The Hodge Laplacian on differential k-forms on an oriented Riemannian manifold is the second-order differential operator
 |
where 
is the exterior derivative and 
is its formal adjoint
with respect to the 
inner product. A differential
k-form is a harmonic form precisely when
it lies in the kernel of 
. The Hodge Laplacian commutes with 
, 
,
and the Hodge star, and on functions it agrees with
the Laplace-Beltrami operator up to
the chosen sign convention.
