nLab
harmonic differential form

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Riemannian geometry

Contents

Definition

A differential form ωΩ n(X)\omega \in \Omega^n(X) on a Riemannian manifold (X,g)(X,g) is called a harmonic form if it is in the kernel of the Laplace operator Δ g\Delta_g of XX in that Δω=(d+d ) 2ω=0\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