The geometry of 4-dimensional, Ricci-flat manifolds which admit a metric

This paper discusses the relationships between the metric, the connection and the curvature tensor of 4-dimensional, Ricci-flat manifolds which admit a metric. It is shown that, with the exception of what are effectively very special cases (and which occur only when the signature is indefinite), these metric, connection and curvature concepts are essentially equivalent for such manifolds. The procedures involved include a description of the holonomy structure of 4-dimensional manifolds admitting metrics of any of the three possible signatures and, in particular, those admitting a metric of neutral signature (+,+,−,−), together with the solutions (in the appropriate cases) of the equation ∇h=0 for a second order symmetric non-degenerate tensor h. The interplay between (full) holonomy and “infinitesimal holonomy” through the “curvature map” and the Ambrose-Singer theorem is stressed. Attention is also drawn to the situation when the Riemann tensor admits an annihilating vector k, that is, Rabcdkd=0. Some related comments regarding the Weyl conformal and projective tensors are also made.