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A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. (English) Zbl 0593.58040

Let M be a compact Riemannian spin manifold, \(n:=\dim M\geq 3\) and L the Yamabe operator with first eigenvalue \(\mu_ 1\). Then any eigenvalue \(\lambda\) of the Dirac operator satisfies \(\lambda^ 2\geq (n/4(n- 1))\mu_ 1.\) Discussion of equality: If \(\mu_ 1=0\), M is conformally equivalent to a Ricci-flat manifold; if \(\mu_ 1>0\), M is Einstein.
Reviewer: U.Simon

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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