Short geodesic segments between two points on a closed Riemannian manifold (Q1034680)

Inspired by a nice theorem of J.-P. Serre (`any two points of a closed Riemannian manifold can be connected by infinitely many geodesics'), in an earlier paper [Electron. Res. Announc. Am. Math. Soc. 13, 13--20, electronic only (2007; Zbl 1113.53026)], the authors formulated the following Conjecture. If \(M\) is an \(n\)-dimensional closed Riemannian manifold of diameter \(d\), then there exists a function \(f(k,n)\) such that any two points of \(M\) can be connected by at least \(k\) geodesics of length at most \(f(k,n)d\). Concerning this conjecture, in the present paper it is shown that if the closed Riemannian manifold in question is homotopy equivalent to the product of \(S^2\) and an arbitrary \((n-2)\)-dimensional manifold, then there exist at least \(k\) distinct geodesics between any two points of \(M\) of length at most \(20k!d\), for every positive integer \(k\).