Proposition: is a smooth manifold.

Proof:

Define and . Also define by and .

Let be the homeomorphism from to defined by and define by .

Note that is an open cover of , and are well-defined homeomorphisms (from onto an open set in ). Then is an atlas of .

The transition function

is differentiable of class . Similarly, is of class . Hence is a smooth manifold.

Proposition:

is a smooth manifold.

Proof:

Define and . Then is an open cover of .

Define by and by .

We can check that . Hence is a homeomorphism from onto an open set in . Similarly, is a homeomorphism from onto an open set in . Thus is an atlas for .

The composite is differentiable of class since both , are of class . Similarly, is of class . Thus is a smooth manifold.

We can also compute the transition function explicitly:

Note that .

Define by and .

We see that is well-defined since if then so that .

Similarly, we have a well-defined inverse defined by and .

We check that (from our previous workings)

are of class . So is a smooth map. Similarly, is smooth. Hence is a diffeomorphism.