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Contents
Context
Riemannian geometry
Differential geometry
Contents
Idea
Given a finite dimensional (pseudo)-Riemannian manifold , the Hodge star operator “completes” a -differential form to the volume form of .
Definition
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
Generalizations
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.)
Properties
Component expression
Let be a (pseudo-)Riemannian manifold of dimension , and locally, on some open subset , let
be a frame of differential 1-forms (a vielbein). For example if is a coordinate chart on , then is such a frame.
With this choice, any differential p-form has a component expansion
for smooth function-components (where here and in the following we use the Einstein summation convention).
In terms of these components, the Hodge dual of is expressed by the following formula:
Here
-
, are the factorials of and , respectively,
-
(the Levi-Civita symbol) is the signature of the permutation
-
is the square matrix of components of the metric tensor in the chosen basis, i.e. such that
-
is the determinant of
-
is the absolute value of the determinant.
Basic properties
Let be a (pseudo-) Riemannian manifold of dimension and let . Then the following holds:
(2) (3) (4)
where denotes the volume form.
(e.g. Frankel 1997 (14.9), (14.5); Padmanabhan 2010 (11.61))
Examples
Hodge star operator 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.
Hodge star operator on Minkowski spacetime
We spell out component expressions for the Hodge star operator on -dimensional Minkowski spacetime.
Conventions
We use Einstein summation convention throughout. With this convention, a generic differential p-form reads
Here denotes the factorial of .
We take the Minkowski metric to be the diagonal matrix of the form
We normalize the Levi-Civita symbol as
(5)
which means that
(6)
We normalize the sign of the volume form as
We write
(8)
for the generalized Kronecker delta, whose value is the signature of the permutation that takes the upper indices to the lower indices, if any such exists, and zero otherwise.
This appears whenever the Levi-Civita symbol is contracted with itself:
(9)
Notice the minus sign in (9), which comes, via (6), from the Minkowski signature.
Definition
We write for the operator of contraction of differential forms with the vector field , hence the linear operator on differential forms with anticommutator
With the volume form as in (7) it follows that (notice the reversion of the index ordering in the contraction operators )
Definition
For a differential p-form
its Hodge dual is:
(11)
where in the second line we used (10).
Properties
Proposition
(Hodge pairing)
For a differential p-form on -dimensional Minkowski spacetime its wedge product with its Hodge dual (11) is
(12)
Proof
We compute as follows:
Here the sign in the last lines arises from the Minkowski signature via (9).
Proposition
(double Hodge dual)
For a differential p-form on -dimensional Minkowski spacetime, its double Hodge dual (11) is
(13)
Proof
We compute as follows:
Here the sign in the last lines arises from the Minkowski signature via (9).
Proof
We compute as follows:
Here the sign in the last lines arises from the Minkowski signature via (9).
References
Lecture notes:
- Hodge theory on Riemannian manifolds , lecture notes (pdf)
Textbook accounts:
A unified perspective in terms of Berezin integration:
Discussion in complex geometry:
- O. Biquard, A. Höring, Kähler geometry and Hodge theory, 2008 (pdf)
With an eye towards application in supergravity and string theory:
Discussion of the Hodge star operator on supermanifolds (in terms of picture changing operators and integral top-forms for integration over supermanifolds):