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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
##### Keywords:
projective space; polarity; connections; intrinsic geometries
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