On the affine geometry of congruences of lines

Congruences, or 2-parameter families of lines in 3-space have a long history, with contributions from Dupin, Hamilton, Weierstrass, Darboux, Eisenhart, Study, Blaschke and others, and applications to the theories of minimal surfaces, Backlund transformations, sine-Gordon equations and many other areas of mathematics. The generic geometry of normal congruences was investigated by Thom and is linked to Arnold's notion of Lagrangian singularities. The geometry of “almost all” normal congruences is well understood, see for example [7] and references therein. In this paper there is described, for the first time, the local affine geometry of almost all general congruences, in particular that associated with their focal sets, less well studied focal planes and middle surfaces. In this work a projective quadric in the projectivised tangent space to the manifold of lines is identified which plays a basic role in understanding the affine geometry of ruled surfaces, congruences and 3-parameter families or complexes. Many of the results generalise to lines in ${\mathbb R}^n, n>3$.