Like the plane and the open cylinder, the open Möbius band admits not only a complete metric of constant curvature 0, but also a complete metric of constant negative curvature, say −1. One way to see this is to begin with the upper half plane (Poincaré) model of the hyperbolic plane ℍ, namely with the Riemannian metric given by . The orientation-preserving isometries of this metric are all the maps of the form , where "a", "b", "c", "d" are real numbers satisfying . Here "z" is a complex number with , and we have identified with endowed with the Riemannian metric that was mentioned. Then one orientation-reversing isometry "g" of is given by , where denotes the complex conjugate of "z". These facts imply that the mapping given by is an orientation-reversing isometry of that generates an infinite cyclic group "G" of isometries. (It can be expressed as , and its square is the isometry , which can be expressed as .) The quotient of the action of this group can easily be seen to be topologically a Möbius band. But it is also easy to verify that it is complete and non-compact, with constant negative curvature equal to −1.