Tangent bundle geometry induced by second order partial differential equations

TL;DR

This paper generalizes tangent bundle decomposition from ordinary differential equations to second order PDEs of connection type, introducing curvature operators and analyzing harmonic map equations.

Contribution

It extends tangent bundle geometry to second order PDEs, incorporating a closed 1-form and transverse vector field for the decomposition.

Findings

Decomposition is intrinsic for ODEs but requires additional data for PDEs.

Introduces natural curvature operators from the decomposition.

Analyzes harmonic map equations within this geometric framework.

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. The harmonic map equation is examined, and in this case both the 1-form and the vector field arise naturally.