The curvature 2 form then follows:

The curvature 2 form is defined as .

If the curvature is self dual, then the Ricci is 0 = ``Ricci flat''. Proof: use bianchi identity and self-duality. turn levi-civita's into delta's.

If spin connections are self-dual than curvature 2-form is self dual.

Given a self dual R, w is self-dual up to a local rotational transformation SO(4).

source

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