Linearly degenerate partial differential equations and quadratic line complexes

A quadratic line complex is a three-parameter family of lines in projective space P-3 specified by a single quadratic relation in the Plucker coordinates. Fixing a point p in P-3 and taking all lines of the complex passing through p we obtain a quadratic cone with vertex at p. This family of cones supplies P-3 with a conformal structure, which can be represented in the form f(ij) (p) dp(i) dp(j) in a system of affine coordinates p = (p(1), p(2), p(3)). With this conformal structure we associate a three-dimensional second-order quasilinear wave equation Sigma(i,j) f(ij) (u(x1), u(x2), u(x3))u(xixj) = 0 whose coefficients can be obtained from f(ij) (p) by setting p(1) = u(x1), p(2) = u(x2), p(3) = u(x3). We show that any partial differential equations (PDE) arising in this way is linearly degenerate, furthermore, any linearly degenerate PDE can be obtained by this construction. This provides a classification of linearly degenerate wave equations into 11 types, labelled by Segre symbols of the associated quadratic complexes. We classify Segre types for which the structure f(ij) (p) dp(i) dp(j) is conformally flat, as well as Segre types for which the corresponding PDE is integrable.