On Four-Dimensional Einstein Manifolds

Abstract In dimensions ≥4, equation (1.1) no longer determines the geometry in a local manner, and this vastly increases both the interest and the difficulty of understanding Einstein manifolds in higher dimensions. In dimensions ≥5, our knowledge of general Einstein manifolds is quite scant, although an impressive array of methods has been developed for producing and classifying Einstein manifolds with special holonomy or isometry groups, Aubin (1976), Besse (1987), Boyer et al. (1994), Joyce (1996), Yau (1977).