2010/10/28 by Abraham D. Smith, Smith, Abraham D.
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #math.AP #math.DG #msc:37K10 #msc:58A15
paper · pdf · doi:10.48550/arxiv.1010.6010
25 pages
arxiv created 2010/10/28 · arxiv updated 2010/10/29
For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of second-order jet space) and are defined by non-degenerate scalar second-order-only (Hessian) PDEs in any number of variables. The fiber group is a conformal orthogonal group that acts on the space of independent variables, and it is a subgroup of the conformal orthogonal group for a semi-Riemannian metric that exists on the PDE. These G-structures are automatically compatible with the definition of hydrodynamic integrability, so they allow contact-invariant analysis of integrability via moving frames and the Cartan--Kaehler theorem. They directly generalize the GL(2)-structures that arise in the case of Hessian hyperbolic equations in three variables as well as several related geometries that appear in the literature on hydrodynamic integrability. Part I primarily discusses the motivation, the definition, and the solution to the equivalence problem, and Part II will discuss integrability in detail.