atlas[`&**`] - Hodge operator
expr - any expression containing p-forms
- Let be vector bundle of p-forms on manifold M of dimension and metric .
- For any integer let us define Hodge operator * as such unique isomorphism of vector bundles * : ---> which has the following property.
- For any which belong to we have where is volume form on M induced by metric .
- Let be the number of in the signature of metric (in the ATLAS package the integer is represented by variable) then the following equations take place:
- - on vector bundle .
The &** procedure allows one to calculate Hodge operator on an expression containing p-forms. In standard mathematical notation &** is * - just Hodge asterisk. If a metric is presented then the Hodge operator is defined completely by the following.
Declare metric of (see atlas[Metric]):
Volume form :
Hodge : p-form -> (n-p)-form
Double Hodge operator: