Fields of geometric objects associated with compiled hyperplane H ( ,L)  -distribution in affine space

Abstract

In the first-order frame a tangentially r-framed hyperband is given in the projective space. For simplicity of presentation, we adapt the frame by the field of the 1st kind normals. The tensor of nonholonomicity of cloth­ing L-planes field is introduced. The vanishing the nonholonomic tensor leads to three different interpretations of the hyperband. With the help of ТL-virtual normals of the 1st and 2nd kind of framed L-planes, we come to the following conclusion: in a third order differential neighborhood the bundle of the hyperband second kind normals generates a one-parameter bundle of ТL-virtual first and second kind normals, which correspond to each other in bijection. We consider focal images associated with the hy­perband, with the help of which the Norden — Timofeev plane of the indicated hyperband is constructed. The geometric interpretation of the object defining the Norden — Timofeev surface was found by R. F. Dom­brovsky for tangentially r-framed surfaces in the projective space. We note that the field of ТL-virtual first kind normals induces the field of the Norden — Timofeev planes, this is the field of the 2nd kind regular hyper­band normals. It is proved that with each the 1st kind ТL-virtual normal is induced a bundle of Cartan planes in the 1st kind normal at a fixed point of the hyperband. In conclusion, we consider the p-structures of the tangent planes field at the base surface of the hyperband.