The types of surfaces which admit nontrivial isometries homotopic to the identity are classified up to diffeomorphism. In dimension three this is done for complete manifolds of constant negative curvature. Three-dimensional visibility manifolds that admit nontrivial isometries homotopic to the identity are shown to be diffeomorphic to a product L x RI.
Norris, Douglas A., "Isometries Homotopic to the Identity" (1989). Mathematics and Computer Science. 4.