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The Dirac spectrum of Bieberbach manifolds. (English) Zbl 0984.58017

Bieberbach manifolds are complete Riemannian manifolds with vanishing curvature.
In the present paper explicit formulas for the eigenvalues of the Dirac operator on three-dimensional compact Bieberbach manifolds are derived for all spin structures. The corresponding \(\eta\)-invariants are computed. From the results it also follows that the flat torus is the only compact three-dimensional spin manifold carrying a non-zero parallel spinor.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C27 Spin and Spin\({}^c\) geometry
58J28 Eta-invariants, Chern-Simons invariants
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References:

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