Yamaguchi, Takao Manifolds of almost nonnegative Ricci curvature. (English) Zbl 0658.53036 J. Differ. Geom. 28, No. 1, 157-167 (1988). This paper deals with the problem of studying relations between Ricci curvature and topology of a Riemannian manifold. Let \({\mathcal M}(n,D)\) be a family of compact connected \(C^{\infty}\) Riemannian n-dimensional manifolds M with sectional curvature K(M)\(\leq 1\) and diam(M)\(\leq D\). The author gives an affirmative answer, for \(M\in {\mathcal M}(n,D)\), to a generalized conjecture of Gromov: There is an \(\epsilon >0\) depending on n and D such that if M satisfies diam(M)\(\leq D\) and \(Ric(M)>-\epsilon\), then M is a fiber bundle over a \(b_ 1(M)\)-torus [see M. Gromov, Structures métriques pour les variétés riemanniennes (Paris, 1980; Zbl 0509.53034)]. He also describes the global geometric structure of \({\mathcal M}(n,D)\) manifolds in the situation that the greatest lower bound of Ricci curvature tends to zero and the first Betti number is large enough. Reviewer: E.D.Rodionov Cited in 1 ReviewCited in 5 Documents MSC: 53C20 Global Riemannian geometry, including pinching Keywords:Ricci curvature; sectional curvature; conjecture of Gromov; Betti number Citations:Zbl 0509.53034 PDFBibTeX XMLCite \textit{T. Yamaguchi}, J. Differ. Geom. 28, No. 1, 157--167 (1988; Zbl 0658.53036) Full Text: DOI