Let 
be a compact oriented Riemannian manifold
without boundary, and let 
be the vector space
of smooth differential k-forms on 
. If 
is the formal adjoint of the
exterior derivative 
with respect to the 
inner product, the Hodge
Laplacian is
 |
(1)
|
A form 
is harmonic if 
,
and the space of harmonic 
-forms is denoted 
.
The Hodge decomposition theorem gives the orthogonal direct sum
 |
(2)
|
Equivalently, every differential 
-form 
can be written
 |
(3)
|
where the three terms 
, 
, and 
are the unique exact, coexact, and harmonic summands, respectively.
In particular, a closed form has no coexact part,
so it can be written uniquely as an exact form plus a harmonic form. It follows that
every de Rham cohomology class has a unique
harmonic representative and
 |
(4)
|
where 
is the 
th
Betti number.
On a compact Kähler manifold, complex differential forms additionally split into types 
. Since the Hodge Laplacian preserves type, the harmonic
representatives give the cohomological Hodge decomposition
 |
(5)
|
where 
is represented by harmonic forms of type 
and complex conjugation
interchanges 
and 
.
The finite-dimensional analogue for cochains on a simplicial complex is the discrete Hodge decomposition used in discrete exterior calculus.
