Elliptic CR-manifolds and shear invariant ODE with additional symmetries

We classify the ODEs that correspond to elliptic CR-manifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can be only 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable istropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold

Explicit description of spherical rigid hypersurfaces in C²

a

Free CR distributions

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions $n$ and codimensions $n^2$ are among the very few possibilities of the so called parabolic geometries. Indeed, the homogeneous model turns out to be \PSU(n+1,n)/P with a suitable parabolic subgroup $P$. We study the geometric properties of such real $(2n+n^2)$-dimensional submanifolds in $\mathbb C^{n+n^2}$ for all $n>1$. In particular we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry

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.Comment: 42 pages, see also http://wwwmaths.anu.edu.au/research.reports/97mrr.htm

CR embeddings of CR manifolds

We improve results of Baouendi, Rothschild and Treves and of Hill and Nacinovich by finding a much weaker sufficient condition for a CR manifold of type (n, k) to admit a local CR embedding into a CR manifold of type (n+ ℓ, k- ℓ). While their results require the existence of a finite dimensional solvable transverse Lie algebra of vector fields, we require only a finite dimensional extension

From the Hubbard to the PPP Model

This paper presents an extension of the Hubbard Model to Pariser-Parr-Pople form. Although the Hubbard model contains most of the essentials of chemical bonding, it is unable to describe excited states with separated charges, such as the lowest 1Bu states of linear polyenes. The PPP model adds longrange electron-electron repulsions to the Hubbard model to remedy this defect. If the long range repulsion integrals are assumed to follow a standard form, all parameters in the model can be evaluated exactly from high accuracy ab initio computations on stretched ethlyene. This yields a model based on the Mataga- Nishimoto form for the long-range integrals which gives excellent agreement with both excitation energies and ground-state bond lengths, but with a significantly smaller value of the one center electron repulsion U than is usually assumed. A major conclusion of this work is that the exact form of the long-range integrals is not so important, but that the value of the one center integral U must be chosen smaller than traditional values. The PPP-MN model is recommended for applications because it contains no adjustable parameters, with all parameter values determined directly from ab initio results. (doi: 10.5562/cca2297

Automorphisms of Nondegenerate CR Quadrics and Siegel Domains: Explicit Description

In this paper we give the complete explicit description of the holomorphicautomorphisms of any nondegenerate CR-quadric Q of arbitrary CRdimensionand codimension and of Siegel domains of second kind with not necessarilyLevi-nondegenerate Silov-boundary.We introduce a family of k-dimensional chains (k = codim Q), the analoguesof one-dimensional Chern-Moser chains for hyperquadrics.We also analyse some different types of rigid quadrics

Partially Integrable Almost CR Manifolds of CR Dimension and Codimension Two

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 seats. 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) x PSU(2,1),B x B), respectively (PSL(3,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