Manifolds of almost nonnegative Ricci curvature
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 \textit{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.
scientific article; zbMATH DE number 4075986