From PDE Systems and Metrics to Generalized Field Theories

TL;DR

This paper establishes a geometric framework connecting solutions of first order PDE systems to generalized harmonic maps, and constructs a related field theory using least squares variational methods on jet bundles.

Contribution

It introduces a novel geometric structure on jet bundles that transforms PDE solutions into generalized harmonic maps and develops an associated field theory.

Findings

Solutions of PDEs can be viewed as generalized harmonic maps.

A natural geometry induced by PDE systems is constructed.

A field theory attached to the PDE system is developed.

Abstract

The paper proved that every $C^2$-solution of a given first order PDEs system, regarded on the jet fibre bundle of order one $J^1(T,M)$, may be viewed as a "generalized harmonic map", via the least squares variational method. Our ideas are structured in the following way: 1) we find a suitable geometrical structure on $J^1(T,M)$ that convert the solutions of the given PDEs system into "generalized harmonic maps"; 2) we build a natural geometry induced by a such PDEs system; 3) we construct a field theory, in a general setting, naturally attached to this PDEs system.