Manifolds without conjugate points and their fundamental groups (Q2447216)

The authors prove that if a compact smooth Riemannian manifold \(M\) has no conjugate points, then it behaves in a sense like a nonpositively curved manifold. Specifically, for any nontrivial element \(g \in \pi_1(M)\), there is a finite index subgroup \(\mathbb{Z} \times \Gamma \subset Z(g)\) of the centralizer \(Z(g) \subset \pi_1(M)\) so that \(\mathbb{Z}\) is generated by \(g\). They use this result to provide new examples of compact manifolds on which every Riemannian metric must have conjugate points. For example, they prove that the unit tangent bundle of any higher genus compact Riemann surface has no metric without conjugate points.