On a relation between holomorphic functions and \(G\)-transformations of \(2n\)-dimensional manifolds (Q1357951)

Let \(F^{2n}\) be a \(2n\)-dimensional submanifold \((n\geq 1)\) of class \(C^3\) in the Euclidean space \(E^{2n+2}\), considered as a complex manifold \({\mathcal F}^n =(F^{2n}, \Sigma)\) with the complex structure \(\Sigma\) obtained by giving the transformations of local coordinates of the type \(z'= \varphi (z)\) where \(\varphi\) is a bi-holomorphic mapping between two domains of \(\mathbb{C}^n\) (see \textit{B. V. Shabat} [`Introduction to complex analysis', Part II (Translations of Mathematical Monographs 110, AMS, Providence) (1992; Zbl 0799.32001)]). The author defines here the \(G\)-transformations of \({\mathcal F}^n\) as its transformations such that \(F^{2n}\) is being transformed into \(\widetilde F^{2n}\) with the preservation of its Grassmann image, i.e., the normal planes \(N_x(F^{2n})\) and \(\widetilde N_{\tilde x} (\widetilde F^{2n})\) of the surfaces \(F^{2n}\) and \(\widetilde F^{2n}\) at the corresponding points \(x\) and \(\widetilde x\) are parallel in \(E^{2n+2}\). Denoting by \(\overline {\mathcal U} (x)\) the field of displacements of the point \(x\) of \(F^{2n}\) under a \(G\)-transformation, the equation of a \(G\)-transformation for \({\mathcal F}^n\) can be written in the form: \[ \bigl(d \overline {\mathcal U} (x),\;\overline n(x) \bigr)= 0, \quad \forall x\in F^{2n}, \tag{1} \] where \(\overline n(x)\) is an arbitrary vector normal to \(F^{2n}\) at \(x\). The author investigates here the local structure of the differential equation (1). It is shown that, under some conditions, equation (1) on \({\mathcal F}^n\) is reduced to the form \[ \overline \partial \varphi(z) =0, \quad z\in D\subset \mathbb{C}^n. \] Some results from the theory of holomorphic functions of several complex variables are applied to obtain the solution of problems for \(G\)-transformations of \({\mathcal F}^n\).