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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
