Abstract
We have frequently encountered relations between curvature and topology. Among others we can mention the Gauss—Bonnet—Blaschke theorem 28, the von Mangoldt—Hadamard—Cartan theorem 72, Myers’ theorem 63, and Synge’s theorem 64. The topic of curvature and topology has been for some time the most popular and highly developed topic in Riemannian geometry.
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How close is close enough can be made explicit.
This connection should not be too surprising, considering the KPn.
We admit not being complete in quite a few more.
Of course there are many details which we have ignored.
As we have seen this is mainly because the injectivity radius goes to zero.
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© 2003 Springer-Verlag Berlin Heidelberg
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Berger, M. (2003). From Curvature to Topology. In: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18245-7_12
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DOI: https://doi.org/10.1007/978-3-642-18245-7_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65317-2
Online ISBN: 978-3-642-18245-7
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