math214:hw10
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math214:hw10 [2020/04/04 09:50] pzhou created |
math214:hw10 [2020/04/12 09:44] (current) pzhou [Problem 4] |
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| $$ A = x dy - y d x $$ | $$ A = x dy - y d x $$ | ||
| Let point $a=(1,0)$, $b=(-1,0)$, and $\gamma_\pm$ be path from $a$ to $b$, going along upper (or lower) semicircle: | Let point $a=(1,0)$, $b=(-1,0)$, and $\gamma_\pm$ be path from $a$ to $b$, going along upper (or lower) semicircle: | ||
| - | $$ \gamma_\pm: [0,1] \to \R^2, \quad t \mapsto (\cos t, \pm \sin t). $$ | + | $$ \gamma_\pm: [0,\pi] \to \R^2, \quad t \mapsto (\cos t, \pm \sin t). $$ |
| Question: compute the parallel transport along $\gamma_+$ and $\gamma_-$. | Question: compute the parallel transport along $\gamma_+$ and $\gamma_-$. | ||
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| Let $M = \R^2$, and $E$ the trivial rank-2 vector bundle on $\R^2$. Let $$ \nabla = d + A, $$ where $d$ is the trivial connection on $L$ and $A$ is the connection 1-form | Let $M = \R^2$, and $E$ the trivial rank-2 vector bundle on $\R^2$. Let $$ \nabla = d + A, $$ where $d$ is the trivial connection on $L$ and $A$ is the connection 1-form | ||
| $$ A = \begin{pmatrix} 1 & 0 \cr 0 & -1 \end{pmatrix} dx + \begin{pmatrix} 0 & 1 \cr -1 & 0 \end{pmatrix} dy $$ | $$ A = \begin{pmatrix} 1 & 0 \cr 0 & -1 \end{pmatrix} dx + \begin{pmatrix} 0 & 1 \cr -1 & 0 \end{pmatrix} dy $$ | ||
| - | Compute the curvature 2-form $F \in \Omega^2(M)\otimes M_2(\R)$. | + | Compute the curvature 2-form $F \in \Omega^2(M)\otimes M_2(\R)$. |
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| + | Optional: Compute the parallel transport along the boundary of the unit square $[0,1]^2$, starting from $(0,0)$ in counter-clockwise fashion. (Hint: you will end up with answer like $\beta^{-1}\alpha^{-1}\beta\alpha$ where $\alpha$ and $\beta$ are in $GL(2, | ||
math214/hw10.1586019026.txt.gz · Last modified: 2020/04/04 09:50 by pzhou
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