Spin manifolds, killing spinors and universality of the Hijazi inequality

Abstract

In terms of the Dirac operator P, we introduce on any field a first-order operator D and show that the operator (Δ−ρ) on the spinors (ρ=(n/4(n−1))R; dim W=n) is positive. By means of a universal formula, we show that, on a compact spin manifold of dimension ≥3, the Hijazi inequality [8] holds for every spinor field such that (Pψ, Pψ) = λ ²(ψ, ψ) (λ=const.). In the limiting case, the manifold admits a Killing spinor which can be evaluated in terms of ψ. Different properties of spin manifolds admitting Killing spinors are proved. D is nothing but the ‘twistor operator’.