Weyl metrizability of 3-dimensional projective structures and CR submanifolds
arXiv:2501.07460 · doi:10.1007/s00209-026-03958-0
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.
Final version; an error in the derivation of equation (2.12) is corrected; section 3.1 is slightly expanded; several parts are improved and minor errors are corrected thanks to the referee report; a reference is added; to appear in Math. Z