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On the intrinsinc geometry of the variety of planes of maximal dimension for polarities of second order. (Russian) Zbl 0561.53028
A correlation in the projective space \(P_ n\) is a polarity if its tensor has the symmetry \(a_{\alpha \beta}=\sigma a_{\beta \alpha}\), \(\sigma^ 2=1\), \(\alpha,\beta,\gamma,...=\overline{0,n}\). Let \(Q_{2k,k}\) be the variety of all zero-planes of maximal dimension. On \(Q_{2k,k}\) connections of the first and the second order are defined. In the paper under review the author gives an interpretation of some differential-geometric properties of \(Q_{2k,k}\) in special cases. Let \(\sigma =1\). If \(k=2\) then \(Q_{4,2}\) is a pair of families of planes generating quadrics \(Q_ 4\) in \(P_ 5\) and the intrinsinc geometries of the normalized variety \(Q_{4,2}\) are projectively Euclidean corresponding to the intrinsic geometries of three-dimensional spaces normalized in the sense of Norden. If \(k=3\) then the connection of the first order is conformally Euclidean. This fact illustrates the principle of H. Cartan. If \(k=2s-1\) then the geometry of the first order is the geometry with some special metric tensors of valency s which define in the space of directions of the tangent space a pair of dual polarities of order \(s\). The analogous results take place in the case \(\sigma =1\).
MSC:
53B15 Other connections
14M99 Special varieties
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