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 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal Lorentzian geometries. The second example shows that every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry. The third example associates to each point equivalence class of 3rd order ODEs a 6-dimensional conformal geometry of neutral signature. The fourth example exhibits the one-to-one correspondence between point equivalent classes of 2nd order ODEs and 4-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.MSC 2000: 34A26, 53B15, 53B30, 53B50.

Abstract. On a natural circle bundle T(M ) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null selfdual 2-planes, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. Over the open set where g is not antiselfdual, the distribution D is (2,3,5) in T(M ). We show that if M is a Cartesian product of two Riemann surfaces (Σ 1 , g 1 ) and (Σ 2 , g 2 ), and if g = g 1 ⊕ (−g 2 ), then the circle bundle T(Σ 1 × Σ 2 ) is just the configuration space for the physical system of two surfaces Σ 1 and Σ 2 rolling on each other. The condition for the two surfaces to roll on each other 'without slipping or twisting' identifies the restricted velocity space for such a system with the tautological distribution D on T(Σ 1 × Σ 2 ). We call T(Σ 1 × Σ 2 ) the twistor space, and D the twistor distribution for the rolling surfaces. Among others we address the following question: "For which pairs of surfaces does the restricted velocity distribution (which we identify with the twistor distribution D) have the simple Lie group G 2 as the group of its symmetries?" Apart from the well known situation when the surfaces Σ 1 and Σ 2 have constant curvatures whose ratio is 1:9, we unexpectedly find three different types of surfaces that when rolling 'without slipping or twisting' on a plane, have D with the symmetry group G 2 . Although we have found the differential equations for the curvatures of Σ 1 and Σ 2 that gives D with G 2 symmetry, we are unable to solve them in full generality so far.

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The equivalence problem for second-order ordinary differential equations (ODEs) given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate three-dimensional Cauchy-Riemann structures. This approach enables an analogue of the Fefferman metrics to be defined. The conformal class of these (split signature) metrics is well defined by each point equivalence class of second-order ODEs. Its conformal curvature is interpreted in terms of the basic point invariants of the corresponding class of ODEs.

A Lorentzian manifold is defined here as a smooth pseudo-Riemannian manifold with a metric tensor of signature (2n + 1, 1). A Robinson manifold is a Lorentzian manifold M of dimension 4 with a subbundle N of the complexification of T M such that the fibers of N → M are maximal totally null (isotropic) and [Sec N, Sec N ] ⊂ Sec N . Robinson manifolds are close analogs of the proper Riemannian, Hermite manifolds. In dimension 4, they correspond to space-times of general relativity, foliated by a family of null geodesics without shear. Such space-times, introduced in the 1950s by Ivor Robinson, played an important role in the study of solutions of Einstein's equations: plane and sphere-fronted waves, the Gödel universe, the Kerr solution, and their generalizations, are among them. In this survey article, the analogies between Hermite and Robinson manifolds are presented in considerable detail.1991 Mathematics Subject Classification. 32C81, 53B30; 32V30, 83C20.