Complete system of finite order for the embeddings of pseudo-Hermitian manifolds into ${\bf C}\sp {N+1}$

Abstract

Let $(M, {\cal V}, \theta)$ be a real analytic $(2n+1)$-dimensional pseudo-hermitian manifold with nondegenerate Levi form and $F$ be a pseudo-hermitian embedding into $\mathbb{C}^{n+1}$. We show under certain generic conditions that $F$ satisfies a complete system of finite order. We use a method of prolongation of the tangential Cauchy-Riemann equations and pseudo-hermitian embedding equation. Thus if $F \in C^{k}(M)$ for sufficiently large $k$, $F$ is real analytic. As a corollary, if $M$ is a real hypersurface in $\mathbb{C}^{n+1}$, then $F$ extends holomorphically to a neighborhood of $M$ provided that $F$ is sufficiently smooth.