Positive pinching, volume and second Betti number
We consider a compact Riemannian manifold of dimension \(n\) with positive sectional curvature \(\sigma>0\). It is known that the topology of \(M\) can heavily depend on whether \(n\) is even or odd. A Riemannian manifold is called \(\delta\)-pinched if its sectional curvature \(\sigma\) satisfies the condition \(\delta\leq\sigma\leq 1\). The aim of the present paper is to study the relation between \(\delta\)-pinching, volume and second Betti number. The main results of this paper can be stated as follows: (i) For odd \(n\geq 3\) and \(0<\delta\leq 1\), there exists a small constant \(\varepsilon(n,\delta)> 0\) such that, if a simply connected \(n\)-manifold \(M\) satisfies \(\delta\leq\sigma\leq 1\) and \(\text{vol}(M)< \varepsilon(n,\delta)\), then the second Betti number, \(b_2(M)\neq 0\). (ii) Let \(M\) be a simply connected odd-dimensional \(n\)-manifold with \(b_2(M)= 0\). If \(M\) admits a \(\delta\)-pinched metric, then \(\text{inj}(M)\geq i(n,\delta)> 0\). In particular, \(M\) has at most \(\#(n,\delta)< \infty\) many possible diffeomorphic types. (iii) Let \(M\) be a compact \(n\)-manifold of finite second homotopy group. Then, for any \(0<\delta\leq 1\), there are at most, if any, finitely many \(n\)-manifolds which are homotopy equivalent to \(M\) and which admit a \(\delta\)-pinched metric.
scientific article; zbMATH DE number 1383538