These are parts of the solutions to exercises from:
Carroll, S. M. (2005). Spacetime and geometry: An introduction to general relativity. Addison Wesley.
- First consider mapping the infinite cylinder to a semi-infinite cylinder, then consider projecting points on it to a plane.
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.
4. First two can be verified directly.
5. Set , . The commutator will be given by
Set . Then , are nowhere vanishing, and their commutator is nowhere vanishing as well.