Abstract
We now give some examples of how to apply the general classification scheme of Chapter III. By Theorem 5.2 of Chapter II, we can pick any group we want for the holonomy group Φ. It is, of course, trivial to see that the only Bieberbach groups with trivial holonomy groups (i.e. Φ = {1}) are the free abelian groups, so the only compact riemannian manifolds with trivial holonomy group are the flat tori. Notice that we did not have to say that the riemannian manifold was “flat” since by Theorem 3.2 of Chapter II, any manifold with a finite (or even merely totally disconnected) holonomy group must have zero curvature.
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© 1986 Springer-Verlag New York Inc.
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Charlap, L.S. (1986). Holonomy Groups of Prime Order. In: Bieberbach Groups and Flat Manifolds. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8687-2_4
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DOI: https://doi.org/10.1007/978-1-4613-8687-2_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-96395-2
Online ISBN: 978-1-4613-8687-2
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