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 fom 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

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