Anti-self-dual differential form (${\Omega }_{-}^2$)

On an oriented Riemannian four-manifold, a two-form $\alpha$ is anti-self-dual when $\ast \alpha =-\alpha$.

On an oriented Riemannian four-manifold, a two-form is anti-self-dual when $\ast \alpha =-\alpha$. Here $\ast$ is the Hodge star operator.

If an exact two-form $\beta =d\alpha$ on a compact oriented Riemannian four-manifold is anti-self-dual, thenso $\beta =0$.