Riemannian Metric and the Hodge-Star in 3D

Euclidean Space

The flat metric is

Given one-forms and , for brevity we will write them as column vectors

and their inner product is given by

Give two-forms and , again we will express them as column vectors

and their inner product is given by

The Hodge star must be such that we must have the following properties satisfied for one-forms

Clearly, since , and , then for a one-form

and for a two-form

General (Non-Euclidean) Space

In this case the metric (inner product between one-forms) is given by

The inner product between two one-forms is then given by

Expanding the matrix expressions

To compute the inner product between two-forms we need to consider terms of the form . The wedge product is an anti-symmetric tensor product, it is given by

The inner product between two forms is then given by

where up to a normalization constant

we can deduce that

The normalization constant will be chosen so that form an orthonormal basis, thefore , and

The inner product between two two-forms then will be given by

where we have used the fact that

Expanding the matrix expression

The hodge-star must be such that

where is

and

The wedge product in 3D between two one-forms is

and between a one-form and a two form is

Now we need to look for a such that (where the subscript in indicates its a k-form).

Clearly one way to satisfy that is

Then the expression for the hodge-star is given by

Now to compute the hodge-star of a two-form, we need to look for such that

Clearly, the following satisfies the above equality

Summary

• zero-form

  

• one-form

  

• two-form

  

• three-form