= Exact anti-self-dual form on a compact four-manifold
Let $M$ be a compact oriented <Riemannian manifold> of dimension four, without boundary. An exact <two-form> $\beta=d\alpha$ that is an <anti-self-dual differential form> must vanish. Indeed the <Generalized Stokes theorem> and the <Hodge star operator> identity $*\beta=-\beta$ give
$$
0=\int_Md(\alpha\wedge d\alpha)=\int_M\beta\wedge\beta=-\|\beta\|_{L^2}^2.
$$
The boundaryless hypothesis is essential. On a Euclidean four-ball, the nonzero constant <two-form> $dx^1\wedge dx^2-dx^3\wedge dx^4$ is both an <exact differential form> and an <anti-self-dual differential form>; the omitted boundary integral need not vanish there.
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