Scalar Curvature, Killing Vector Fields and Harmonic One-Forms on Compact Riemannian Manifolds

It is well known that no non-trivial Killing vector field existson a compact Riemannian manifold of negative Ricci curvature;analogously, no non-trivial harmonic one-form exists on a compactmanifold of positive Ricci curvature. One can consider the following,more general, problem. By reducing the assumption on the Riccicurvature to one on the scalar curvature, such vanishing theoremscannot hold in general. This raises the question: "What informationcan we obtain from the existence of non-trivial Killing vectorfields (or, respectively, harmonic one-forms)?" This paper givesanswers to this problem; the results obtained are optimal. 2000Mathematics Subject Classification 53C20 (primary), 53C24 (secondary).