection; Cartan curvature; Parabolic geometry; Fefferman construction; parabolic geometry},
language = {eng},
number = {5},
pages = {1896-1913},
title = {Free CR distributions},
url = {http://eudml.org/doc/269158},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Gerd Schmalz
AU - Jan Slovák
TI - Free CR distributions
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1896
EP - 1913
AB - 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.
LA - eng
KW - Cartan connection; Cartan curvature; Parabolic geometry; Fefferman construction; parabolic geometry
UR - http://eudml.org/doc/269158
ER -

References

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  1. [1] Armstrong S., Free 3-distributions: holonomy, Fefferman constructions and dual distributions, preprint available at http://arxiv.org/abs/0708.3027 
  2. [2] Čap A., Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math., 2005, 582, 143–172 Zbl1075.53022
  3. [3] Čap A., Slovák J., Parabolic Geometries I, Math. Surveys Monogr., 154, American Mathematical Society, Providence, 2009 Zbl1183.53002
  4. [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 Zbl1208.53047
  5. [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 Zbl0847.32015
  6. [6] Schmalz G., Slovák J., The geometry of hyperbolic and elliptic CR-manifolds of codimension two, Asian J. Math., 2000, 4(3), 565–598 Zbl0972.32025
  7. [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 Zbl0972.32025
  8. [8] Šilhan J., A real analog of Kostant’s version of the Bott-Borel-Weil theorem, J. Lie Theory, 2004, 14(2), 481–499 Zbl1090.17010
  9. [9] Tanaka N., On the equivalence problem associated with simple graded Lie algebras, Hokkaido Math. J., 1979, 8(1), 23–84 
  10. [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 Zbl0812.17018

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