The Hodge star on an 
-dimensional
Riemannian manifold 
with a chosen manifold
orientation is the linear map 
determined by
 |
where 
and 
are differential k-forms,
is their pointwise
inner product, and 
is the volume form determined by the Riemannian
metric 
and the chosen manifold orientation. Thus,
changing either changes the Hodge star. The result 
is called the Hodge dual
of 
. This name is also sometimes used
for the operator itself.
For a positively oriented orthonormal coframe, the Hodge star maps the wedge product of any 
basis covectors to the signed
wedge product of the complementary basis covectors. Applying the Hodge star twice to a differential
k-form on a Riemannian manifold
gives
 |
