用户: Solution/ 习题: 规范场/Example Sheet 2

Question 1. Consider a group $G$ and a set $M$.

•	Let $$\begin{array}{rl}G\times M& \to M\\ (g,p)& \mapsto g\cdot p\end{array}$$be a left action. Then $$\begin{array}{rl}M\times G& \to M\\ (p,g)& \mapsto p\ast g=g^{-1}\cdot p\end{array}$$defines a right action of $G$ on $M$.
•	Find an example of a left action $$\begin{array}{rl}G\times M& \to M\\ (p,g)& \mapsto p\ast g=g\cdot p\end{array}$$does not define a right action of $G$ on $M$.

Question 2. Consider the left action of $SU(2)$ on ${\mathbb{C}}^2$, defined by matrix multiplication, $$\begin{array}{rl}SU(2)\times {\mathbb{C}}^2& \to {\mathbb{C}}^2\\ (g,v)& \mapsto gv\end{array}$$and vectors $${\tau }_a=-\frac{\iota {\sigma }_a}{2}\in \mathfrak{su}(2),a=1,2,3.$$

•	Determine the fundamental vector fields ${\tilde{\tau }}_a$ on ${\mathbb{C}}^2$.
•	Show, by direct calculation, that $$[{\tilde{\tau }}_a,{\tilde{\tau }}_b]=\widetilde{\left[{\tau }_a,{\tau }_b\right]},\forall a,b\in \{1,2,3\}.$$
•	Calculate directly $l_{A\ast }({\tilde{\tau }}_1)$ with $$A=\left(\begin{array}{cc}r& -\bar{r}\\ r& \bar{r}\end{array}\right)\in SU(2),r=\frac{1}{2}-\frac{1}{2}\iota .$$
•	Compare $l_{A\ast }({\tilde{\tau }}_1)$ with $\widetilde{{\mathrm{Ad}}_A{\tau }_1}$.

Question 3. Let $G,H$ be Lie groups with a Lie group homomorphism $\varphi :G\to H$. Suppose that $G$ acts continuously on the right on smooth manifold $M$ and $H$ acts continuously on the right on smooth manifold $N$. Consider a $\varphi$-equivariant map $f$, i.e. $$f(p\cdot g)=f(p)\cdot \varphi (g),\forall p\in M,g\in G.$$Prove

•	If $f$ is continuous, then $f$ induces a continuous map $$f_{\varphi }:M/G\to N/H.$$
•	If $\varphi$ is an isomorphism and $f$ is a homeomorphism, then $f_{\varphi }$ is a homeomorphism.

Question 4. Consider the half-plane $$H=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$$and the special linear group $$SL(2,\mathbb{R})=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):ad-bc=1\text{ with }a,b,c,d\in \mathbb{R}\right\}$$

•	Show that the map $$\begin{array}{rl}SL(2,\mathbb{R})\times H& \to H\\ (A,z)& \mapsto \frac{az+b}{cz+d}\end{array}$$is a well-defined left-action of $SL(2,\mathbb{R})$ on $H$.
•	Prove that this action is transitive and that it defines a diffeomorphism between $H$ and $SL(2,\mathbb{R})/SO(2)$.

Question 5. Consider the Möbius strip $$M=\left\{\left(e^{\iota \varphi },r{e}^{\iota \varphi /2}\right):\varphi \in \left[0,2\pi \right],r\in \left[-1,1\right]\right\}\subset {\mathbb{S}}^1\times \mathbb{C}$$Define the projection $\pi :M\to {\mathbb{S}}^1$ by $\pi ={\mathrm{pr}}_1{|}_M:{\mathbb{S}}^1\times \mathbb{C}\to {\mathbb{S}}^1$. Denote the map $f_n:{\mathbb{S}}^1\to {\mathbb{S}}^1$, $f_n(z)=z^n$ for $n\in \mathbb{Z}$.

•	Show that $\pi :M\to {\mathbb{S}}^1$ is a fibre bundle with general fibre $[-1,1]$.
•	Prove that the boundary $\partial M$ is connected and that the bundle $\pi :M\to {\mathbb{S}}^1$ is not trivial.
•	Prove that the image of any smooth section $s:{\mathbb{S}}^1\to M$ intersects the zero section $z:{\mathbb{S}}^1\to M$, $z(\alpha )=(\alpha ,0)$.
•	Show that the pull-back bundle $f_n^{\ast }M$ is isomorphic to the bundle $M_n\to {\mathbb{S}}^1$ given by $$M_n:=\left\{\left(e^{\iota \psi },r{e}^{\iota n\psi /2}\right):\psi \in \left[0,2\pi \right],r\in \left[-1,1\right]\right\}\subset {\mathbb{S}}^1\times \mathbb{C}.$$
•	Determine those $n\in \mathbb{Z}$ for which $f_n^{\ast }M$ is trivial and those for which it is non-trivial.

Question 6. Let $G\to P\stackrel{\pi }{\to }M$ be a principal bundle and $f:N\to M$ a smooth map between smooth manifolds. Prove that the pullback $f^{\ast }P$ has the canonical structure of a principal $G$-bundle over $N$.

Question 7. Let $F\to E\stackrel{\pi }{\to }M$ and $F^{\prime }\to E^{\prime }\stackrel{\pi }{\to }M$ be fibre bundles and $H:E\to E^{\prime }$ a bundle morphism. Suppose that $H$ maps every fibre of $E$ diffeomorphically onto a fibre of $E^{\prime }$. Show that $H$ is a bundle isomorphism.

Question 8. Let $G\to P\stackrel{\pi }{\to }M$ and $G^{\prime }\to P^{\prime }\stackrel{\pi }{\to }M$ be principle bundles. Suppose $f:G\to G^{\prime }$ is a Lie group isomorphism. Show that every $f$-equivariant bundle morphism $H:P\to P^{\prime }$ is a diffeomorphism.

Question 9. Let $E\to M$ be a $\mathbb{K}$-vector bundle of rank $m$ with a positive definite bundle metric. Suppose $F\subset E$ is a vector subbundle. Prove that the orthogonal complement $F^{\perp }$ is a vector subbundle of $E$ and that $F\oplus F^{\perp }$ is isomorphic to $E$.

Question 10. Let $G\to P\stackrel{\pi }{\to }M$ be a principle bundle and $\rho :G\to \mathrm{GL}(V)$, ${\rho }_i:G\to \mathrm{GL}(V_i)$ for $i=1,2$ representations on $V,V_i$. Consider the associated vector bundles $$E=P{\times }_{\rho }V,E_i{\times }_{{\rho }_i}{V}_i.$$

•	Show that the dual bundle $E^{\ast }$, the direct sum $E_1\oplus E_2$ and the tensor product $E_1\otimes E_2$ are isomorphic to vector bundles associated to $P$.
•	Determine the corresponding representations of $G$ and the vector bundle isomorphisms.

Question 11. Let $E\to M$ be a complex vector bundle of rank $n\ge 2$. Show that $E$ is associated to a principal $SU(n)$-bundle over $M$ if and only if ${\Lambda }^{n}E$ is a trivial complex line bundle.

Question 12. Let $E=P{\times }_{\rho }V$ be an associated vector bundle and $\alpha$ a section of the adjoint bundle $\mathrm{Ad}(P)$. Prove that $\alpha$ defines a canonical endomorphism of the vector bundle $E$.