Hodge's decomposition theorem saids that on a compact orientable Riemannian manifold any $k$-form can be decomposed uniquely as the sum of an exact, a co-exact and a harmonic form:

$\Omega^k(M)=d\Omega^{k-1}(M) \oplus \delta\Omega^{k+1}(M) \oplus \mathcal{H}^k(M)$

Is there a similar decomposition if the metric is indefinite? What is the closest such result we know? Are there any references on this?

**Added:**

I know that if the metric is indefinite Hodge's theorem doesn't hold. This is why my question is whether there is something similar (not identical) for the indefinite case.