3-dimensional Cauchy-Riemann structures and 2nd order ordinary differential equations

The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman 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

Twistor geometry of a pair of second order ODEs

We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature $(2, 2)$. We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in particular. Using this framework, we give a new derivation of the Wilczynski invariants for a system of ODEs whose solution space is endowed with a conformal structure. We explain how to reconstruct the conformal structure directly from the integral curves, and present new examples of systems of ODEs with point symmetry algebra of dimension four and greater which give rise to anti--self--dual structures with conformal symmetry algebra of the same dimension. Some of these examples are $(2, 2)$ analogues of plane wave space--times in General Relativity. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature.Comment: Final version to appear in the Communications in Mathematical Physics. The procedure of recovering a system of torsion-fee ODEs from the heavenly equation has been clarified. The proof of Prop 7.1 has been expanded. Dedicated to Mike Eastwood on the occasion of his 60th birthda

Equivalence of variational problems of higher order

We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication by a nonzero constant, and modulo divergence; the equivalence problem for the special class of rank 2 distributions associated with underdetermined ODEs z'=f(x,y,y',..., y^{(n)}); the equivalence problem for variational ODEs of order 2n. This leads to new results such as the fundamental system of invariants for all these problems and the explicit description of the maximally symmetric models. The central role in all three equivalence problems is played by the geometry of self-dual curves in the projective space of odd dimension up to projective transformations via the linearization procedure (along the solutions of ODE or abnormal extremals of distributions). More precisely, we show that an object from one of the three equivalence problem is maximally symmetric if and only if all curves in projective spaces obtained by the linearization procedure are rational normal curves.Comment: 20 page

Third order ODEs and four-dimensional split signature Einstein metrics

We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables

Simple and collective twisted symmetries

After the introduction of $\lambda$-symmetries by Muriel and Romero, several other types of so called "twisted symmetries" have been considered in the literature (their name refers to the fact they are defined through a deformation of the familiar prolongation operation); they are as useful as standard symmetries for what concerns symmetry reduction of ODEs or determination of special (invariant) solutions for PDEs and have thus attracted attention. The geometrical relation of twisted symmetries to standard ones has already been noted: for some type of twisted symmetries (in particular, $\lambda$ and $\mu$-symmetries), this amounts to a certain kind of gauge transformation. In a previous review paper [G. Gaeta, "Twisted symmetries of differential equations", {\it J. Nonlin. Math. Phys.}, {\bf 16-S} (2009), 107-136] we have surveyed the first part of the developments of this theory; in the present paper we review recent developments. In particular, we provide a unifying geometrical description of the different types of twisted symmetries; this is based on the classical Frobenius reduction applied to distribution generated by Lie-point (local) symmetries.Comment: 40 pages; to appear in J. Nonlin. Math. Phys. 21 (2014), 593-62