Advanced Studies in Pure Mathematics

Partially Integrable Almost CR Manifolds of CR Dimension and Codimension Two

Andreas Čap and Gerd Schmalz

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Abstract

We extend the results of [11] on embedded CR manifolds of CR dimension and codimension two to abstract partially integrable almost CR manifolds. We prove that points on such manifolds fall into three different classes, two of which (the hyperbolic and the elliptic points) always make up open sets. We prove that manifolds consisting entirely of hyperbolic (respectively elliptic) points admit canonical Cartan connections. More precisely, these structures are shown to be exactly the normal parabolic geometries of types $(PSU(2, 1) \times PSU(2, 1), B \times B)$, respectively $(PSL(3, \mathbb{C}), B)$, where $B$ indicates a Borel subgroup. We then show how general tools for parabolic geometries can be used to obtain geometric interpretations of the torsion part of the harmonic components of the curvature of the Cartan connection in the elliptic case.

Article information

Source
Lie Groups, Geometric Structures and Differential Equations — One Hundred Years after Sophus Lie, T. Morimoto, H. Sato and K. Yamaguchi, eds. (Tokyo: Mathematical Society of Japan, 2002), 45-77

Dates
Received: 25 September 2000
Revised: 27 June 2001
First available in Project Euclid: 1 January 2019

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1546368688

Digital Object Identifier
doi:10.2969/aspm/03710045

Mathematical Reviews number (MathSciNet)
MR1980896

Zentralblatt MATH identifier
1041.32023

Citation

Čap, Andreas; Schmalz, Gerd. Partially Integrable Almost CR Manifolds of CR Dimension and Codimension Two. Lie Groups, Geometric Structures and Differential Equations — One Hundred Years after Sophus Lie, 45--77, Mathematical Society of Japan, Tokyo, Japan, 2002. doi:10.2969/aspm/03710045. https://projecteuclid.org/euclid.aspm/1546368688


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