I want to find a example of a manifold that has positive scalar curvature but is not half conformally flat.
Does there exists such manifolds?
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Take $S^2\times S^2$ with the standard product metric. It has positive scalar curvature, it can't be conformally flat because it's simply connected and compact and yet not the $4$-sphere, and it can't be conformally half-flat because switching orientation on one of the two factors gives an isometric manifold but switches self-dual and anti-self dual parts of the Weyl curvature (so if either part vanished, the other would as well, implying that the metric is conformally flat, which, as has been established, cannot be the case).