These are parts of the solutions to exercises from:

Carroll, S. M. (2005). Spacetime and geometry: An introduction to general relativity. Addison Wesley.

Solutions:

First consider mapping the infinite cylinder to a semi-infinite cylinder, then consider projecting points on it to a plane.

Let ,

.

is the map desired whose image is an open set .

2. No. must be k-dimension manifolds. Furthermore, the dimension of a manifold is unique.

Suppose that is a subset of a manifold , if there are two charts with different dimension and $latex n$, then there exists , , . Then is a diffeomorphism from to . Hence m must be equal to n.

Note: Here I use such an assertion( J. Milnor, Topology from Differentiable Viewpoint, P4).

Assertion. If is a diffeomorphism between open sets and , then must equal , and the linear mapping

must be nonsingular.

Proof. The composition is the identity map of U; hence is the identity map of . Similarly is the identity map of . Thus has a two-sided inverse, and it follows that .

By the way, can also be seen from the , denote the linear map and respectively.

3. Trivial.

4. First two can be verified directly.

Composition formula:

.

.

Transformation:

.

5. Set , . The commutator will be given by

.

Set . Then , are nowhere vanishing, and their commutator is nowhere vanishing as well.