On intermediate Ricci curvature and fundamental groups
Synge's theorem states that a closed Riemannian \(n\)-manifold with positive sectional curvature is orientable if either \(n\) is even and has fundamental group of order 1 or 2. This result is extended here for any manifold with positive Ricci curvature provided the first systole (i.e., the length of the shortest closed noncontractible curve) is sufficiently large. Synge's theorem is implied by Weinstein's result which states that any isometry \(f\) of a compact, oriented, Riemannian \(n\)-manifold with positive sectional curvature has a fixed point, provided \(n\) is even (resp. odd) and \(f\) preserves (resp. reverses) orientation. An analogous result is given here. If \(f\) is an isometry of a complete, oriented, Riemannian \(n\)-manifold with \(\text{Ric}_k\geq k\), whose minimal displacement is sufficiently large, then \(f\) preserves (resp. reverses) orientation provided \(n\) is odd (resp. even).
scientific article; zbMATH DE number 1043161