math214:04-15
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| ====== 2020-04-15, Wednesday ====== | ====== 2020-04-15, Wednesday ====== | ||
| - | ==== Cartan' | + | ===== Cartan' |
| Pick any **Orthonormal Frame ** $X_\alpha$ of $TM$, and choose its dual frame $\theta^\alpha$ of $T^*M$. Introduce a collection of 1-forms using covariant derivatives | Pick any **Orthonormal Frame ** $X_\alpha$ of $TM$, and choose its dual frame $\theta^\alpha$ of $T^*M$. Introduce a collection of 1-forms using covariant derivatives | ||
| $$ \nabla X_\alpha = \omega_\alpha^\beta X_\beta $$ | $$ \nabla X_\alpha = \omega_\alpha^\beta X_\beta $$ | ||
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| $$X_i = \frac{1}{h_i} \d_{x^i}, \theta^i = h_i dx^i $$ | $$X_i = \frac{1}{h_i} \d_{x^i}, \theta^i = h_i dx^i $$ | ||
| Then, we can get $\omega_i^j$ by solving | Then, we can get $\omega_i^j$ by solving | ||
| - | $$ d \theta^i = -\d_j(h_i) dx^i \wedge dx^j $$ | + | $$ d \theta^i = d(h_i) \wedge |
| + | Now, one need to figure out what is $\omega_i^j$ in each specific cases. Once that is done, one can easily get the curvature. | ||
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| + | ===== Aside: Ehresmann connection, Connection on Principal Bundle ===== | ||
| + | Another more geometric notion of connection is called Ehresmann connection, it applies to fiber bundle. Let $\pi: E \to M$ be a fiber bundle. A connection is a splitting of | ||
| + | $$T_p E = T_p E_h \oplus T_p E_v $$ | ||
| + | where $h$ stands for horizontal and $v$ for vertical. $TE_v$ is canonical, it is the tangent space to the fiber. $T_p E_h$ contains information. | ||
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| + | If $P$ is a principal $G$-bundle, then we want the choice of horizontal subspace to be invariant under the (right) action of $G$. | ||
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| + | Parallel transport along a path $\gamma: [0,1] \to M$ on $M$ gives a diffeomorphism | ||
| + | $$ P_\gamma: E_{\gamma(0)} \to E_{\gamma(1)}. $$ | ||
| + | In the case of $G$-bundle, this diffeomorphism commute with the right $G$-action on the fiber. | ||
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| + | If $V$ is a represenation of $G$, we may form the associated bundle $P \times_G V$, where each fiber over $b \in M$ is $$ P_b \times_G V = \{ (p , v) \in P_b \times V \} / (p g, v) \sim (p, g \cdot v) \cong V $$ | ||
| + | since $G$ acts on $P_b$ freely and transitively ($P_b$ is said to be a $G$-torsor). Then, $P_\gamma$ also induces parallel transport on $P \times_G V$. | ||
math214/04-15.1586935776.txt.gz · Last modified: 2020/04/15 00:29 by pzhou
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