Line congruences as surfaces in the space of lines

The space of lines in R³ can be viewed as a four dimensional homogeneous space of the group of Euclidean motions, E(3). Line congruences arise in the classical method of transforming one surface to another by lines. These transformations are particularly interesting if some geometric property of the original surface is preserved. Line congruences, then, are two parameter families of lines and can be studied as surfaces in the space of lines. In this paper, we use the method of moving frames to study line congruences. We calculate the first order invariants of line congruences for which there are two real focal surfaces, and give the geometric meaning of these invariants. We look specifically at the case where the two first order invariants are constant and give a simple proof of Bäcklund's Theorem which relates to the transformation of one pseudospherical surface, a surface of constant negative Gaussian curvature, to another. These transformations are of interest since pseudospherical surfaces correspond to solutions to the sine-Gordon equation. We also give a proof of Bianchi's permutability theorem for pseudospherical surfaces in this context. Finally, we use the results of these theorems to generate some pseudospherical surfaces. All of these concepts and results are understood in terms of the structure equations of the line congruence.