Prolongation closes an overdetermined differential equation into a first-order system by naming its higher derivatives as new variables, which is possible exactly when the equation is of finite type. The closed system is a connection whose parallel sections correspond one-to-one with solutions, so the rank of the prolonged bundle bounds the solution space from above - attained in the flat case, obstructed by curvature otherwise. Applied to projectively invariant equations on a projective manifold (an equivalence class of connections with the same unparametrised geodesics), prolongation reconstructs the projective tractor connection and, through the Leibniz rule rather than a second prolongation, its dual cotractor connection. The worked case is the projective metrisability equation - vanishing of the trace-free part of the covariant derivative of a symmetric contravariant two-tensor - whose non-degenerate solutions are the metrics whose Levi-Civita connection lies in the projective class. Prolonging it produces a connection on the symmetric square of the tangent bundle carrying Weyl and Cotton curvature terms, and that connection coincides with the tractor connection on the same bundle if and only if the metric is Einstein.
McNaughton's "Projective Geometry and PDE Prolongation" is a BSc(Hons) dissertation written at the University of Auckland in 2021 under Rod Gover and posted to arXiv in 2024. Prolongation is the proc…