Geometry of gravitational instantons

We discuss various geometric formulations of the equations of gravitational instantons and their generalization to arbitrary finite dimensional space-times. We show that the Palatini formalism generically leads to exactly the gravitational instanton equations which include as a special case the classical vacuum Einstein equations. We introduce the notion of fm Einstein pair to clarify the distinction between the Palatini and the metric variational equations. Our approach naturally extends to coupled field equations which can be thought of as arising from topological obstruction to the existence of Einstein pairs.