Chapter 4 Einstein metrics in dimension four

Abstract

This chapter presents selected results concerning Einstein metrics on four-manifolds and focuses on dimension four, which is the main area of interest for Einstein metrics, both in differential geometry and mathematical physics. Two main areas are addressed concerning four-dimensional Einstein metrics: (1) the global properties of compact Einstein four-manifolds, which are Riemannian in the sense that their metrics are positive definite, and (2) the local properties of Einstein metrics on four-dimensional manifolds in Riemannian as well as Lorentzian and neutral cases; the italicized terms refer to the sign patterns – + + + and – – + +, of which the former characterizes spacetime metrics in general relativity. The Einstein condition discussed in the chapter is a fairly complicated system of nonlinear second-order partial differential equations imposed on the local component functions of the metric g . A simple method of constructing Einstein four-manifolds is through the conformal changes of products of surface metrics.