σk-Scalar curvature and eigenvalues of the Dirac operator

Abstract

On a four-dimensional closed spin manifold (M 4, g), the eigenvalues of the Dirac operator can be estimated from below by the total σ2-scalar curvature of M 4 as follows: \(\lambda^4 \geq \frac{32}{3} \frac{\int_{M^4} \sigma_2 (g) {\rm d} {\rm vol} (g)}{{\rm vol} (M^4, g)}\) Equality implies that (M 4, g) is a round sphere and the corresponding eigenspinors are Killing spinors.