Affine rigidity of Levi degenerate tube hypersurfaces

TL;DR

This paper investigates a special class of 5-dimensional CR-hypersurfaces with Levi degeneracy, showing that those with vanishing CR-curvature are affinely equivalent to the tube over the future light cone, revealing a rigidity property.

Contribution

It establishes affine rigidity for Levi degenerate tube hypersurfaces with zero CR-curvature within a specific CR-structure class, building on previous CR-structure reductions.

Findings

Hypersurfaces with vanishing CR-curvature are affinely equivalent to the tube over the future light cone.

CR-structures in C(2,1) are reducible to so(3,2)-valued absolute parallelisms.

The result characterizes the geometric structure of Levi degenerate hypersurfaces with flat CR-curvature.

Abstract

Let C(2,1) be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In a recent article, we proved that the CR-structures in C(2,1) are reducible to so(3,2)-valued absolute parallelisms. In the present paper, we apply this result to study tube hypersurfaces in C^3 that belong to C(2,1) and whose CR-curvature identically vanishes. Every such hypersurface is shown to be affinely equivalent to an open subset of the tube over the future light cone.