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On the first eigenvalue of the Dirac operator on 6-dimensional manifolds. (English) Zbl 0577.58034

Let \((M^ n,g)\) be a closed Riemannian spin-manifold with positive scalar curvature R and let \(R_ 0\) denote its minimum. If \(\Lambda^{\pm}\) is the first positive or negative eigenvalue of the Dirac operator on M, then \[ \sqrt{(n/(n-1))R_ 0}\leq | \Lambda^{\pm}| \] and if equality holds then M must be an Einstein space. The authors give the first example of \((M^ n,g)\) with n even different from the sphere realizing the lower bound as an eigenvalue.
Reviewer: K.Wojciechowski

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds
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[1] J.E. D’Atri, H.K. Nickerson: Geodesic symmetries in spaces with special curvature tensors. J. Diff. Geom. 9 (1974), 251-262. · Zbl 0285.53019
[2] Th. Friedrich: Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skarlarkrümmung. Math. Nachr. 97(1980), 117-146. · Zbl 0462.53027
[3] Th. Friedrich: A remark on the first eingenvalue of the Dirac operator on 4-dimensional manifolds. Math. Nachr. 102 (1981), 53-56. · Zbl 0481.53039
[4] Th. Friedrich, R. Grunewald: On Einstein metrics on the twistor space of a four-dimensional Riemannian manifold. Math. Nachr. (to appear) · Zbl 0572.53037
[5] D. Husemoller: Fibre bundles New York 1966
[6] A. Ikeda: Formally self adjointness for the Dirac operator on homogenous spaces. Osaka J. of Math. 12(1975), 173-185. · Zbl 0317.58019
[7] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry, vol. II. New York, London, Sydney 1969 · Zbl 0175.48504
[8] J. Milnor: Spin-structures on manifolds. L’Enseignement Mathématique IX (1963), 198-203 · Zbl 0116.40403
[9] S. Sulanke: Der erste Eigenwert des Dirac-Operators auf S ? 5 . Math. Nachr. 99(1980), 259-271 · Zbl 0479.53040
[10] M. Wang, W. Ziller: On Normal Homogenous Einstein Manifolds. Preprint (1984)
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