Normal and tangent ranks of CR mappings
Note with \(T^ pM\), the usual tangent space of the real manifold \(M \subset \mathbb{C}^ N\) at \(p \in M\), by \(T_ p^ cM \equiv T_ pM \cap iT_ pM\) its complex tangent space. \(M\) is said to be generic if \(T_ pM + iT_ pM = \mathbb{C}^ N\) holds for all \(p \in M\). Note by \(F_ * : T M \to TM'\) the differential of a \(C^ 1\) map \(F : M \to M'\). The main results are the following. Theorem 1: Let \(F : M \to M'\) be a CR mapping of class \(C^{2, \alpha}\) between generic manifolds of class \(C^{2, \alpha}, \alpha > 0\), and let \(p\) be a minimal point of \(M\), such that \(F_ * (T_ q M) + T_{F(q)}^ c M' = T_{F(q)} M'\) holds for \(q\) arbitrarily close to \(p\). Then \(F_ * (T_ pM) \not \subset T_{F(p)}^ cM'\) unless CR functions on \(M'\) extend holomorphically to a common full neighborhood of \(F(p)\) in \(\mathbb{C}^ N\). In any case, \(F(-p)\) is minimal in \(\{M' \}\). Minimality is understood in the sense of \textit{A. E. Tumanov} [Math. USSR, Sb. 64, No. 1, 129-140 (1989); translation from Mat. Sb., Nov. Ser. 136(178), No. 1(5), 128-139 (1988; Zbl 0692.58005)]. Theorem 1 generalizes a result for hypersurfaces obtained by \textit{M. S. Baouendi} and \textit{L. Preiss-Rothschild} [Invent. Math. 111, No. 2, 331- 348 (1993; Zbl 0781.32021)]. Theorem 2: Let \(F : M \to M'\) be a homeomorphic CR mapping of class \(C^{2, \alpha}\) between CR manifolds of class \(C^{2, \alpha}\) with the same CR dimension. Then \(F\) is a diffeomorphism at all minimal points of \(M\). This theorem is closely related with \textit{S. Pinchuk}'s results on CR homeomorphism [Sib. Math. J. 15, 909-917 (1974; Zbl 0289.32011); Indiana Univ. Math. J. 41, No. 1, 1-16 (1992; Zbl 0766.32021)]. Theorem 3: Let \(M,M'\) be CR manifolds of class \(C^{2, \alpha}\), \(\alpha > 0\), and \(F : M \to M'\) a CR mapping of class \(C^{2, \alpha}\). Suppose that, for some \(p \in M\), \(p' = F(p)\) is minimal and \(F_ * (T_ p^ c M) = T^ c_{p'} M'\). Then \(F\) has surjective differential at \(p\) and \(p\) is minimal. Theorem 3 permits to give a definition of minimality of \(p \in M\) which may be much easier to handle than the classic one.
scientific article; zbMATH DE number 727830
scientific article