These identities have many implications. For instance, the two operators

(14)

and

(15)

(called Laplacians because they are elliptic operators) satisfy . At this point, assume that is also a compact
manifold. Along with Hodge's theorem, this
equality of Laplacians proves the Hodge decomposition.
The operators
and
commute with these Laplacians. By Hodge's theorem,
they act on cohomology, which is represented by harmonic
forms. Moreover, defining