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On a relation between holomorphic functions and $$G$$-transformations of $$2n$$-dimensional manifolds. (English. Russian original) Zbl 0879.53053
Russ. Math. 40, No. 8, 37-46 (1996); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1996, No. 8(411), 39-48 (1996).
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 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 $${\mathcal F}^n$$.
##### MSC:
 53C56 Other complex differential geometry 32C20 Normal analytic spaces 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces