self-dual manifolds exist only for dim(M)=4, M riemannian, and M with no torsion. The dual is .

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 psfile jl@crush.caltech.edu index