Further Results

Abstract

In the first section of this chapter we consider tube hypersurfaces in ℂⁿ⁺¹ locally CR-equivalent to a quadric Q _g where the Hermitian form g is degenerate. For g ≢ 0 we show that every tube hypersurface of this kind is real-analytic and ex- tends to a closed non-singular real-analytic tube hypersurface in ℂⁿ⁺¹ represented as the direct sum of a complex linear subspace of ℂⁿ⁺¹ and a closed spherical tube hypersurface lying in a complementary complex subspace. For g ≡ 0 such a hy- persurface is an open subset of a real affine hyperplane in ℂⁿ⁺¹. Thus, the study of tube hypersurfaces locally CR-equivalent to Levi-degenerate quadrics reduces to the study of spherical tube hypersurfaces. In the second section we briefly describe the approach to the problem of affine classification of spherical tube hypersurfaces recently proposed by Fels and Kaup in [41], [42]. Many results of this chapter apply to CR-manifolds of arbitrary CR-codimension.