# 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  