Proposition: is a smooth manifold.
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.
is a smooth manifold.
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.