Obstructions to toric integrable geodesic flows in dimension 3 (Q2431741)

The author proves the following result: Let \(Q\) be a \(3\)-dimensional compact, connected Riemannian manifold admitting a toric integrable geodesic flow. Then, the cosphere bundle \(S(T^{*}Q)\) is diffeomorphic to either \(S^{3}\times{S^{2}}\) if \(Q\) is simply connected or \(T^{3}\times{S^{2}}\) if the fundamental group of \(Q\) infinite. If the fundamental group of \(Q\) is finite and nontrivial, then \(S(T^{*}Q)\) is homotopy equivalent to \((S^{3}/{\Gamma})\times {S^{2}}\), where \(\Gamma\) is a finite cyclic group.