nLab harmonic differential form

Contents

Context

Riemannian geometry

Riemannian geometry

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 16:49:32. See the history of this page for a list of all contributions to it.