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math214:final [2020/05/04 13:47]
pzhou 		math214:final [2020/05/06 22:46] (current)
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 $$\gdef\gfrak{\mathfrak g}$$  $$\gdef\gfrak{\mathfrak g}$$ 
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 +Due Date: May 10th (Sunday) 11:59PM. Submit online to gradescope. 
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 ** 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.  ** 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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   * (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.     * (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) Show that the new connection is torsionless.
-  * (5 pt) Let $G$ be a Lie group. Let $\nabla$ be the flat connection on $TG$ where the left-invariant vector fields are flat sections.  Compute the torsion of this connection.  +  * (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.  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. 
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 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 $\dot \gamma(0) = \dot \gamma(1)$.  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 and $g$ any smooth Riemannian metric. Is it true that there is always exists a closed geodesic on $M$? +  * (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. ))    * (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 $S^1$.  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 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. +  * (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 $\gamma(0)=\gamma(1)=p$.    * (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 $\gamma(0)=\gamma(1)=p$. 
  
math214/final.1588625231.txt.gz · Last modified: 2020/05/04 13:47 by pzhou

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