Affine rigidity of Levi degenerate tube hypersurfaces

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 our 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 C₃ that belong to C_2,1 and whose CR-curvature identically vanishes. By explicitly solving the zero CR-curvature equations up to affine equivalence, we show that every such hypersurface is affinely equivalent to an open subset of the tube M₀ over the future light cone {(x₁, x₂, x₃) ∈ double-struck R³ | x₁² + x₂² - x₃² = 0, x₃ > 0}. Thus, if a tube hypersurface in the class C_2,1 locally looks like a piece of M₀ from the point of view of CR-geometry, then from the point of view of affine geometry it (globally) looks like a piece of M₀ as well. This rigidity result is in stark contrast to the Levi nondegenerate case, where the CR-geometric and affine-geometric classifications significantly differ.