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Zbl 0758.53022
Cheeger, Jeff; Fukaya, Kenji; Gromov, Mikhael
Nilpotent structures and invariant metrics on collapsed manifolds. (English)
[J] J. Am. Math. Soc. 5, No.2, 327-372 (1992). ISSN 0894-0347; ISSN 1088-6834/e

Let $M$ be a complete Riemannian $n$-manifold of bounded curvature. For any $\varepsilon > 0$, the $\varepsilon$-collapsed part of $M$ is defined as the set ${\cal C}\sp n(\varepsilon)$ of points at which the injectivity radius of the exponential map is $<\varepsilon$. The authors study the structure of the $\varepsilon$-collapsed part of a manifold $M$ for suitably small $\varepsilon$. The main results show that the local geometry of ${\cal C}\sp n(\varepsilon)$ is encoded partially in the symmetry properties of a nearby metric. More precisely, a given metric can be closely approximated by one that admits a sheaf of nilpotent Lie algebras of local Killing vector fields pointing in all sufficiently collapsed directions of ${\cal C}\sp n(\varepsilon)$. This sheaf is called the nilpotent Killing structure. A detailed construction of this nilpotent Killing structure is presented, its properties are established and some applications to the description of collapses of a Riemannian manifold $M$ are indicated.
[D.V.Alekseevsky (Moskva)]

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

Keywords: injectivity radius; nilpotent Killing structure

Cited in: Zbl 1155.53023 Zbl 1200.53057 Zbl 1062.53031 Zbl 1023.53022 Zbl 0982.53030 Zbl 0912.57012 Zbl 0887.53049 Zbl 0830.53037

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