On the complete system of finite order for CR mappings and its application (Q1268681)

Let \(F: M \to \widetilde M\) be an arbitrary CR mapping between CR manifolds. A function \(F\) is said to satisfy a complete system of order \(K\) if, for each multi-index \(\alpha\) with \(| \alpha| = K\), there exists a real analytic function \(H_{\alpha}\) such that \(D^{\alpha}F=H_{\alpha}(z,D^{\beta}F)\), \(|\beta|\leq K-1\). The paper is related to the following problem: If the system of vectors derived from the mapping \(F\) and the tangential Cauchy-Riemann vector fields on \(M\) satisfy certain conditions, then \(F\) satisfies a complete system of finite order. The author proves the following theorem: Let \(M\) and \(\widetilde M\) be real hypersurfaces in \(\mathbb C^{n+1}\) and \((f,f_{n+1}):M\to \widetilde M\) a CR mapping. Suppose that \(\widetilde M\) has a nondegenerate Levi form at the origin and that the origin in \(M\) is a point of type \(l(<\infty)\). Consider the following two cases: (I) \(M\) has a nondegenerate Levi form at the origin \((l = 2)\), or \(M\) has a degenerate Levi form at the origin and \(n = 1\). (II) \(M\) has a degenerate Levi form at the origin and \(n \geq 2\). In case (I), if \((f,f_{n+1})\) satisfies the Hopf lemma property at the origin, then it satisfies a complete system of order \(l + 1\). In case (II), if \((f,f_{n+1})\) satisfies \(sp\langle f_1,\dots,f_n\rangle_{\mathbb C}\not\ni 0\) (mod \(\mathcal I^{m+1}\)), then it satisfies a complete system of finite order (for the special notations see p. 618 of the paper).