Construction of Local Conservation Laws by Generalized Isometric Embeddings of Vector Bundles

TL;DR

This paper employs Cartan-Kähler theory to derive local conservation laws from covariantly closed forms and extends classical isometric embedding results to vector bundles of rank two.

Contribution

It introduces a method to construct conservation laws using generalized isometric embeddings of vector bundles, broadening the scope of classical geometric embedding theorems.

Findings

Constructed conservation laws for divergence-free energy-momentum tensors.

Extended isometric embedding results to vector bundles of rank two.

Applied Cartan-Kähler theory to new geometric contexts.

Abstract

This article uses Cartan-K\"ahler theory to construct local conservation laws from covariantly closed vector valued differential forms, objects that can be given, for example, by harmonic maps between two Riemannian manifolds. We apply the article's main result to construct conservation laws for covariant divergence free energy-momentum tensors. We also generalize the local isometric embedding of surfaces in the analytic case by applying the main result to vector bundles of rank two over any surface.