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?
Best Answer
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.