Tangent bundle geometry induced by second order partial differential equations

Abstract

We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs ‘of connection type’. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent variables, together with a transverse local vector field. The resulting decomposition provides several natural curvature operators. We give three examples to indicate possible applications of this theory.