Some special geometry in dimension six. (English) Zbl 1047.53018

Čap, Andreas; Eastwood, Michael

Bureš, Jarolím (ed.), The proceedings of the 22nd winter school “Geometry and physics”, Srní, Czech Republic, January 12–19, 2002. Palermo: Circolo Matemàtico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 71, 93-98 (2003).

Motivated by the study of CR-submanifolds of codimension \(2\) in \(\mathbb{C}^4\), the authors consider here a \(6\)-dimensional oriented manifold \(M\) equipped with a \(4\)-dimensional distribution. Under some non-degeneracy condition, two different geometric situations can occur. In the elliptic case, one constructs a canonical almost complex structure on \(M\); the hyperbolic case leads to a canonical almost product structure. In both cases the only local invariants are given by the obstructions to integrability for these structures. The local ’flat’ models are a \(3\)-dimensional complex contact manifold and the product of two \(3\)-dimensional real contact manifolds, respectively.
For the entire collection see [Zbl 1014.00011].

Reviewer: Eric Boeckx (Leuven)

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
32V05	CR structures, CR operators, and generalizations

Keywords:

CR-manifolds; almost contact structure; almost product structure; complex contact manifold