Reduction of a problem for classification of tube quadrics to the solution of a system of differential equations of special form
type: Item
instance of: scholarly article
title: Reduction of a problem for classification of tube quadrics to the solution of a system of differential equations of special form
zbMATH Open document ID: 0684.32014
DOI: 10.1007/BF01157928
published in: Mathematical Notes
publication date: 1989-01-01T00:00:00Z
review text: The tube hypersurface \[ (1)\quad x_ 0=F(x_ 1,x_ 2,...,x_ n)\quad in\quad {\mathbb{C}}^{n+1} \] is called hyperquadric if its equation has the form \[ (2)\quad x_ 0=\sum^{k}_{j=1}| z_ j|^ 2- \sum^{n}_{j=k+1}| z_ j|^ 2,\quad z_ j=x_ j+iy_ j \] and is called spherical if it is locally biholomorphically equivalent to (2). The author proves the following Theorem: The hypersurface of kind (1) is spherical if and only if it is affinely equivalent to the hypersurface of the same kind (1), for which the corresponding function F is analytic at the neighbourhood of the origin and satisfies the system of partial differential equation of the kind \[ \frac{\partial^ 2F}{\partial x_ i\partial x_ j}=\sum^{n}_{\alpha =1}\frac{\partial F}{\partial x_{\alpha}}(D_ i^{\alpha}\frac{\partial F}{\partial x_ j}+D_ j^{\alpha}\frac{\partial F}{\partial x_ i}+C^{\alpha}_{ij})+h_{ij} \] with initial conditions \(F_{x_ i}(0)=0\), \(i=1,...,n.\) In case \(k=n\) the necessity of the conditions of this theorem was stated by \textit{P. C. Yang} [Am. J. Math. 104, NO. 5, 1005-1024 (1982; Zbl 0514.32018)].
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32A07
zbMATH DE Number: 4120523
zbMATH Keywords: spherical hypersurface
zbMATH Keywords: hyperquadric
author: A. V. Isaev
reviewed by: Abduhamid Džuraev
MaRDI profile type: Publication
cites work: Automorphism of Tube Domains
cites work: Automorphisms of Tube Domains and Spherical Hypersurfaces
cites work: Q3800361
cites work: Real hypersurfaces in complex manifolds
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description: scientific article; zbMATH DE number 4120523