Contigious hyperquadrics of coequipped hyperbands sHm

Abstract

We consider hyperquadrics that are internally connected to coequipped hyperbands in the projective space. Specifically, a hyperquadric Qn1 tangent to a hyperplane at the point is called a contiguous hyper quadric of a hyperband if it has a second-order contact with the base surface of the hyperband. In a the third order differential neighborhood of the forming element of the hyperband, two two-parameter bundles of fields of adjoining hyperquadrics are internally invariantly joined, their equations are given in a dot frame. The set of hyperquadrics such that the plane and the plane of Cartan are conjugate with respect to hyperquadric Qn1 is considered. The condition is shown under which the normal of the 2nd kind and the Cartan plane are conjugate with respect to the hyperquadric Qn1 . In addition, the following theorem is proved: normalization of a coequipped regular hyperband has a semi-internal equipment if and only if its normals of the first and second kind are polarly conjugate with respect to the hyperquadric.