# What's the term for elements in (image of codifferential/adjoint exterior differential aka 3rd part of hodge decomposition theorem )?

Part of the Hodge Decomposition Theorem is that smooth $$k$$-forms on compact oriented Riemannian (smooth) $$m-$$manifold $$(M,g)$$ (I think $$M$$ need not be connected, but assume connected if need be or you want) are decomposed as

$$\Omega^k(M) = B^kM \bigoplus \mathscr H^k(M,g) \bigoplus image(\delta_{k+1}),$$ where

$$\Omega^k(M):=$$ smooth $$k$$-forms on $$M$$ (I think $$g$$ is not needed here.)

$$\mathscr H^k(M,g):=ker(\Delta_k):=$$ harmonic $$k$$-forms on $$(M,g)$$, where $$\Delta_k$$ is Laplace operator (or Hodge-Laplace operator or Laplacian or Hodge-Laplacian or whatever)

$$B^kM:=image(d_{k-1}):=$$ exact $$k$$-forms on $$M$$, aka image of exterior derivative/differential $$d_{k-1}: \Omega^{k-1}M \to \Omega^{k}M$$ (I think $$g$$ is not needed here.)

$$image(\delta_{k+1}):=$$ image of codifferential/coderivative $$\delta_{k+1}: \Omega^{k+1}M \to \Omega^{k}M$$ of exterior derivative/differential $$d_{k-1}: \Omega^{k-1}M \to \Omega^{k}M$$ (I guess $$\delta_{k+1}$$ is like 'exterior codifferential/coderivative' or 'co-exterior differential/derivative' or whatever; I believe $$g$$ is needed here.)

Question 1: What is/are the term/s for elements in $$image(\delta_{k+1})$$?

Guess: After typing up all this I weirdly came up with the simple answer,

'$$\delta$$-exact $$k$$-forms',

in that elements of $$B^kM$$ are actually 'exact' in the sense of being '$$d$$-exact'. It's like how for forms on Riemann surfaces we have $$d$$-closed vs $$\partial$$-closed vs $$\overline \partial$$-closed.

• Question 2: Any other term besides $$\delta$$-exact?

I think that co-closed (for lying in the kernel of $$\delta$$) and co-exact (for lying in the image of $$\delta$$) is quite common terminology.