# Free CR distributions

Open Mathematics (2012)

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

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topGerd 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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- [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] 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
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- [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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