- expr - any expression containing p-forms
- If a metric is presented then the Hodge operator is defined completely by the following:
- - Let pM be vector bundle of p-forms on manifold M of dimension n=dim(M) and metric g.
- - For any integer 0 < p ≤ n let us define Hodge operator * as such unique isomorphism of vector bundles * : pM ---> n-pM which has the following property.
- - For any , which belong to p we have (*())=g(, )g where g is volume form on M induced by metric g.
- Let s be the number of -1 in the signature of metric g (in the ATLAS package the integer is represented by sgn variable) then the following equations take place:
- - *1=g and *g=1
- - (*)2=-1p(n-p)+s on vector bundle pM.
Basic Examples (1)