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Displaying similar documents to “Report on M. Gromov's almost flat manifolds”

Kikkawa loops and homogeneous loops

Michihiko Kikkawa (2004)

Commentationes Mathematicae Universitatis Carolinae

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In H. Kiechle's publication ``Theory of K-loops'' [3], the name Kikkawa loops is given to symmetric loops introduced by the author in 1973. This concept started from an analogical imagination of sum of vectors in Euclidean space brought up on a sphere. In 1975, this concept was extended by him to the more general concept of homogeneous loops, and it led us to a non-associative generalization of the theory of Lie groups. In this article, the backstage of finding these concepts will be...

On the finiteness of the fundamental group of a compact shrinking Ricci soliton

Zhenlei Zhang (2007)

Colloquium Mathematicae

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Myers's classical theorem says that a compact Riemannian manifold with positive Ricci curvature has finite fundamental group. Using Ambrose's compactness criterion or J. Lott's results, M. Fernández-López and E. García-Río showed that the finiteness of the fundamental group remains valid for a compact shrinking Ricci soliton. We give a self-contained proof of this fact by estimating the lengths of shortest geodesic loops in each homotopy class.