PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 91, Number 2. June 1984 NATURALLY REDUCTIVE METRICS OF NONPOSITIVE RICCI CURVATURE CAROLYN GORDON AND WOLFGANG ZILLER1 Abstract. The main theorem states that every naturally reductive homogeneous Riemannian manifold of nonpositive Ricci curvature is symmetric. As a corollary, every noncompact naturally reductive Einstein manifold is symmetric. A homogeneous space G/77 is called naturally reductive if there exists a decom- position g = t) A p with ad(b)p c p and ([X,Y],Z) + (Y, [X, Z]> = 0 for all X, Y, Ze p. The goal of this paper is to prove the following: Theorem. Every naturally reductive Riemannian manifold of nonpositive Ricci curvature is symmetric. This strengthens a result of E. Deloff [D] asserting that every naturally reductive homogeneous manifold of nonpositive sectional curvature is symmetric. It also has the following consequence. A metric is called Einstein if there exists a constant E such that RiciX, Y) = E(X, Y); E is called the Einstein constant. Corollary. Every naturally reductive homogeneous Einstein manifold with nonposi- tive Einstein constant is symmetric. In particular, every noncompact naturally reductive Einstein manifold is symmet- ric. This is in sharp contrast to the compact case, where naturally reductive metrics provide a rich source of Einstein metrics [DZ, WZ]. To establish some preliminaries, let M be a connected homogeneous Riemannian manifold, C7 a transitive group of isometries of M and 77 the isotropy subgroup of G at a point p e M. For convenience, we assume G acts effectively on M, i.e., only e acts as the identity transformation on M. Denote by g and fj the Lie algebras of G and 77, and by f an Ad(77)-invariant complement of f) in g. M is naturally identified with the tangent space TpM. Under this identification the Riemannian structure defines an Ad(77)-invariant inner product ( , ) on p. The metric is called naturally reductive (with respect to G and p) if ([X, Y]v, Z] A (Y,[X, Z]„) = 0 for all X,Y, Z e p, where [A; F]p is the p component of [A", Y\. Let , . g = p A [p, p], 5 = Í) n g and G = the subgroup of G with Lie algebra g. Received by the editors May 9, 1983. 1980 Mathematics Subject Classification. Primary 53C30. 'The second author was partially supported by a grant from the National Science Foundation. ©1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page 287 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use