Decomposition of 2-Form on 4-Dimensional Riemannian Manifold

Theorem

Let $M$ be a $4$-dimensional oriented Riemannian manifold.

Let $\omega$ be a $2$-form on $M$.

Then $\omega$ can be written uniquely as a sum of a self-dual $2$-form and an anti-self-dual $2$-form.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems