New Examples of Einstein Metrics in Dimension Four

The holonomy group of a metric g at a point p of a manifold M is the group of all linear transformations in the tangent space of p defined by parallel translation along all possible loops starting at p 1 . It is obvious that a connection can only be the Levi-Civita connection of a metric g if the holonomy group is a subgroup of the generalized orthogonal group corresponding to the signature of g 1–3 . At any point p ∈ M, and in some coordinate system about p, the set of matrices of the form