Lecture Notes

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====== Final ======
$$\gdef\gfrak{\mathfrak g}$$ 
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Due Date: May 10th (Sunday) 11:59PM. Submit online to gradescope. 

** Policy **: You can use textbook and your notes. There should be no discussion or collaborations, since this is suppose to be a exam on your own understanding. If you found some question that is unclear, please let me know via email. 
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1. (15 pt) Let $G$ be a Lie group, $\gfrak = T_e G$ its Lie algebra. Let $TG$ be identified with $G \times \gfrak$ by
$$ G \times \gfrak \to TG, \quad (g, X) \mapsto (L_g)_* X $$  
Endow $TG$ with the natural induced Lie group structure, 
$$\rho: TG \times TG \to TG $$
such that if $\gamma_1, \gamma_2: (-\epsilon, \epsilon) \to G$ are two curves in $G$, then 
$$ \rho(\dot \gamma_1(0),  \dot \gamma_2(0)) = (d/dt)|_{t=0} (\gamma_1(t) \gamma_2(t)). $$ 
Write down the product law of $TG$ using identification with $G \times \gfrak$, i.e.
$$ (g, X) \cdot (h, Y) = ? $$

2. (15 pt) Let $\C$ acts on $\C^p \RM \{0\} \times \C^q \RM \{0\} $ by 
$$ t \cdot (z_1, \cdots, z_p; w_1, \cdots, w_q) \mapsto (e^{it} z_1, \cdots, e^{it} z_p; e^t w_1, \cdots, e^t w_q) $$
Show that the action is free, and the quotient is diffeomorphic to $S^{2p-1} \times S^{2q-1}$. 


3. (20 pt) Let $M$ be a smooth manifold, $\nabla$ be a connection on $TM$. Recall the torsion is defined as
$$ T: TM \times TM \to TM, \quad T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y]. $$ 
  * (10 pt) Show that the following formula defines a new connection  $$\widetilde \nabla_X Y = \nabla_X Y - (1/2) T(X, Y)$$ where $X, Y$ are any vector fields.  
  * (5 pt) Show that the new connection is torsionless.
  * (5 pt) Let $G$ be a Lie group. Let $\nabla$ be the flat connection ((a connection $\nabla$ is flat if the associated curvature $F_\nabla = 0$.)) on $TG$ where the left-invariant vector fields are flat sections((a section $s$ is flat, if $\nabla s = 0$)).  Compute the torsion of this connection.  

4.(15 pt) Let $\pi: S^3 \to S^2$ the Hopf fibration. Let $\omega$ be a 2-form on $S^2$ such that $[\omega] \in H^2(S^2)$ is non-zero. 
  * (10 pt) Show that there exists a 1-form $\alpha \in \Omega^1(S^3)$, such that  $$ d\alpha = \pi^* \omega. $$
  * (5 pt) Suppose $\omega$ is the volume form on $S^2$ from the round metric, can you give an explicit construction of such an $\alpha$ on $S^3$, and $\alpha \wedge d\alpha$ is a non-vanishing 3-form on $S^3$? 

5. (10 pt) Let $(M, g)$ be a Riemannian manifold, a closed geodesic is a geodesic $\gamma: [0, 1] \to M$ such that $\gamma(0)=\gamma(1)$ and $\dot \gamma(0) = \dot \gamma(1)$. 
  * (7 pt) Let $M$ be a genus $g\geq 1$ surface, smooth, compact without boundary, orientable 2-dimensional manifold ((see wiki https://en.wikipedia.org/wiki/Genus_g_surface)) and $g$ any smooth Riemannian metric. Is it true that there is always exists a closed geodesic on $M$? 
  * (3 pt) Let $M = S^2$ and $g$ any smooth Riemannian metric. Is it true that there is always exists a closed geodesic on $M$? Try your best to give an argument.  ((J. Franks proved that, if $S^2$ is equipped with a metric with positive Gaussian curvature, then there are infinitely many closed geodesics. Here we are asking for a much simpler version. )) 

6. (15 pt) Let $K \In \R^3$ be a knot, that is, a smooth embedded submanifold in $\R^3$ diffeomorphic to $S^1$. 
  * (10 pt) Can you construct a geodesically complete metric ((For any point $p \in M$, the exponential map exists for the entire $T_p M$, https://en.wikipedia.org/wiki/Geodesic_manifold )) on $M = \R^3 \RM K$? i.e. a metric  such that for any $p \in M$, $v \in T_p M$, the geodesics with initial condition $(p,v)$ exists for inifinite long time? Describe your construction explicitly. 
  * (5 pt) Assume such metric exists, prove that for any point $p \in M$, there are infinitely many distinct geodesics $\gamma: [0,1] \to M$ with $\gamma(0)=\gamma(1)=p$. 

7. (10 pt) Let $G = SU(2)$. Let  $\nabla^{L}$ (resp. $\nabla^{R}$) be the flat connection on $TG$, where the flat sections are left (resp. right)-invariant vector fields. Prove that there is no 1-parameter family of flat connections $\nabla^{(t)}$ connecting $\nabla^{L}$ and $\nabla^{R}$, i.e.  $\nabla^{(0)} = \nabla^{L}$ and $\nabla^{(1)} = \nabla^{R}$
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