The sphere theorem (Q2709147)

The isometries of compact Riemannian manifolds with Ricci curvature greater than \(n-1\), with the sphere in \((\mathbb R^n, can)\) 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). \textit{J. Cheeger} and \textit{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 \((S^n)\). NEWLINENEWLINEThis 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.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00015].