math214:04-01
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| $$\nabla: \Omega^0(M, E) \to \Omega^1(M, E). $$ | $$\nabla: \Omega^0(M, E) \to \Omega^1(M, E). $$ | ||
| In general, we define | In general, we define | ||
| - | $$ \Omega^k(M, E) = C^\infty( \wedge^k(T^*E) \ot E) $$ | + | $$ \Omega^k(M, E) = C^\infty( \wedge^k(T^*M) \ot E) $$ |
| as $k$-forms with coefficient in $E$. | as $k$-forms with coefficient in $E$. | ||
| Line 68: | Line 68: | ||
| ==== Tensor, Hom and Dual ==== | ==== Tensor, Hom and Dual ==== | ||
| Let $E_1, E_2$ be two vector bundles on $M$. Recall that we can define the following | Let $E_1, E_2$ be two vector bundles on $M$. Recall that we can define the following | ||
| - | $$ E_1 \ot E_2, \quad \Hom(E_1, E_2), \quad | + | $$ E_1 \ot E_2, \quad \Hom(E_1, E_2) $$ |
| - | as vector bundles on $M$, where the fiber satisfies $(E_1 \ot E_2)_p = (E_1)_p \ot (E_2)_p$ and $\Hom(E_1, E_2)_p = \Hom((E_1)_p, | + | as vector bundles on $M$, where the fiber satisfies $(E_1 \ot E_2)_p = (E_1)_p \ot (E_2)_p$ and $\Hom(E_1, E_2)_p = \Hom( (E_1)_p, (E_2)_p)$. We define the dual bundle of a vector bundle $E$ as $E^\vee := \Hom(E, \underline{\R})$ where |
| $\underline{\R} = \R \times M$ is the trivial bundle. | $\underline{\R} = \R \times M$ is the trivial bundle. | ||
math214/04-01.1585721815.txt.gz · Last modified: 2020/03/31 23:16 by pzhou
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