Weyl metrizability of 3-dimensional projective structures and CR submanifolds

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.