Reduction of a problem for classification of tube quadrics to the solution of a system of differential equations of special form
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)].
scientific article; zbMATH DE number 4120523
scientific article