The geometry of hyperbolic and elliptic CR-manifolds of codimension two

Abstract

Abstract. The general theory of parabolic geometries is applied to the study of the normal Cartan connections for all hyperbolic and elliptic 6-dimensional CR-manifolds of codimension two. The geometric meaning of the individual components of the torsion is explained and the chains of dimensions one and two are discussed.There have been many attempts to use some ideas going back up to Cartan, in order to understand the geometry of CR-manifolds. In the codimension one cases, the satisfactory solution had been worked out in the seventies, see [22,23,8], but the higher codimensions have not been understood yet in a comparable extent. In this paper, the recent general theory of the so called parabolic geometries is applied. In particular, we use the approach developed in [4,21], see also [24,27] for earlier results. Relying on recent achievements by the authors, a clean and quite simple construction of the normal Cartan connection is presented. This Cartan connection replaces the absolute parallelisms from [9] by more powerful geometric tools and it enables the detailed study of geometrical and analytical properties of the CR structures. Consequently the resulting geometric picture is much more transparent and surprising new results are obtained.The main advantage of our approach is the fully coordinate-free handling of the normal Cartan connection and its curvature. Thus we are able to translate the cohomological properties of the structure algebras into full geometrical understanding of the curvature obstruction, without writing down the curvature components explicitly. The initial section introduces the CR structures and provides a brief exposition of distinguished second order osculations of the surfaces by quadrics. Then we observe, that this osculation transfers enough data from the quadric to apply the general construction of normal Cartan connections, due to [24,4]. This leads easily to the main Theorems 1.2 and 1.3. In fact, the Cartan connections are constructed also for certain abstract CRmanifolds though the embedded ones have many distinguished properties. The third section is devoted to the exposition of the generalities 1 2 GERD SCHMALZ AND JAN SLOVÁK on parabolic geometries modelled over |2|-graded algebras and provides the proof of the existence of the normal connections.Next we study the local geometry of the hyperbolic points in detail. We recover easily all known facts from [9], but we go much further. In particular, we identify the complete geometric obstructions against the integrability of the almost product structure on the tangent bundle (Theorem 3.5), the integrability of the almost complex structure on the tangent CR space (Theorem 3.6), and the compatibility of the almost product and almost complex structures (3.8). It turns out that the latter two obstructions always vanish on the embedded hyperbolic CR-structures which results in automatic vanishing of several algebraic brackets. In particular, the whole hyperbolic CR-manifold M ⊂ C 4 is a product of two 3-dimensional CR-manifolds if an...

Generic one-step bracket-generating distributions of rank four

“…If P ⊂ Λ 2 R 4 is a linear subspace, one can restrict the wedge product to P . (4,5), (4,9) and (4, 10) and there are two generic types in bidimensions (4, 6), (4, 7) and (4,8).…”

“…If the wedge product restricts to a linear subspace P ⊂ Λ 2 R 4 of dimension 10 − n with signature [t, s], then the canonical projection Λ 2 R 4 → Λ 2 R 4 /P onto the quotient is in O. In this picture, the model algebras for the types (4, 10) and (4,9) are immediately deduced. In dimension ten, we have the free algebra R 4 ⊕ Λ 2 R 4 , which can be realized as the negative graded part of a grading on g = so(9) such that the first cohomology…”

“…Any generic (4, 6)-distribution endowed with an additional almost complex structure on the subbundle H, which is compatible with the Levi bracket in an appropriate sense, is in fact equivalent to a parabolic geometry (see [6, p. 443-455]). This is related to CR-structures of dimension and codimension two, see [5] and [9].…”

“…However, our description is more explicit and leads directly to a nice presentation of model algebras. In this picture, one can easily deduce model algebras of generic (4,9) and (4, 10)-types. The remaining generic types are described through suitable generalizations of the real Heisenberg algebra.…”

Correspondence spaces and twistor spaces for parabolic geometries

“…From their definition, real parabolic geometries of type (G, Q) are six dimensional smooth manifolds endowed with an almost complex structure and two complementary complex line bundles H + , H − ⊂ T M , such that, with H = H + ⊕ H − , the tensorial map L : H ⊗ H → T M/H induced by the Lie bracket is complex bilinear and non-degenerate. Building on earlier work in [21] it has been shown in [8] that flipping the almost complex structure on the subbundle H + leads to an equivalence of categories between the category of regular normal parabolic geometries of type (G, Q) and the category of elliptic partially integrable almost CR manifolds of CR dimension and codimension two. Since we are dealing with the underlying real Lie algebra of a complex Lie algebra, the structure of torsions and curvatures is rather complicated.…”