Causal geometries and third-order ordinary differential equations

Understand

We discuss contact invariant structures on the space of solutions of a third-order ordinary differential equation.

• Associated to any third-order differential equation modulo contact transformations, Chern introduced a degenerate conformal Lorentzian metric on the space of 2-jets of functions of one variable.
• When the Wuenschmann invariant vanishes, the degenerate metric descends to a proper conformal Lorentzian metric on the space of solutions.
• In the general case, when the Wuenschmann invariant is not zero, we define the notion of a causal geometry, and show that the space of solutions supports one.