Nilpotent structures and invariant metrics on collapsed manifolds
TL;DR: In this article, a complete Riemannian manifold of bounded curvature, called Mn, was considered, and the complementary set was defined as the e-collapsed part of the manifold.
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Abstract: Let Mn be a complete Riemannian manifold of bounded curvature, say IKI 0, we put Mn = W n(c) U Fn (c), where ?1n (e) consists of those points at which the injectivity radius of the exponential map is > e The complementary set, &"n (c) is called the e-collapsed part of Mn If x E 4 (n), r < E, then the metric ball Bx (r) is quasi-isometric, with small distortion, to the flat ball B0(r) in the Euclidean space, Rn After slightly
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Citations
Chapter 8 Sphere theorems
Katsuhiro Shiohama
- 01 Jan 2000
TL;DR: In this paper, the uniqueness and finiteness of topological types of certain classes of Riemannian manifolds determined by geometric quantities are discussed. And the differentiable pinching problem is discussed.
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Classification of negatively pinched manifolds with amenable fundamental groups
Igor Belegradek,Vitali Kapovitch +1 more
TL;DR: In this paper, a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups is given, namely the M\"obius band, and the products of a line with the total spaces of flat vector bundles over closed infranilmanifolds.
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Complete Calabi-Yau metrics from P^2 # 9 \bar P^2
TL;DR: In this paper, it was shown that the Ricci-flat Kahler metrics in de Rham cohomology classes admit curvature tensors at an exponential rate if the curvature estimate from Cheeger-Tian is improved.
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Collapsed manifolds with pinched positive sectional curvature
TL;DR: In this paper, it was shown that a closed 5-manifold of pinched curvature is homeomorphic to a spherical space form if it satisfies one of the following conditions: δ = 1/4 and the fundamental group is a non-cyclic group of order ≥ C, a constant.
Pinching estimates for negatively curved manifolds with nilpotent fundamental groups
Igor Belegradek,Vitali Kapovitch +1 more
TL;DR: In this paper, it was shown that a ≥ k answering a question of Gromov admits a complete Riemannian metric of sectional curvature within [−a2,−1] whose fundamental group contains a k-step nilpotent subgroup of finite index.
References
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Discrete subgroups of Lie groups
M. S. Raghunathan
- 01 Jan 1972
TL;DR: In this paper, the existence of lattices in semisimple Lie groups has been studied and the density theorem of Borel has been proved for non-potent Lie groups.
2K
Deforming the metric on complete Riemannian manifolds
TL;DR: Soit (M,g ij (x)) une variete de Riemann a n dimensions complete non compacte de tenseur de complexe riemannien {R ijkl } satisfaisant: |R Ijkl | 2 ≤k 0 sur M, ou 0 0 dependant seulement de n, m and k 0 telle que l'equation d'evolution δg Ij (ex,t)|δt=−2R iJ (x,t), δt =−2
Finiteness theorems for riemannian manifolds.
TL;DR: In this article, it was shown that if one puts arbitrary fixed bounds on the size of certain geometrical quantities associated with a riemannian metric, then the set of diffeomorphism classes of compact n-dimensional manifolds admitting a metric for which these bounds are satisfied is finite.
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