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On the geometry of the k-quasiaffine space. (Russian) Zbl 0677.53008

Webs and quasigroups, Interuniv. thematic Collect. sci. Works, Kalinin 1988, 147-155 (1988).
[For the entire collection see Zbl 0632.00006.]
A projective space \(P_ n\), in which a subspace \(P^{\infty}\) of dimension \(n-k-1\) (called its plane at infinity) is fixed, is called a k- quasi-affine space \(A^ k_ n\). The basic element of a space \(A^ k_ n\) is a k-plane which does not cross \(P^{\infty}\), and its group of transformations is a subgroup of the group of transformations of the space \(P_ n\), which transforms \(P^{\infty}\) into itself. The space \(A^ k_ n\) permits a mapping on an affine space \(A_ m\) where \(m=(k+1)(n-k);\) in a hyperplane at infinity of \(A_ m\) a Segre surface \(S(k,n-k-1)\) is fixed. Such space is called a Segre-affine space \(SA_ m\). For this mapping a k-plane of a space \(A^ k_ n\) is mapped into a point of a space \(SA_ m\). The mapping \(A^ k_ n\to SA_ m\) is a generalization of the stereographic projection of a sphere into a plane. A correspondence between straight lines and curves of different types of a space \(SA_ m\) and one-parameter sets of k-planes of a space \(A^ k_ m\) is established.
Reviewer: M.A.Akivis

MSC:

53A15 Affine differential geometry
53A40 Other special differential geometries

Citations:

Zbl 0632.00006