Weyl metrizability of 3-dimensional projective structures and CR submanifolds

TL;DR

This paper explores the conditions under which 3-dimensional projective structures are Weyl metrizable, linking this property to CR submanifolds, and extends Beltrami's theorem to higher dimensions.

Contribution

It provides a geometric interpretation of Weyl metrizability in 3D projective structures via CR submanifolds and generalizes Beltrami's theorem to higher dimensions.

Findings

Weyl metrizability corresponds to certain CR submanifolds in 7D structures.

In 3D, flat projective structures are Weyl metrizable only with flat conformal structures.

Beltrami's theorem extends to higher dimensions for conformal structures.

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 locally flat projective structure is Weyl metrizable exclusively with respect to a locally flat conformal structure. In higher dimensions it is shown that conformal Beltrami theorem remains true as well.