# nLab harmonic differential form

cohomology

### Theorems

#### Riemannian geometry

Riemannian geometry

# Contents

## Definition

A differential form $\omega \in \Omega^n(X)$ on a Riemannian manifold $(X,g)$ is called a harmonic fom if it is in the kernel of the Laplace operator $\Delta_g$ of $X$ in that $\Delta \omega = (d + d^\dagger)^2 \omega = 0$.

## Properties

### Relation to Dolbeault cohomology

On a compact Kähler manifold the Hodge isomorphism (see there) identifies harmonic differential forms with Dolbeault cohomology classes.

### Relation to Hodge theory

For the moment see at Hodge theory

## References

Revised on June 5, 2014 04:16:26 by Urs Schreiber (92.68.97.89)