Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
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

Highlights
Master Server

### Zentralblatt MATH Berlin [Germany]

© FIZ Karlsruhe GmbH

Zentralblatt MATH master server is maintained by the Editorial Office in Berlin, Section Mathematics and Computer Science of FIZ Karlsruhe and is updated daily.

Other Mirror Sites

Copyright © 2013 Zentralblatt MATH | European Mathematical Society | FIZ Karlsruhe | Heidelberg Academy of Sciences