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Weyl metrizability of 3-dimensional projective structures and CR submanifolds

Omid Makhmali

TL;DR

The paper develops a CR-geometric framework to study Weyl metrizability of 3D oriented projective structures by embedding the problem into the twistor CR bundle $ ext{T}^{[ abla]}$, where Weyl metrizability corresponds to Levi-nondegenerate CR submanifolds transverse to the Levi kernel with a fundamental binary quartic that is zero or has a quadruple root. This CR characterization yields concrete corollaries: in dimension three, a flat projective structure is Weyl metrizable only with flat conformal structures, and a similar rigidity holds in higher dimensions; it also provides a CR description of the Einstein–Weyl condition via descent to minitwistor space. The main method combines Cartan theory of parabolic geometries, the twistor construction, and a detailed analysis of the fundamental quartic invariants to relate Weyl structures to CR submanifolds. The results illuminate how conformal and projective geometries interlock through CR and twistor geometry, offering precise criteria for metrizability and rigidity with potential extensions to related parabolic-geometric settings.

Abstract

A projective structure is Weyl metrizable if it has a representative that preserves a conformal structure. We interpret Weyl metrizability of 3-dimensional projective structures as certain 5-dimensional nondegenerate CR submanifolds in a class of 7-dimensional 2-nondegenerate CR structures. As a corollary, it follows that in dimension three Beltrami's theorem extends to conformal structures, i.e. a flat projective structure is Weyl metrizable exclusively with respect to a flat conformal structure. In higher dimensions it is shown that conformal Beltrami theorem remains true as well.

Weyl metrizability of 3-dimensional projective structures and CR submanifolds

TL;DR

The paper develops a CR-geometric framework to study Weyl metrizability of 3D oriented projective structures by embedding the problem into the twistor CR bundle , where Weyl metrizability corresponds to Levi-nondegenerate CR submanifolds transverse to the Levi kernel with a fundamental binary quartic that is zero or has a quadruple root. This CR characterization yields concrete corollaries: in dimension three, a flat projective structure is Weyl metrizable only with flat conformal structures, and a similar rigidity holds in higher dimensions; it also provides a CR description of the Einstein–Weyl condition via descent to minitwistor space. The main method combines Cartan theory of parabolic geometries, the twistor construction, and a detailed analysis of the fundamental quartic invariants to relate Weyl structures to CR submanifolds. The results illuminate how conformal and projective geometries interlock through CR and twistor geometry, offering precise criteria for metrizability and rigidity with potential extensions to related parabolic-geometric settings.

Abstract

A projective structure is Weyl metrizable if it has a representative that preserves a conformal structure. We interpret Weyl metrizability of 3-dimensional projective structures as certain 5-dimensional nondegenerate CR submanifolds in a class of 7-dimensional 2-nondegenerate CR structures. As a corollary, it follows that in dimension three Beltrami's theorem extends to conformal structures, i.e. a flat projective structure is Weyl metrizable exclusively with respect to a flat conformal structure. In higher dimensions it is shown that conformal Beltrami theorem remains true as well.
Paper Structure (19 sections, 14 theorems, 115 equations)

This paper contains 19 sections, 14 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Outline of the article and main results
  3. Conventions
  4. Conformal structures and nondegenerate CR geometry
  5. Parabolic geometries and Weyl structures
  6. Conformal geometry in dimension 3
  7. CR geometry in dimension 5
  8. Twistor CR manifolds and their Petrov type
  9. Projective structures and 2-nondegenerate CR geometry
  10. Projective structures in dimension three
  11. Almost para-CR structure on the correspondence space
  12. The twistor bundle and its 2-nondegenerate CR structure
  13. Weyl metrizability and CR submanifolds
  14. Projective class of a conformal Weyl structure
  15. CR submanifolds as holomorphic sections of the Levi kernel
  16. ...and 4 more sections

Key Result

Theorem 2.5

A 5-dimensional CR structure with Levi form of signature (1,1) is a Cartan geometry $(\mathcal{Q}\to N,\phi)$ of type $(\mathrm{SU}(2,2),P)$ where $P$ is the stabilizer of a null line in $\mathbb{C}^4.$ The Cartan connection and curvature can be expressed as which are $\mathfrak{su}(2,2)$-valued with respect to the Hermitian form where $\alpha=\alpha^1+\mathrm{i}\alpha^2$ and $\beta=\beta^1+\mat

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Proposition 2.9
  • proof
  • ...and 33 more