Congruences of horospheres, horocycles, equidistant surfaces, and equidistant curves in the three-dimensional Lobachevskij space (Q1896737)

The three-dimensional space of Lobachevskij is represented by the interior points of the non-ruled and non-degenerate absolute quadric \(Q_0\) of the projective space \(P_3\). A horosphere \(Q\) is a non- ruled and non-degenerate quadric lying inside \(Q_0\). \(Q\) is tangent to \(Q_0\) in the unique point \(A_0\). For arbitrary choice of \(M \in Q\) the pole \(A_M\) with respect to \(Q_0\) of a tangent plane to \(Q\) in \(M\) belongs to the straight line \(A_0 M\). The congruence \(O^*\) (two- parametric family) of horospheres is considered. The frame of \(O^*\) is found. The focal manifold of \(O^*\) on the generator \(Q\) contains two imaginary ideal straight lines and a unique actual point \(F\). The line congruence of \(\ell = A_0 F\) and the congruence of the lines \(\ell^*\) which are polar-conjugate to \(\ell\) with respect to \(Q_0\) are studied, too. Some special congruences \(O^*\) are introduced and their properties are found. Similarly, the properties of the congruences of horocycles, of equidistant surfaces and equidistant curves in Lobachevskij three-dimensional space are studied. The existence problems of these congruences are solved.