. 42) where the center IR consists of those elements of N3 for which x = y = 0. As a Lie group, Jf 3 admits a Riemannian metric invariant under left multiplication (see Marenich  for its description and equivariant compactification by St) and becomes a line bundle over a 2-cell -- R2, see for details Goldman .
Then Io is simultaneously a closed and an open subset in [0, 1] because d is a local homeomorphism and M (together with M) is a complete metric space. This shows that Io = [0, 1] and hence proves the (X, g)-completeness of the manifold M. One can use standard elementary arguments to prove the following assertions: (2) (3) # (4) q (5). So we will complete the proof by showing that (1) = (3). To prove that (1) = (3), we first note that the universal covering space X of the manifold X satisfies (2), (3), (4) and (5).
G-6. SL2-geometry. The space SL2R denotes the universal covering for the 3dimensional Lie group SL2 JR of all 2 x 2 real matrices with determinant 1. The group SL2 ll$ is itself a Lie group and so admits a metric invariant under left (or right) multiplication. However, we will describe the metric on the space SL2 R (which is homeomorphic to an open solid torus B2 x S1) in a more useful way. Namely, we consider a natural Riemannian metric on the unit tangent bundle Tl H2 of the hyperbolic plane H2.