The almost cyclicity of the fundamental groups of positively curved manifolds
In this paper, the author studies the fundamental groups of positively curved manifolds by using the notion of pure \(F\)-structure, which is a generalization of torus action. An invariant pure \(F\)-structure on a Riemannian manifold \(M\) consists of a flat \(T^k\)-bundle with non-trivial holonomy acting on \(M\) such that the induced vector fields are local Killing vector fields. We have a notion of orbit, and the rank of the action is then defined as the dimension of an orbit of the smallest dimension. The author proves the following result. Main Theorem. Let \(M\) be a compact positively curved \(n\)-manifold. Suppose, in addition, that \(M\) admits an invariant positive rank pure \(F\)-structure. Then the fundamental group of \(M\) has a cyclic subgroup with index less than a constant depending only on \(n\). As a consequence, he obtains the main result of the paper. Theorem 0.2. There exists a small constant, \(v(n,\delta)>0\), depending on \(n\), and \(0<\delta \leq 1\) such that, if a \(\delta\)-pinched \(n\)-manifold has volume smaller than \(v(n,\delta)\), then the fundamental group has a finite cyclic subgroup with index less than \(w_n\), a constant depending only on \(n\). In a previous paper the author has proved the existence of a finite cyclic subgroup with index less than a constant depending on \(n\) and \(\delta\); thus, this result is substantially improved in the paper under review.
scientific article; zbMATH DE number 931983