J. Cheeger; K. Fukaya; M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. A.M.S. 5 (1992), 327--372  Home/Search   Document Not in Database   Summary   Related Articles   Check

....point under a lower curvature and upper diameter bound. In particular, in collapsing under lower curvature bounds manifolds of almost nonnegative curvature play the same basic role as almost flat manifolds do in the Cheeger Fukaya Gromov theory of collapse with bounded curvature (cf. Yam] FY] [CFG]) Let us briefly outline the construction of the metrics needed to prove Theorem A. If a connected closed smooth manifold M admits a cohomogeneity one action, then the orbit space is either a circle or a compact interval (cf. Mo] In the first case, M is the total space of a homogeneous bundle ....

J. Cheeger, K. Fukaya and M. Gromov, Nilpotent Structures and Invariant Metrics on Collapsed Manifolds, J. Amer. Math. Soc. 5,2 (1992) 327--372

....M t contains a stable subsequence. C. The proof of the Gluing Theorem is divided into two parts. Recall that a metric g with bounded sectional curvature whose injectivity radii are small everywhere gives rise to a certain topological structure on M , a so called N structure of positive rank (cf. [CFG] and section 1) In the case where M is simply connected and of bounded diameter, such a structure is actually given by an almost isometric smooth effective global torus action on M with empty fixed point set, whose orbits, roughly speaking, contain the directions in which the injectivity radii of ....

....Collapse Theorem and the Limit of Covering Geometry Theorem are given in section 4. We would like to thank Slava Matveyev for help with section 3. 1. Preliminaries We gather here several notions and results about collapsed manifolds and Alexandrov spaces. As general references we mention [BGP] [CFG], Fu] GLP] Ro] and [Ro2] A. Collapsed manifolds and N structures. Let M = M m ; g) be a Riemannian manifold of dimension m and let FM = F (M m ) denote its bundle of orthonormal frames. When fixing a bi invariant metric on O(m) the Levi Civita connection of g gives rise to a ....

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J. Cheeger; K. Fukaya; M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. A.M.S 5 (1992), 327-372.

....assume that dim C i m. We can view C i as a cone over its space of directions, C i = C ( Sigma i ) where Sigma i is an Alexandrov space of curvature 1 or dim Sigma i = 1. Sigma i can be viewed as a unit sphere in C i . Consider a neighbourhood U oe Sigma i ae C i . From the results of [CFG] (see section 1 of [PRT] where also further references can be found) we have an N structure : E ffl U , where E ffl is a subset of (M; fflg) containing the set S 1;ffl = fx 2 (M; fflg) jpxj = 1g. Since E ffl is homotopically equivalent to N i , it follows that E ffl is simply connected. ....

....= 1) and consider a spherical neighbourhood of U x x. Consider the preimage V ffl = Gamma1 (U x ) ae E ffl and let e V ffl be its universal Riemannian covering. Then the T k action induces an almost isometric R k Theta F action on e V ffl , where F is a finite Abelian group. From [CFG] one has a uniform bound for the injectivity radius of e V ffl , so that, as ffl 0, e V ffl converges to a flat manifold e V 0 with boundary and isometric R k Theta F action (for the convergence claim see the first part of Lemma 2.1.4 in [PRT] Since the interior of e V 0 is flat, ....

J. Cheeger; K. Fukaya; M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. A.M.S 5 (1992), 327-372.

....Jurgen Jost and Wolfgang Ziller. They all know what for. 1. Preliminaries In this section, we introduce some notation and review relevant results. For basic notions and results about collapsed manifolds, equivariant) Hausdorff convergence, and Alexandrov spaces the reader is referred to [BGP] [CFG], Fu] and [GLP] Let M = M m ; g) be a Riemannian manifold of dimension m and let FM = F (M m ) denote its bundle of orthonormal frames. When fixing a bi invariant metric on O(m) the Levi Civita connection of g gives rise to a canonical metric on FM , so that the projection FM M becomes ....

....is said to have positive rank if its fibres are nontrivial, i.e. if they have positive dimension. A pure N structure j : FM B over a Riemannian manifold (M; g) gives rise to a sheaf on FM whose local sections restrict to local right invariant vector fields on the fibres of j; see [CFG]. If the local sections of this sheaf are local Killing fields for the metric g, then g is said to be invariant for the N structure (and j is then also sometimes referred to as pure nilpotent Killing structure for g) Theorem 1.1 ( CFG] Ro1] Let for m 2 and D 0 M(m;D) denote the class of ....

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J. Cheeger; K. Fukaya; M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. A.M.S 5 (1992), 327-372.

....0 . Remark 9. In some cases, it could be sufficient to have a local S 1 fibration (e.g. if we have local Lie groups rather than proper ones) the analysis of obstructions to a local S 1 fibration of a manifold is considerably more complicate than for global S 1 fibrations; see Gromov et al. [18]. Remark 10. Considerations not so different from the above do also apply for periodic or quasiperiodic solutions to the dynamics in orbit space. The number of conditional constants of motion would now be (k Gamma q) for q periodic solutions in Omega Gamma and the coefficients for the dynamics ....

J. Cheeger, K. Fukaya and M. Gromov, "Nilpotent structures and invariant metrics on collapsed manifolds"; J. Am. Math. Soc. 5 (1992), 327

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J. Cheeger; K. Fukaya; M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. A.M.S. 5 (1992), 327--372

....V 1 (C) # (C) is trivial. Therefore, the T k(C) bration over W 1 (C) # (C) which is a small deformation of the restriction of the T k(C) bration of V 1 (C) # (C) over W 1 (C) # (C) is also trivial. This completes the proof of Theorem 0.4. 36 6. An example In [CFG] and [CG2] an F structure is constructed on any n manifold with jKj 1 and inj x (n) for all x. Even if the curvature of the compact manifold, M n is nonpositive, this structure need not be injective; see Example 6.1. Thus the existence of an injective substructure (let al..one a ....

....with jKj 1 and inj x (n) for all x. Even if the curvature of the compact manifold, M n is nonpositive, this structure need not be injective; see Example 6.1. Thus the existence of an injective substructure (let al..one a Cr structure) does not follow from the general considerations of [CFG] and [CG2] which rely only on the boundedness of the curvature and are local in nature. Those of the present paper depend on the nonpositivity of the curvature and the compactness of the underlying manifold. Example 6.1. Take a noncompact complete hyperbolic 3 manifold, N 3 , of nite volume ....

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J. Cheeger; K. Fukaya; M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. A.M.S., 5 (1992), 327-372.

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Cheeger, J.,Fukaya, K., & Gromov, M. (1992). Nilpotent structures and invariant metrics on collapsed manifolds. J. of the American math. society, 5, 327-372.

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