Partially integrable almost CR manifolds of CR dimension and codimension two. (English) Zbl 1041.32023

Čap, Andreas; Schmalz, Gerd

Morimoto, Tohru (ed.) et al., Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie. Based on the international conference on the occasion of the centennial after the death of Sophus Lie (1842–1899), Kyoto and Nara, Japan, December 1999. Tokyo: Mathematical Society of Japan (ISBN 4-931469-21-3/hbk). Adv. Stud. Pure Math. 37, 45-77 (2002).

A smooth manifold \(M\) with some complex subbundle \(HM\) of the tangent bundle \(TM\) is called an almost CR manifold. Let QM be the quotient bundle \(TM/HM\). Using the bracket of vector fields we can define a bilinear skew symmetric bundle map \({\mathcal L}: HM\times HM\to QM\) which is called the Levi-bracket of \(M\).
Using the Levi-bracket the authors first define partially integrable abstract almost CR manifolds and try to extend the results of M. Schmalz and J. Slovák [Asian J. Math. 4, 565–598 (2003; Zbl 0972.32025)] on embedded CR manifolds of CR dimension and codimension 2 to abstract partially integrable almost CR manifolds. The authors first prove that for nondegenerate partially integrable almost CR manifolds of the above type the possible Levi brackets fall into three different classes, called hyperbolic, elliptic, and exceptional. Next, the authors show that the manifolds all of whose points are hyperbolic (resp. elliptic) are exactly the normal parabolic geometries of type (\(\text{PSU}(2, 1)\times \text{PSU}(2,1)\), \(B\times B\)) (resp. (\(\text{PSL}(3,\mathbb{C}), B)\)), where \(B\) denotes a Borel subgroup.
In fact, in the hyperbolic case, the authors prove that there is a canonical principal bundle \(p:{\mathcal G}\to M\) with structure group \(B\times B\), where \(B\) is the Borel subgroup in \(\text{PSU}(2, 1)\) endowed with a normal Cartan connection \(\omega\in \Omega^1({\mathcal G},{\mathfrak s}{\mathfrak u}(2,1)\times {\mathfrak s}{\mathfrak u}(2,1))\). Similar results are obtained in the elliptic case. Furthermore, in the elliptic case the authors give geometric interpretations to the torsion part of the harmonic components of the curvature of the Cartan connection. For example the authors show that torsion-free elliptic manifolds are automatically real analytic and therefore embeddable.
For the entire collection see [Zbl 1011.00031].

Reviewer: Akihiko Morimoto (Nagoya)

MSC:

32V05	CR structures, CR operators, and generalizations
53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)

Keywords:

almost CR manifolds; Levi-bracket; parabolic geometry; Cartan connections

Citations:

Zbl 0972.32025