AFFINE RIGIDITY OF LEVI DEGENERATE TUBE HYPERSURFACES

作者：Isaev Alexander^*

来源：Journal of Differential Geometry, 2016, 104(1): 111-141.

DOI：10.4310/jdg/1473186540

Let C-2,C-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,C-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,C-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 M-0 over the future light cone {(x(1), x(2), x(3)) is an element of R-3 vertical bar x(1)(2) + x(2)(2) - x(3)(2) = 0, x(3) > 0}. Thus, if a tube hypersurface in the class C-2,C-1 locally looks like a piece of M-0 from the point of view of CR-geometry, then from the point of view of affine geometry it (globally) looks like a piece of M-0 as well. This rigidity result is in stark contrast to the Levi nondegenerate case, where the CR-geometric and affine-geometric classifications significantly differ.

• 出版日期2016-9