Klein manifolds representing line manifolds in an elliptic 3-space (Q1097525)

The manifold of all straight lines in an elliptic 3-space \(S_ 3\) is mapped by the Klein mapping onto the Klein quadric \(Q\) \(2_ 4\) of the Klein elliptic space \(S_ 5\) (the metric of which is defined by the metric of \(S_ 3)\) in a projective space \(P_ 5\). Then a line complex resp. a line congruence is represented by a surface of dimension 3 resp. 2 (its Klein image) in \(Q\) \(2_ 4\). A line complex is called inflectional iff both centres of each of its lines (generators) are inflection centres, a simple and a double one. The osculating spaces of the first and second order of the Klein images of these inflectional line complexes and of some associated line congruences are studied and their properties are found up to motions in the space \(S_ 5\). All invariants to be found, belonging to a line manifold M in \(S_ 3\), are interpreted in terms of the Gauss curvature and the mean curvature vector of the Klein image of M in \(Q\) \(2_ 4\subset S_ 5\).