unique decomposition 
where 
is harmonic. 
. This can also be
written using the laplacian as 
.
, M Riemannian
Theorem:
unique decomposition where is harmonic. . This can also be
written using the laplacian as .
. This can be inverted to get 
as long as there are no 0 mode oscillations.
all 3 terms are 0. This can be shown by taking the inner product
of 
with itself.
Can define a projection operator 
. Then use projection operator to
solve for 
, then show that you can invert to get 
.
closed there is then there is an equivalence 
.
(isomorphism)