Linear Complexes

Abstract

Having identified the set of lines with the set of points of a certain quadric contained in projective five-space (the Klein quadric), we are able to apply concepts of projective geometry to the set of lines. The basic element in a projective space are its subspaces, ‘linear mainfolds’ of points, defined by linear equation in the points’ homogeneous coordinates. Accordingly in this chapter we are going to study ‘linear manifolds’ of lines. These correspond to intersection of the Klein quadric with subspaces, and they are defined by linear equations in the lines’ Plücker coordinates. The basic object will be the so-called linear line complex, a three-dimensional linear manifold of lines defined by one linear equation. It is on the one hand connected to null polarities of projective three-space, and has on the other hand several nice interpretations in Euclidean geometry, kinematics and statics. Other linear manifolds of lower dimension are intersections of complexes. Unlike projective subspaces of the same dimension, all of which ‘look the same’, i.e., are projectively equivalent, linear manifolds of lines exhibit a slightly more complicated behaviour.