# nLab Hodge-Maxwell theorem

Contents

### Context

#### Differential cohomology

differential cohomology

# Contents

## Idea

The Hodge theorem in the language of electromagnetism. Over a Riemann surface this may be regarded as simple case of the Narasimhan-Seshadri theorem.

## Statement

Let $(X,g)$ be a compact oriented Riemannian manifold of dimension $n$. Write $\star$ for the corresponding Hodge star operator.

Then for every exact differential n-form $j$ of degree $n-k-1$ there is in each de Rham cohomology class of degree 2 a unique representative closed 2-form $F$

$\mathbf{d} F = 0$

such that

$\mathbf{d}\star F = j \,.$

Reading this as Maxwell's equations on $(X,g)$ then $g$ is the field of gravity, $F$ is the Faraday tensor measuring the field strength of the electromagnetic field and $j$ is the electric current.

## References

The term “Hodge-Maxwell theorem” in the above form appears in

Last revised on December 11, 2020 at 17:43:50. See the history of this page for a list of all contributions to it.