Chains of path geometries on surfaces: theory and examples (Q6635141)

Path geometries equivalently encode systems of second order ODEs on a smooth manifold as a geometric structure on the projectivized tangent bundle. In the special case of a single equation, this structure is defined on a \(3\)-manifold and in this case, it is equivalent to a para-CR structure. This is a contact structure together with a decomposition of the contact subbundle into a sum of two line bundles, which makes additional tools available. In particular, there is an analog of the Fefferman construction of a conformal structure on the total space of a bundle with one dimensional fibers over a \(3\)-dimensional CR structure. Moreover, there is a notion of chains, which are distinguished unparametrized curves in directions transverse to the contact distribution. They can be realized as the non-constant projections of null geodesics of the Fefferman conformal structure.\N\NAfter reviewing basic material about the relevant path geometries, and the Fefferman construction for CR structures, the authors give a construction of the Fefferman conformal class associated to a path geometry analogous to \textit{J. Lee}'s construction [Trans. Am. Math. Soc. 296 411--429 (1986; Zbl 0595.32026)] in the CR case. This is then used to explicitly derive the differential equation characterizing chains. In particular, this shows that the chains project to paths of the initial path geometry if and only if this path geometry comes from a projective structure, so the initial equation can be written as the (unparametrized) geodesic equation for a connection. A substantial part of the article (about the second half) is devoted to the discussion of several examples of path geometries, for which the results are made explicit.