Fields of geometric objects associated with compiled hyperplane H ( ,L) -distribution in affine space
- DOI
- 10.5922/0321-4796-2020-52-10
- Pages
- 97-116
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.
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