is the unique (up to isometry) complete simply-connected 2-dimensional Riemann manifold of constant sectional curvature -1.

- It is diffeomorphic to as a topological space (or, indeed, isometric as a Riemannian manifold, if we give a left-invariant metric): to show this, we note that has isometry group , and the subgroup of isometries which stabilize any given is isomorphic to . 211 more words