Hodge Dual - Hodge Star On Manifolds

Hodge Star On Manifolds

One can repeat the construction above for each cotangent space of an n-dimensional oriented Riemannian or pseudo-Riemannian manifold, and get the Hodge dual (n−k)-form, of a k-form. The Hodge star then induces an L2-norm inner product on the differential forms on the manifold. One writes

for the inner product of sections and of . (The set of sections is frequently denoted as . Elements of are called exterior k-forms).

More generally, in the non-oriented case, one can define the hodge star of a k-form as a (n−k)-pseudo differential form; that is, a differential forms with values in the canonical line bundle.