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MiS Preprint
50/2006
$\sigma_k$-scalar curvature and eigenvalues of the Dirac operator
Guofang Wang
Abstract
On a $4$-dimensional closed spin manifold $(M^4,g)$, the eigenvalues of the Dirac operator can be estimated from below by the total $\sigma_2$-scalar curvature of $M^4$ as follows \[\lambda^4\ge \frac {32}3\frac{\int_{M^4} \sigma_2(g) dvol(g)}{vol(M^4,g)}.\] Equality implies that $(M^4,g)$ is a round sphere and the corresponding eigenspinors are Killing spinors.
