Pfäffle, Frank The Dirac spectrum of Bieberbach manifolds. (English) Zbl 0984.58017 J. Geom. Phys. 35, No. 4, 367-385 (2000). 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. Reviewer: Christian Bär (Hamburg) Cited in 30 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C27 Spin and Spin\({}^c\) geometry 58J28 Eta-invariants, Chern-Simons invariants Keywords:Dirac operator; Bieberbach manifolds; eta invariant; parallel spinors; spectrum PDFBibTeX XMLCite \textit{F. Pfäffle}, J. Geom. Phys. 35, No. 4, 367--385 (2000; Zbl 0984.58017) Full Text: DOI References: [1] C. Bär, Das Spektrum von Dirac-Operatoren, Bonner Math. Schr. 217 (1991).; C. Bär, Das Spektrum von Dirac-Operatoren, Bonner Math. Schr. 217 (1991). [2] Bär, C., The Dirac operator on space forms of positive curvature, J. Math. Soc. Jpn., 48, 69-83 (1996) · Zbl 0848.58046 [3] Bieberbach, L., Über die Bewegungsgruppe der Euklidischen Räume 1, Math. Ann., 70, 297-336 (1911) · JFM 42.0144.02 [4] Bieberbach, L., Über die Bewegungsgruppe der Euklidischen Räume 2, Math. Ann., 72, 400-412 (1912) · JFM 43.0186.01 [5] L.S. Charlap, Bieberbach Groups and Flat Manifolds, Springer, New York, 1986.; L.S. Charlap, Bieberbach Groups and Flat Manifolds, Springer, New York, 1986. · Zbl 0608.53001 [6] Friedrich, T., Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten, Colloq. Math., 44, 277-290 (1981) · Zbl 0487.53016 [7] Friedrich, T., Zur Abhängigkeit des Dirac-Operators von der Spin-Struktur, Colloq. Math., 47, 57-62 (1984) · Zbl 0542.53026 [8] T. Friedrich, Dirac-Operatoren in der Riemannschen Geometrie, Vieweg, Braunschweig, 1997.; T. Friedrich, Dirac-Operatoren in der Riemannschen Geometrie, Vieweg, Braunschweig, 1997. · Zbl 0887.58060 [9] Hantzsche, W.; Wendt, H., Dreidimensionale euklidische Raumformen, Math. Ann., 110, 593-611 (1935) · JFM 60.0517.02 [10] T. Sakai, Riemannian Geometry, American Mathematical Society, Providence, RI, 1996.; T. Sakai, Riemannian Geometry, American Mathematical Society, Providence, RI, 1996. [11] E.T. Whittaker, G.N. Watson, A Course in Modern Analysis, 4th Edition, Cambridge University Press, London, 1927.; E.T. Whittaker, G.N. Watson, A Course in Modern Analysis, 4th Edition, Cambridge University Press, London, 1927. · JFM 53.0180.04 [12] J.A. Wolf, Spaces of Constant Curvature, McGraw-Hill, New York, 1967.; J.A. Wolf, Spaces of Constant Curvature, McGraw-Hill, New York, 1967. · Zbl 0162.53304 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.