RESUMEN

Let (M,g) be a compact, connected Riemannian manifold of dimension n ≥ 3. We denote with Ric_g, S_g and γ the Ricci curvature, the scalar curvature and conformal class of g, respectively. Then using techniques introduced  by A. Lichnerowicz we present a proof of Hijazi’s following result. If Sg ≥ 0 and (M,g) is a spin manifold, there is a constant μ ≥ 0, depending only on n and γ, such that every eigenvalue λ of the Diracoperator acting on spinor fields over M satisfies λ² vol (M,g)^2/n ≥ μ. In an appendix we prove that if S_g is constant and (M,g) is not conformally diffeomorphic to a standard n-sphere, then if γ contains another metric with constant scalar curvature, there is a positive function p on M such that (Ric – (Sg/n)g) (Ñp, Ñp) is negative somewhere on M.