Electromagnetic resolution of curvature and gravitational instantons

We study the electromagnetic resolution of the Riemann curvature on a space -time manifold nl with metric g into its electric and magnetic parts relati ve to a unit timelike vector with respect to g. There exists a duality tran sformation between the active and passive electric parts which leaves invar iant a subclass of field equations which correspond to the generalized grav itational instanton equations. We also discuss various geometric formulatio ns of the equations of gravitational instantons and their generalization wh ich includes as a special case the classical vacuum Einstein equations. Sam e special solutions and their physical significance are also considered.