Gravitational instantons admit hyper-Kahler structure

We construct the explicit form of three almost-complex structures that a Ri emannian manifold with self-dual curvature admits and show that their Nijen huis tensors vanish so that they are integrable. This proves that gravitati onal instantons with self-dual curvature admit hyper-Kahler structure. In o rder to arrive at the three vector-valued 1-forms defining almost-complex s tructure, we give a spinor description of real four-dimensional Riemannian manifolds with Euclidean signature in terms of two independent sets of two- component spinors. This is a version of the original Newman-Penrose formali sm that is appropriate to the discussion of the mathematical, as well as ph ysical properties of gravitational instantons. We shall build on the work o f Goldblatt who first developed an NP formalism for gravitational instanton s but we shall adopt it to differential forms in the NP basis to make the f ormalism much more compact. We shall show that the spin coefficients, conne ction 1-form, curvature 2-form, Ricci and Bianchi identities, as well as th e Maxwell equations naturally split up into their self-dual and anti-self-d ual parts corresponding to the two independent spin frames. We shall give t he complex dyad as well as the spinor formulation of the almost-complex str uctures and show that they reappear under the guise of a triad basis for th e Petrov classification of gravitational instantons. Completing the work of Salamon on hyper-Kahler structure, we show that the vanishing of the Nijen huis tensor for all three almost-complex structures depends on the choice o f a self-dual gauge for the connection which is guaranteed by virtue of the fact that the curvature 2-form is self-dual for gravitational instantons.