Paraconformal structures, ordinary differential equations and totally geodesic manifolds

2013/10/25 by Wojciech Kryński, Wojciech Krynski, Krynski, Wojciech
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Differential Geometry Research #Nonlinear Waves and Solitons #math.DG

paper · pdf · doi:10.48550/arxiv.1310.6855

arxiv created 2015/07/03 · arxiv updated 2015/07/07

Abstract

We construct point invariants of ordinary differential equations that generalise the Cartan invariants of equations of order two and three. The vanishing of the invariants is equivalent to the existence of a totally geodesic paraconformal structure which consist of a paraconformal structure, an adapted GL(2,\R)-connection and a two-parameter family of totally geodesic hypersurfaces on the solution space. The structures coincide with the projective structures in dimension 2 and with the Einstein-Weyl structures of Lorentzian signature in dimension 3. We show that the totally geodesic paraconformal structures in higher dimensions can be described by a natural analogue of the Hitchin twistor construction. We present a general example of Veronese webs which correspond to the hyper-CR Einstein-Weyl structures in dimension 3. The Veronese webs are described by a hierarchy of integrable systems.

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