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\begin{document}

self-dual manifolds exist only for dim(M)=4, M riemannian, and M with
no torsion. The dual is \( \widetilde{R}_{abgd}\equiv \frac{1}{2}\epsilon _{ab}^{a'b'}R_{a'b'cd} \).

The curvature 2 form then follows: \( \widetilde{R}_{ab}=\frac{1}{2}\epsilon _{ab}^{a'b'}R_{a'b'} \)

The curvature 2 form is defined as \( R_{ab}=\frac{1}{2}R_{abgd}\theta ^{g}\wedge \theta ^{d} \).

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).

\end{document}