The Structure of Compact Ricci-flat Riemannian Manifolds

where k is the first Betti number b^M), T is a flat riemannian λ -torus, M~ is a compact connected Ricci-flat (n — λ;)-manifold, and Ψ is a finite group of fixed point free isometries of T x M' of a certain sort (Theorem 4.1). This extends Calabi's result on the structure of compact euclidean space forms ([7] see [20, p. 125]) from flat manifolds to Ricci-flat manifolds. We use it to essentially reduce the problem of the construction of all compact Ricci-flat riemannian ^-manifolds to the construction in dimensions < n and in dimension n to the case of manifolds with bx = 0 (see § 4). We also use it to prove (Corollary 4.3) that any compact connected Ricci-flat manifold M has a finite normal riemannian covering T X N —> M where T is a flat riemannian torus, dim T > bx(M), and N is a compact connected simply connected Ricci-flat riemannian manifold. This extends one of the Bieberbach theorems [4], [20, Theorem 3.3.1] from flat manifolds to Ricci-flat manifolds, and reduces the question of whether compact Ricci-flat manifolds are flat to the simply connected case. J. Cheeger and D. Gromoll have pointed out to us that this extension also follows from their proof of [8, Theorem 6]. Our direct proof however uses considerably less machinery than their deeper considerations of manifolds of nonnegative curvature. As a consequence of these results, we can give a variety of sufficient topological conditions for Ricci-flat riemannian /i-manifolds M to be flat. For example, if the homotopy groups πk(M) = 0 for k > 1, or the universal covering of M is acyclic (Theorem 4.6), or M has a finite topological covering by a