Differential forms

$V_{1} \otimes V_{2} \otimes \dots \otimes V_{n} ≅ {V_{1}^{*} \times \dots \times V_{n}^{*} multilinear K}$ , where $K$ is the base field.

[!Definition]
$⋀^{K} V^{*} := {V \times \dots \times V multilinear, alternating K}$ .

$⋀^{0} V^{*} = K$ .
$⋀^{1} V^{*} = V^{*}$ .
$⋀^{2} V^{*}$ is the space of skew symmetric forms.

[!Definition]
Let $i = 1, 2$ . Let $w_{i} \in \otimes_{m_{i}} V^{*}$ .

$w_{1} \land w_{2} (v_{1}, \dots, v_{m_{1} + m_{2}}) = \sum_{r \in Sh (m_{1}, m_{2})} (- 1)^{r} w_{1} (v_{r_{1}}, \dots, v_{r_{m_{1}}}) w_{2} (v_{r_{m_{1} + 1}}, \dots, v_{r_{m_{1} + m_{2}}})$ .
$Sh (m_{1}, m_{2}) = {r \in S_{m_{1} + m_{2}} : r_{1} < \dots < r_{m_{1}}, r_{m_{1} + 1} < \dots < r_{m_{1} + m_{2}}}$ .
$(- 1)^{r} := sgn (r)$ .

$w_{1} \land w_{2} \land w_{3}$ is similarly defined.

[!Proposition]
$(w_{1} \land w_{2}) \land w_{3} = w_{1} \land (w_{2} \land w_{3})$ .
$w_{1} \land w_{2} = (- 1)^{m_{1} m_{2}} w_{2} \land w_{1}$ .

[!Lemma]
$Sh (m_{1}, m_{2}, m_{3}) \sim Sh (m_{1} + m_{2}, m_{3}) \times Sh (m_{1}, m_{2})$ .

There is a natural inclusion

⋀ m V^{*} i ⨂ m V^{*} .

Define $Alt : ⨂^{m} V^{*} \to ⨂^{m} V^{*}$ by

Alt (w) = \frac{1}{∣ S _{m} ∣} σ \in S_{m} \sum (- 1)^{σ} (σ . w),

where $(σ . w) (v_{1}, \dots, v_{m}) = w (v_{σ (1)}, \dots, v_{σ (m)})$ .

[!Proposition]
$Alt$ is a projector onto $⋀^{m} V^{*}$ , that is,

$Im (Alt) = ⋀^{m} V^{*}$ .
$Alt \circ Alt = Alt$

NoNotes

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LEC CAL2 10

Differential forms

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