Universality of the Divergence

TL;DR

This paper confirms that under certain topological conditions, operators on the module of smooth sections of the tangent bundle of a smooth manifold can be characterized by two axioms, extending algebraic understanding of geometric structures.

Contribution

It proves that operators on the module of smooth sections are uniquely characterized by two axioms for manifolds with trivial first cohomology group, generalizing previous algebraic characterizations.

Findings

Operators are characterized by two axioms when H^1(M, R) = {0}

Extends algebraic characterization to a broad class of manifolds

Supports the universality of divergence in geometric analysis

Abstract

Algebraists asked whether or not an operator on the module of smooth sections of the tangent bundle over the commutative ring of smooth functions of a smooth (orientable) manifold (can be any piece of a compact or a complete manifold) can be characterized by two axioms. In this note we confirm this for any smooth manifold M under the assumption that H^1(M, R) = {0}.