# nLab harmonic differential form

cohomology

### Theorems

#### Riemannian geometry

Riemannian geometry

## Basic definitions

• Riemannian manifold

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

• geodesic

• Levi-Civita connection

• ## Theorems

• Poincaré conjecture-theorem
• ## Applications

• gravity

• # Contents

## Definition

A differential form $\omega \in \Omega^n(X)$ on a Riemannian manifold $(X,g)$ is called a harmonic form 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

Last revised on June 25, 2018 at 12:49:32. See the history of this page for a list of all contributions to it.