1. To show that f is well-defined you need to show that is one to one, so that . This is easy since sinh is one to one.

2. To show that f is locally isometric, show that first fundamental forms of the two surfaces are the same:

3. To show that f is not globally isometric, choose the two points (u,v)=(0,0) and

The two points on H is (0,0,0) and . The distance is since the z-axis lies on H.

The two points on C is (1,0,0) and (1,0,0), the same point. So the distance is 0.