RESUMEN
Let (M,g) be a compact, connected Riemannian
manifold of dimension n ≥ 3. We denote with Ricg, Sg
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 Sg 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.