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Zbl 0727.53043
Cheeger, Jeff; Gromov, Mikhael
Collapsing Riemannian manifolds while keeping their curvature bounded. II. (English)
[J] J. Differ. Geom. 32, No.1, 269-298 (1990). ISSN 0022-040X

[Part I, cf. ibid. 23, 309-346 (1986; Zbl 0606.53028).] \par In part I of this paper, the authors introduced the concept of an F- structure, which generalizes the notion of torus actions. They showed that if a compact manifold admits an F-structure, then it also admits Riemannian metrics with bounded curvature and arbitrarily small injectivity radius. \par In the present paper the converse is proved. More generally, any complete Riemannian manifold may be decomposed into two sets: (1) The set of points with ``small'' injectivity radius, and (2) the set of points with not so small injectivity radius. Here ``small'' is measured in terms of pointwise curvature bounds. The set (2) has controlled geometry and topology and is not discussed further. The set (1) on the other hand admits an F-structure. \par Other remarkable such ``collapsing'' results have been obtained independently by {\it K. Fukaya} in e.g. Differ. Geom. 25, 139-156 (1987; Zbl 0606.53027); J. Math. Soc. Japan 41, 333-356 (1989; Zbl 0703.53042)]. Recent joint efforts of all three authors have resulted in a more complete picture in which ``all collapsing directions'' are taking into account. In this work the (flat) F-structures are replaced with a notion of (nilpotent) N-structures.
[K.Grove]

MSC 2000:: *53C20 Riemannian manifolds (global)

Keywords: F-structure; bounded curvature; injectivity radius; collapsing

Citations: Zbl 0606.53028; Zbl 0606.53027; Zbl 0703.53042

Cited in: Zbl 1239.53052 Zbl 1098.57010 Zbl 1086.53050 Zbl 1051.53034 Zbl 0930.53040 Zbl 0830.53037 Zbl 0735.53033

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