Differential equations and conformal structures

We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of third order ODEs considered modulo contact transformations of variables and (local) three-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of third order ODEs satisfying the Wuenschmann and the Cartan conditions define a three-dimensional Lorentzian-Einstein-Weyl geometry. The third example associates to each point equivalence class of third order ODEs a six-dimensional conformal geometry of neutral signature. The fourth example exhibits the one-to-one correspondence between point equivalent classes of second order ODEs and four-dimensional conformal Fefferman-like metrics of neutral signature. The fifth example shows the correspondence between undetermined ODEs of the Monge type and conformal geometries of signature (3, 2). The Cartan normal conformal connection for these geometries is reducible to the Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group G2. All the examples are deeply rooted in Elie Cartan's works on exterior differential systems.