nLab harmonic differential form

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 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)