Two Hausdorff spaces X and Y are bijectively related if there exist continuous bijections \(f: X\to Y\) and \(g: Y\to X\). The authors show that there exist non-homeomorphic connected bijectively related n-manifolds for each \(n\geq 2\). Starting from this fact, they mention many properties with respect to reversibility for connected manifolds. The following is one of typical results: If the manifold M has only compact boundary components and if M is simply connected, then M is reversible, that is, the only continuous self-bijections \(f: M\to M\) are the homeomorphisms.