A vector field induced de Rham-Hodge theory on manifolds

TL;DR

This paper develops a new de Rham-Hodge theory framework on manifolds induced by a vector field, defining related operators and extending the theory to manifolds with boundary.

Contribution

It introduces a novel vector field induced de Rham-Hodge framework, including new operators and boundary conditions, expanding classical theory on manifolds.

Findings

Established a vector field induced Hodge Laplacian on differential forms.

Extended the framework to manifolds with boundary using specific boundary conditions.

Provided remarks on the properties and implications of the new framework.

Abstract

We introduce a de Rham-Hodge framework induced by a vector field on a compact, oriented smooth manifold. By utilizing a vector field induced isomorphism on differential forms, we define a vector field induced Hodge $L^2$-inner product, codifferential, and Hodge Laplacian on differential forms. We then establish the resulting de Rham-Hodge theory for closed manifolds and extend it to manifolds with boundary by imposing certain vector field induced boundary conditions. We also include some remarks on this resulting framework.