Submaximally Symmetric CR-Structures

Abstract

Hypersurface type CR-structures with non-degenerate Levi form on a manifold of dimension \((2n+1)\) have maximal symmetry dimension \(n^2+4n+3\). We prove that the next (submaximal) possible dimension for a (local) symmetry algebra is \(n^2+4\) for Levi-indefinite structures and \(n^2+3\) for Levi-definite structures when \(n>1\). In the exceptional case of CR-dimension \(n=1\), the submaximal symmetry dimension 3 was computed by E. Cartan.