Codiffferential

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• Codifferential

$$\delta = (-1)^{nk + n + 1}s\, {\star d\star} = (-1)^k\,{\star^{-1}d\star},$$ where d is the exterior derivative or differential, and s=+1 for Riemannian manifolds, $n$ is the dimension of the manifold.

Here we have $$d:\Omega^k(M)\rightarrow \Omega^{k+1}(M),$$ $$\delta:\Omega^k(M)\rightarrow \Omega^{k-1}(M).$$