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.
is a diffeomorphism between open sets
and
, then
must equal
, and the linear mapping
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
.
![[X,Y]^{\mu}=X^{\mu}\partial_{\lambda}Y^{\mu}-Y^{\lambda}\partial_{\lambda}X^{\mu}](https://s0.wp.com/latex.php?latex=%5BX%2CY%5D%5E%7B%5Cmu%7D%3DX%5E%7B%5Cmu%7D%5Cpartial_%7B%5Clambda%7DY%5E%7B%5Cmu%7D-Y%5E%7B%5Clambda%7D%5Cpartial_%7B%5Clambda%7DX%5E%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 "[X,Y]^{\mu}=X^{\mu}\partial_{\lambda}Y^{\mu}-Y^{\lambda}\partial_{\lambda}X^{\mu}")
![[X,Y]^{\mu}=[X,Y](x^{\mu})=X(Y(x^{\mu}))-Y(X(x^{\mu}))=X(Y^{\mu})-Y(X^{\mu})\\ =X^{\mu}\partial_{\lambda}Y^{\mu}-Y^{\lambda}\partial_{\lambda}X^{\mu}](https://s0.wp.com/latex.php?latex=%5BX%2CY%5D%5E%7B%5Cmu%7D%3D%5BX%2CY%5D%28x%5E%7B%5Cmu%7D%29%3DX%28Y%28x%5E%7B%5Cmu%7D%29%29-Y%28X%28x%5E%7B%5Cmu%7D%29%29%3DX%28Y%5E%7B%5Cmu%7D%29-Y%28X%5E%7B%5Cmu%7D%29%5C%5C+%3DX%5E%7B%5Cmu%7D%5Cpartial_%7B%5Clambda%7DY%5E%7B%5Cmu%7D-Y%5E%7B%5Clambda%7D%5Cpartial_%7B%5Clambda%7DX%5E%7B%5Cmu%7D&bg=ffffff&fg=000000&s=0&c=20201002 "[X,Y]^{\mu}=[X,Y](x^{\mu})=X(Y(x^{\mu}))-Y(X(x^{\mu}))=X(Y^{\mu})-Y(X^{\mu})\\ =X^{\mu}\partial_{\lambda}Y^{\mu}-Y^{\lambda}\partial_{\lambda}X^{\mu}")
![[X,Y]^{\mu'}=X^{\mu'}\partial_{\lambda'}Y^{\mu'}-Y^{\lambda'}\partial_{\lambda'}X^{\mu'}\\ =X^{\mu}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}}Y^{\mu})-Y^{\lambda}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}}X^{\mu})\\ =X^{\lambda}Y^{\mu}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}})+\frac{\partial x^{\mu'}}{\partial x^{\mu}}x^{\lambda}\partial_{\lambda}Y^{\mu}-X^{\lambda}Y^{\mu}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}})-\frac{\partial x^{\mu'}}{\partial x^{\mu}}Y^{\lambda}\partial_{\lambda}X^{\mu} \\ = \frac{\partial x^{\mu'}}{\partial x^{\mu}}(X^{\mu}\partial_{\lambda}Y^{\mu}-Y^{\lambda}\partial_{\lambda}X^{\mu})= \frac{\partial x^{\mu'}}{\partial x^{\mu}}([X,Y]^{\mu})](https://s0.wp.com/latex.php?latex=%5BX%2CY%5D%5E%7B%5Cmu%27%7D%3DX%5E%7B%5Cmu%27%7D%5Cpartial_%7B%5Clambda%27%7DY%5E%7B%5Cmu%27%7D-Y%5E%7B%5Clambda%27%7D%5Cpartial_%7B%5Clambda%27%7DX%5E%7B%5Cmu%27%7D%5C%5C++%3DX%5E%7B%5Cmu%7D%5Cpartial_%7B%5Clambda%7D%28%5Cfrac%7B%5Cpartial+x%5E%7B%5Cmu%27%7D%7D%7B%5Cpartial+x%5E%7B%5Cmu%7D%7DY%5E%7B%5Cmu%7D%29-Y%5E%7B%5Clambda%7D%5Cpartial_%7B%5Clambda%7D%28%5Cfrac%7B%5Cpartial+x%5E%7B%5Cmu%27%7D%7D%7B%5Cpartial+x%5E%7B%5Cmu%7D%7DX%5E%7B%5Cmu%7D%29%5C%5C+%3DX%5E%7B%5Clambda%7DY%5E%7B%5Cmu%7D%5Cpartial_%7B%5Clambda%7D%28%5Cfrac%7B%5Cpartial+x%5E%7B%5Cmu%27%7D%7D%7B%5Cpartial+x%5E%7B%5Cmu%7D%7D%29%2B%5Cfrac%7B%5Cpartial+x%5E%7B%5Cmu%27%7D%7D%7B%5Cpartial+x%5E%7B%5Cmu%7D%7Dx%5E%7B%5Clambda%7D%5Cpartial_%7B%5Clambda%7DY%5E%7B%5Cmu%7D-X%5E%7B%5Clambda%7DY%5E%7B%5Cmu%7D%5Cpartial_%7B%5Clambda%7D%28%5Cfrac%7B%5Cpartial+x%5E%7B%5Cmu%27%7D%7D%7B%5Cpartial+x%5E%7B%5Cmu%7D%7D%29-%5Cfrac%7B%5Cpartial+x%5E%7B%5Cmu%27%7D%7D%7B%5Cpartial+x%5E%7B%5Cmu%7D%7DY%5E%7B%5Clambda%7D%5Cpartial_%7B%5Clambda%7DX%5E%7B%5Cmu%7D++%5C%5C+%3D%C2%A0%5Cfrac%7B%5Cpartial+x%5E%7B%5Cmu%27%7D%7D%7B%5Cpartial+x%5E%7B%5Cmu%7D%7D%28X%5E%7B%5Cmu%7D%5Cpartial_%7B%5Clambda%7DY%5E%7B%5Cmu%7D-Y%5E%7B%5Clambda%7D%5Cpartial_%7B%5Clambda%7DX%5E%7B%5Cmu%7D%29%3D%C2%A0%5Cfrac%7B%5Cpartial+x%5E%7B%5Cmu%27%7D%7D%7B%5Cpartial+x%5E%7B%5Cmu%7D%7D%28%5BX%2CY%5D%5E%7B%5Cmu%7D%29&bg=ffffff&fg=000000&s=0&c=20201002 "[X,Y]^{\mu'}=X^{\mu'}\partial_{\lambda'}Y^{\mu'}-Y^{\lambda'}\partial_{\lambda'}X^{\mu'}\\ =X^{\mu}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}}Y^{\mu})-Y^{\lambda}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}}X^{\mu})\\ =X^{\lambda}Y^{\mu}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}})+\frac{\partial x^{\mu'}}{\partial x^{\mu}}x^{\lambda}\partial_{\lambda}Y^{\mu}-X^{\lambda}Y^{\mu}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}})-\frac{\partial x^{\mu'}}{\partial x^{\mu}}Y^{\lambda}\partial_{\lambda}X^{\mu} \\ = \frac{\partial x^{\mu'}}{\partial x^{\mu}}(X^{\mu}\partial_{\lambda}Y^{\mu}-Y^{\lambda}\partial_{\lambda}X^{\mu})= \frac{\partial x^{\mu'}}{\partial x^{\mu}}([X,Y]^{\mu})")
![[V_{1} , V_{2} ] = [\partial _{x} A(x, y)]\partial _{x} + [\partial _{x} B(x, y)]\partial _{y}](https://s0.wp.com/latex.php?latex=%5BV_%7B1%7D+%2C+V_%7B2%7D+%5D+%3D+%5B%5Cpartial+_%7Bx%7D+A%28x%2C+y%29%5D%5Cpartial+_%7Bx%7D+%2B+%5B%5Cpartial+_%7Bx%7D+B%28x%2C+y%29%5D%5Cpartial+_%7By%7D&bg=ffffff&fg=000000&s=0&c=20201002 "[V_{1} , V_{2} ] = [\partial _{x} A(x, y)]\partial _{x} + [\partial _{x} B(x, y)]\partial _{y}")
![d(\omega\wedge\eta)_{\mu_1\dots\mu_{p+q+1}}\\=\frac{(p+q)!(p+q+1)}{p!q!}\partial_{[\mu_1}\omega_{[\mu_2\dots\mu_{p+1}}\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]}\\=\frac{(p+q)!(p+q+1)}{p!q!}[(\partial_{[\mu_1}\omega_{[\mu_2\dots\mu_{p+1}}).\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]}+\omega_{[\mu_{2}\dots\mu_{p+1}}.(\partial_{[\mu_1}\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]})]\\=\frac{(p+q)!(p+q+1)}{p!q!}[(\partial_{[\mu_1}\omega_{[\mu_2\dots\mu_{p+1}}).\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]}+(-1)^p\omega_{[\mu_{1}\dots\mu_{p}}.(\partial_{[\mu_{p+1}}\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]})]\\=(d\omega)\wedge\eta+(-1)^p\omega\wedge(d\eta)](https://s0.wp.com/latex.php?latex=d%28%5Comega%5Cwedge%5Ceta%29_%7B%5Cmu_1%5Cdots%5Cmu_%7Bp%2Bq%2B1%7D%7D%5C%5C%3D%5Cfrac%7B%28p%2Bq%29%21%28p%2Bq%2B1%29%7D%7Bp%21q%21%7D%5Cpartial_%7B%5B%5Cmu_1%7D%5Comega_%7B%5B%5Cmu_2%5Cdots%5Cmu_%7Bp%2B1%7D%7D%5Ceta_%7B%5Cmu_%7Bp%2B2%7D%5Cdots%5Cmu_%7Bp%2Bq%2B1%7D%5D%5D%7D%5C%5C%3D%5Cfrac%7B%28p%2Bq%29%21%28p%2Bq%2B1%29%7D%7Bp%21q%21%7D%5B%28%5Cpartial_%7B%5B%5Cmu_1%7D%5Comega_%7B%5B%5Cmu_2%5Cdots%5Cmu_%7Bp%2B1%7D%7D%29.%5Ceta_%7B%5Cmu_%7Bp%2B2%7D%5Cdots%5Cmu_%7Bp%2Bq%2B1%7D%5D%5D%7D%2B%5Comega_%7B%5B%5Cmu_%7B2%7D%5Cdots%5Cmu_%7Bp%2B1%7D%7D.%28%5Cpartial_%7B%5B%5Cmu_1%7D%5Ceta_%7B%5Cmu_%7Bp%2B2%7D%5Cdots%5Cmu_%7Bp%2Bq%2B1%7D%5D%5D%7D%29%5D%5C%5C%3D%5Cfrac%7B%28p%2Bq%29%21%28p%2Bq%2B1%29%7D%7Bp%21q%21%7D%5B%28%5Cpartial_%7B%5B%5Cmu_1%7D%5Comega_%7B%5B%5Cmu_2%5Cdots%5Cmu_%7Bp%2B1%7D%7D%29.%5Ceta_%7B%5Cmu_%7Bp%2B2%7D%5Cdots%5Cmu_%7Bp%2Bq%2B1%7D%5D%5D%7D%2B%28-1%29%5Ep%5Comega_%7B%5B%5Cmu_%7B1%7D%5Cdots%5Cmu_%7Bp%7D%7D.%28%5Cpartial_%7B%5B%5Cmu_%7Bp%2B1%7D%7D%5Ceta_%7B%5Cmu_%7Bp%2B2%7D%5Cdots%5Cmu_%7Bp%2Bq%2B1%7D%5D%5D%7D%29%5D%5C%5C%3D%28d%5Comega%29%5Cwedge%5Ceta%2B%28-1%29%5Ep%5Comega%5Cwedge%28d%5Ceta%29&bg=ffffff&fg=000000&s=0&c=20201002 "d(\omega\wedge\eta)_{\mu_1\dots\mu_{p+q+1}}\\=\frac{(p+q)!(p+q+1)}{p!q!}\partial_{[\mu_1}\omega_{[\mu_2\dots\mu_{p+1}}\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]}\\=\frac{(p+q)!(p+q+1)}{p!q!}[(\partial_{[\mu_1}\omega_{[\mu_2\dots\mu_{p+1}}).\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]}+\omega_{[\mu_{2}\dots\mu_{p+1}}.(\partial_{[\mu_1}\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]})]\\=\frac{(p+q)!(p+q+1)}{p!q!}[(\partial_{[\mu_1}\omega_{[\mu_2\dots\mu_{p+1}}).\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]}+(-1)^p\omega_{[\mu_{1}\dots\mu_{p}}.(\partial_{[\mu_{p+1}}\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]})]\\=(d\omega)\wedge\eta+(-1)^p\omega\wedge(d\eta)")
![d*F=3\partial_{[\mu}F_{\nu\rho ]}=1/2[\partial_{\mu}(\epsilon^{\nu_1\nu_2}_{\nu\rho}F_{\nu_1\nu_2})+\partial_{\nu}(\epsilon^{\nu_1\nu_2}_{\rho\mu}F_{\nu_1\nu_2})+\partial_{\rho}(\epsilon^{\nu_1\nu_2}_{\mu\nu}F_{\nu_1\nu_2})]=\epsilon^{\sigma}_{\mu\nu\rho}J_{\rho}=*J](https://s0.wp.com/latex.php?latex=d%2AF%3D3%5Cpartial_%7B%5B%5Cmu%7DF_%7B%5Cnu%5Crho+%5D%7D%3D1%2F2%5B%5Cpartial_%7B%5Cmu%7D%28%5Cepsilon%5E%7B%5Cnu_1%5Cnu_2%7D_%7B%5Cnu%5Crho%7DF_%7B%5Cnu_1%5Cnu_2%7D%29%2B%5Cpartial_%7B%5Cnu%7D%28%5Cepsilon%5E%7B%5Cnu_1%5Cnu_2%7D_%7B%5Crho%5Cmu%7DF_%7B%5Cnu_1%5Cnu_2%7D%29%2B%5Cpartial_%7B%5Crho%7D%28%5Cepsilon%5E%7B%5Cnu_1%5Cnu_2%7D_%7B%5Cmu%5Cnu%7DF_%7B%5Cnu_1%5Cnu_2%7D%29%5D%3D%5Cepsilon%5E%7B%5Csigma%7D_%7B%5Cmu%5Cnu%5Crho%7DJ_%7B%5Crho%7D%3D%2AJ&bg=ffffff&fg=000000&s=0&c=20201002 "d*F=3\partial_{[\mu}F_{\nu\rho ]}=1/2[\partial_{\mu}(\epsilon^{\nu_1\nu_2}_{\nu\rho}F_{\nu_1\nu_2})+\partial_{\nu}(\epsilon^{\nu_1\nu_2}_{\rho\mu}F_{\nu_1\nu_2})+\partial_{\rho}(\epsilon^{\nu_1\nu_2}_{\mu\nu}F_{\nu_1\nu_2})]=\epsilon^{\sigma}_{\mu\nu\rho}J_{\rho}=*J")
