Let (M,g) be a spin manifold with Ricci curvature Ric viewed as an endomorphism of the tangent bundle. The Dirac-Ricci operator DR acting on spinors may be defined as:
DR(ψ)=∑iRic(ei).∇ei(ψ)
with ψ a spinor and (ei) is an orthonormal basis of the tangent bundle.
The Dirac-Lichnerowicz-Ricci formula is:
DR2=−ΔR+α
with:
ΔR=∑i,jg(Ric(ei),Ric(ej))[∇ei∇ej−∇∇eiej]
α is a lower term; it is a scalar if ∇Ric=0. If M is an Einstein manifold, then the Dirac-Ricci operator is reduced to the Dirac operator. Can we have bounds over the first proper value of the Dirac-Ricci operator?