M\"obius-flat hypersurfaces in projective space

TL;DR

This paper develops a unified theory of Moebius-flat hypersurfaces in projective space, connecting conformal and centro-affine geometries, and characterizes those with flat metrics using polynomial conserved quantities.

Contribution

It introduces a comprehensive framework for Moebius-flat hypersurfaces in projective space, extending previous examples and linking to Lie sphere geometry.

Findings

Unified class of hypersurfaces with flat induced conformal structure for n > 3

Extension of Akivis-Konnov's example

Characterization of hypersurfaces with flat centro-affine metric

Abstract

I give a theory of Moebius-flat hypersurfaces in n-dimensional projective space, analogous to that in conformal geometry. This unifies the classes of hypersurfaces with flat induced conformal structure (n > 3) and a classically studied class of surfaces (n = 3). I extend an example of Akivis-Konnov, and use polynomial conserved quantities to characterise hypersurfaces with flat centro-affine metric among Moebius-flat hypersurfaces. The theory has an obvious counterpart in Lie sphere geometry.