V. V. Ezhov | Searcharxiv

arXiv1998

Invariants of elliptic and hyperbolic CR-structures of codimension 2

We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves ``almost'' like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of ``semi-flat'' manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection. In addition, for real-analytic manifolds we describe certain local normal forms that do not require passing to bundles, but in many ways agree with the structure of the parallelism.

math.CV↗

arXiv1998

Canonical isomorphism of two Lie algebras arising in CR-geometry

We show that the maximal prolongation of a certain algebra associated with a non-degenerate Hermitian form on ${\Bbb C}^n\times{\Bbb C}^n$ with values in ${\Bbb R}^k$ is canonically isomorphic to the Lie algebra of infinitesimal holomorphic automorphisms of the corresponding quadric in ${\Bbb C}^{n+k}$. This fact creates a link between different approaches to the equivalence problem for Levi-nondegenerate strongly uniform CR-manifolds.

math.CV↗