Abstract
On a Riemannian spin manifold, we give a lower bound for the square of the eigenvalues of the Dirac operator by the smallest eigenvalue of the conformal Laplacian (the Yamabe operator). We prove, in the limiting case, that the eigenspinor field is a killing spinor, i.e., parallel with respect to a natural connection. In particular, if the scalar curvature is positive, the eigenspinor field is annihilated by harmonic forms and the metric is Einstein.
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Communicated by A. Jaffe
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Hijazi, O. A conformal lower bound for the smallest eigenvalue of the Dirac operator and killing spinors. Commun.Math. Phys. 104, 151–162 (1986). https://doi.org/10.1007/BF01210797
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DOI: https://doi.org/10.1007/BF01210797