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Zbl 1078.53518
Aubry, Erwann
The sphere theorem. (Théorème de la sphère.) (French)
[A] Seminar on spectral theory and geometry. 1999--2000. St. Martin d'Hères: Université de Grenoble I, Institut Fourier. Sémin. Théor. Spectr. Géom. 18, 125-156 (2000).

The isometries of compact Riemannian manifolds with Ricci curvature greater than $n-1$, with the sphere in $(\Bbb R^n, can)$ with the canonical metric depend on the invariants diameter, volume, radius, and spectrum. (For inequalities see the theorems of Myers, Bischop, Lichnerowicz, and for equalities theorems of Obata and Cheng). {\it J. Cheeger} and {\it T. H. Colding} [J. Differ. Geom. 46, No. 3, 406--480 (1997; Zbl 0902.53034)] proved: There exists a $\delta=\delta(n)>0$ so that all compact Riemannian manifolds $(M^n, g)$ with Ricci curvature greater than $n-1$, satisfying $\text{Vol}(M^n, g)\geq (1-\delta(n) )\text{Vol}(S^n)$ are diffeomorphic to $(S^n)$. This article is an excellent presentation of the proof of Cheeger and Colding in a form allowing the extension of the result to the remaining invariants. It ends with an appendix about several properties of a manifold with positive Ricci curvature.
[S. Noaghi (Deva)]

MSC 2000:: *53C21 Methods of Riemannian geometry (global)
53C20 Riemannian manifolds (global)
53C23 Global topological methods (a la Gromov)

Keywords: sphere theorem; compact Riemannian manifold

Citations: Zbl 0902.53034

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