Given a finite dimensional (pseudo)-Riemannian manifold , the Hodge star operator “completes” a -differential form to the volume form of .
Let be an oriented -dimensional smooth manifold endowed with a (pseudo)-Riemannian metric . For , write for the vector space of -forms on .
Hodge inner product
The metric naturally induces a nondegenerate symmetric bilinear form
If is compact then the integral of this against the volume form exists. This is the Hodge inner product
Hodge star operator
The Hodge star operator is the unique linear function
defined by the identity
where is the volume form induced by .
Therefore in terms of the Hodge operator the Hodge inner product reads
If is a local basis on and is the dual basis, so that , then
where is the sign of the permutation and is the determinant of in the local basis.
Basic properties (Basis-independent formulas)
Let be a Riemannian manifold of dimension and let . Then
On a Kähler manifold
On a Kähler manifold of dimension the Hodge star operator acts on the Dolbeault complex as
(notice the exchange of the role of and ). See e.g. (Biquerd-Höring 08, p. 79). See also at Serre duality.
The metric is used in two places in the specification of the Hodge operator: in the inner product on forms and in the volume form. If is equipped only with a volume form (not necessarily coming from a metric), then the Hodge operator still takes -forms to -vector fields. If the manifold is not oriented, then the metric only gives a volume pseudoform, but the Hodge operator still takes -forms to -pseudoforms. Finally, if is equipped with only a volume pseudoform (which is equivalent to an absolutely continuous Radon measure on ), then the Hodge operator takes -forms to -pseudovector fields. (Of course, in every case, one might apply the operator to pseudoforms or multivector fields to begin with.)
Some useful basic formulas are listed in
- Hodge theory on Riemannian manifolds , lecture notes (pdf)
Discussion in complex geometry includes
- O. Biquard, A. Höring, Kähler geometry and Hodge theory, 2008 (pdf)