On an oriented Riemannian -manifold, the Hodge star is determined byIt maps -forms to -forms and satisfies on -forms.
The codifferential is the formal adjoint of the exterior derivative. On -forms in dimension , one convention is
A differential form is harmonic when it lies in the kernel of the Hodge Laplacian . On a compact manifold, this is equivalent to and .
On a compact oriented Riemannian manifold,as an -orthogonal direct sum. Every de Rham cohomology class consequently has a unique harmonic representative.
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The Hodge star operator is a mathematical operator used extensively in differential geometry and algebraic topology, particularly in the context of differential forms on Riemannian manifolds. It acts on differential forms and is used to relate forms of different degrees.