二階微分形式
📂幾何学 二階微分形式 概要 二項演算 ∧ \wedge ∧ 1次微分形式 を定義した感覚で、微分多様体 M M M 2次形式 を定義する。
微分多様体が難しいと感じるなら、M = R n M = \mathbb{R}^{n} M = R n
ビルドアップ 1次形式 ω \omega ω
ω : M → T ∗ M p ↦ ω p
\begin{align*}
\omega : M &\to T^{\ast}M
\\ p &\mapsto \omega_{p}
\end{align*}
ω : M p → T ∗ M ↦ ω p
これは、n n n M M M p p p 余接空間 の要素 ω p ∈ T p ∗ M \omega_{p} \in T_{p}^{\ast}M ω p ∈ T p ∗ M ω p \omega_{p} ω p T p M T_{p}M T p M 双対空間 の要素なので、以下のような汎関数 である。
w p : T p M → R
w_{p} : T_{p}M \to \mathbb{R}
w p : T p M → R
つまり、「1次」形式は点 p p p p p p ω p \omega_{p} ω p
ウェッジ積 関数 φ : T p M × T p M → R \varphi : T_{p}M \times T_{p}M \to \mathbb{R} φ : T p M × T p M → R 双線形 な交代関数 としよう。
φ ( v 1 , v 2 ) = − φ ( v 2 , v 1 ) , v i ∈ T p M
\varphi (v_{1}, v_{2}) = - \varphi (v_{2}, v_{1}),\quad v_{i} \in T_{p}M
φ ( v 1 , v 2 ) = − φ ( v 2 , v 1 ) , v i ∈ T p M
このようなφ \varphi φ Λ 2 ( T p ∗ M ) \Lambda^{2} (T_{p}^{\ast}M) Λ 2 ( T p ∗ M )
Λ 2 ( T p ∗ M ) : = { φ : T p M × T p M → R ∣ φ is bilinear and alternate }
\Lambda^{2} (T_{p}^{\ast}M) := \left\{ \varphi : T_{p}M \times T_{p}M \to \mathbb{R}\ | \ \varphi \text{ is bilinear and alternate} \right\}
Λ 2 ( T p ∗ M ) := { φ : T p M × T p M → R ∣ φ is bilinear and alternate }
さて、T p ∗ M T_{p}^{\ast}M T p ∗ M Λ 2 ( T p ∗ M ) \Lambda^{2} (T_{p}^{\ast}M) Λ 2 ( T p ∗ M ) ∧ : T p ∗ M × T p ∗ M → Λ 2 ( T p ∗ M ) \wedge : T_{p}^{\ast}M \times T_{p}^{\ast}M \to \Lambda^{2} (T_{p}^{\ast}M) ∧ : T p ∗ M × T p ∗ M → Λ 2 ( T p ∗ M ) Λ 2 ( T p ∗ M ) \Lambda^{2} (T_{p}^{\ast}M) Λ 2 ( T p ∗ M ) T p ∗ M T_{p}^{\ast}M T p ∗ M φ 1 , φ 2 ∈ T p ∗ M \varphi_{1}, \varphi_{2} \in T_{p}^{\ast}M φ 1 , φ 2 ∈ T p ∗ M Λ 2 ( T p ∗ M ) \Lambda^{2} (T_{p}^{\ast}M) Λ 2 ( T p ∗ M ) ∧ \wedge ∧ ∧ \wedge ∧ ウェッジ積 または外積と呼ばれる。TeXコードは\wedge）
φ 1 ∧ φ 2 ∈ Λ 2 ( T p ∗ M )
\varphi_{1} \wedge \varphi_{2} \in \Lambda^{2} (T_{p}^{\ast}M)
φ 1 ∧ φ 2 ∈ Λ 2 ( T p ∗ M )
( φ 1 ∧ φ 2 ) ( v 1 , v 2 ) = − ( φ 1 ∧ φ 2 ) ( v 2 , v 1 ) , v i ∈ T p M
(\varphi_{1} \wedge \varphi_{2}) (v_{1}, v_{2}) = - (\varphi_{1} \wedge \varphi_{2}) (v_{2}, v_{1}),\quad v_{i} \in T_{p}M
( φ 1 ∧ φ 2 ) ( v 1 , v 2 ) = − ( φ 1 ∧ φ 2 ) ( v 2 , v 1 ) , v i ∈ T p M
∧ \wedge ∧
( φ 1 ∧ φ 2 ) ( v 1 , v 2 ) : = det [ ϕ i ( v j ) ]
(\varphi_{1} \wedge \varphi_{2})(v_{1}, v_{2}) := \det \left[ \phi_{i}(v_{j}) \right]
( φ 1 ∧ φ 2 ) ( v 1 , v 2 ) := det [ ϕ i ( v j ) ]
この時、i i i j j j ウェッジ積∧ \wedge ∧ 。
交代性 ( φ 1 ∧ φ 2 ) ( v 1 , v 2 ) = det [ φ i ( v j ) ] = ∣ φ 1 ( v 1 ) φ 1 ( v 2 ) φ 2 ( v 1 ) φ 2 ( v 2 ) ∣ = − ∣ φ 1 ( v 2 ) φ 1 ( v 1 ) φ 2 ( v 2 ) φ 2 ( v 1 ) ∣ by property of determinant = − ( φ 1 ∧ φ 2 ) ( v 2 , v 1 )
\begin{align*}
(\varphi_{1} \wedge \varphi_{2})(v_{1}, v_{2}) =&\ \det \left[ \varphi_{i}(v_{j}) \right]
\\ =&\ \begin{vmatrix} \varphi_{1}(v_{1}) & \varphi_{1}(v_{2}) \\ \varphi_{2}(v_{1}) & \varphi_{2}(v_{2}) \end{vmatrix}
\\ =&\ - \begin{vmatrix} \varphi_{1}(v_{2}) & \varphi_{1}(v_{1}) \\ \varphi_{2}(v_{2}) & \varphi_{2}(v_{1}) \end{vmatrix} & \text{by property of determinant}
\\ =&\ - (\varphi_{1} \wedge \varphi_{2})(v_{2}, v_{1})
\end{align*}
( φ 1 ∧ φ 2 ) ( v 1 , v 2 ) = = = = det [ φ i ( v j ) ] φ 1 ( v 1 ) φ 2 ( v 1 ) φ 1 ( v 2 ) φ 2 ( v 2 ) − φ 1 ( v 2 ) φ 2 ( v 2 ) φ 1 ( v 1 ) φ 2 ( v 1 ) − ( φ 1 ∧ φ 2 ) ( v 2 , v 1 ) by property of determinant
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線形性 for a ∈ R \text{for } a\in \mathbb{R} for a ∈ R
( φ 1 ∧ φ 2 ) ( a v 1 + v 2 , w ) = ∣ φ 1 ( a v 1 + v 2 ) φ 1 ( w ) φ 2 ( a v 1 + v 2 ) φ 2 ( w ) ∣ = ∣ a φ 1 ( v 1 ) + φ 1 ( v 2 ) φ 1 ( w ) a φ 2 ( v 1 ) + φ 2 ( v 2 ) φ 2 ( w ) ∣ by linearity of φ i = ∣ a φ 1 ( v 1 ) φ 1 ( w ) a φ 2 ( v 1 ) φ 2 ( w ) ∣ + ∣ φ 1 ( v 2 ) φ 1 ( w ) φ 2 ( v 2 ) φ 2 ( w ) ∣ by property of determinant = a ∣ φ 1 ( v 1 ) φ 1 ( w ) φ 2 ( v 1 ) φ 2 ( w ) ∣ + ∣ φ 1 ( v 2 ) φ 1 ( w ) φ 2 ( v 2 ) φ 2 ( w ) ∣ by property of determinant = a ( φ 1 ∧ φ 2 ) ( v 1 , w ) + ( φ 1 ∧ φ 2 ) ( v 2 , w )
\begin{align*}
& (\varphi_{1} \wedge \varphi_{2})(av_{1} + v_{2}, w) \\[1em]
=&\ \begin{vmatrix} \varphi_{1}(av_{1}+v_{2}) & \varphi_{1}(w) \\ \varphi_{2}(av_{1}+v_{2}) & \varphi_{2}(w) \end{vmatrix} \\[1em]
=&\ \begin{vmatrix} a\varphi_{1}(v_{1}) + \varphi_{1}(v_{2}) & \varphi_{1}(w) \\ a\varphi_{2}(v_{1}) + \varphi_{2}(v_{2}) & \varphi_{2}(w) \end{vmatrix} & \text{by linearity of } \varphi_{i} \\[1em]
=&\ \begin{vmatrix} a\varphi_{1}(v_{1}) & \varphi_{1}(w) \\ a\varphi_{2}(v_{1}) & \varphi_{2}(w) \end{vmatrix} + \begin{vmatrix}\varphi_{1}(v_{2}) & \varphi_{1}(w) \\ \varphi_{2}(v_{2}) & \varphi_{2}(w) \end{vmatrix} & \text{by property of determinant} \\[1em]
=&\ a\begin{vmatrix} \varphi_{1}(v_{1}) & \varphi_{1}(w) \\ \varphi_{2}(v_{1}) & \varphi_{2}(w) \end{vmatrix} + \begin{vmatrix} \varphi_{1}(v_{2}) & \varphi_{1}(w) \\ \varphi_{2}(v_{2}) & \varphi_{2}(w) \end{vmatrix} & \text{by property of determinant} \\[1em]
=&\ a(\varphi_{1} \wedge \varphi_{2})(v_{1}, w) + (\varphi_{1} \wedge \varphi_{2})(v_{2}, w)
\end{align*}
= = = = = ( φ 1 ∧ φ 2 ) ( a v 1 + v 2 , w ) φ 1 ( a v 1 + v 2 ) φ 2 ( a v 1 + v 2 ) φ 1 ( w ) φ 2 ( w ) a φ 1 ( v 1 ) + φ 1 ( v 2 ) a φ 2 ( v 1 ) + φ 2 ( v 2 ) φ 1 ( w ) φ 2 ( w ) a φ 1 ( v 1 ) a φ 2 ( v 1 ) φ 1 ( w ) φ 2 ( w ) + φ 1 ( v 2 ) φ 2 ( v 2 ) φ 1 ( w ) φ 2 ( w ) a φ 1 ( v 1 ) φ 2 ( v 1 ) φ 1 ( w ) φ 2 ( w ) + φ 1 ( v 2 ) φ 2 ( v 2 ) φ 1 ( w ) φ 2 ( w ) a ( φ 1 ∧ φ 2 ) ( v 1 , w ) + ( φ 1 ∧ φ 2 ) ( v 2 , w ) by linearity of φ i by property of determinant by property of determinant
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基底 これで、T p ∗ M T_{p}^{\ast}M T p ∗ M { ( d x j ) p } j \left\{ (dx_{j})_{p} \right\}_{j} { ( d x j ) p } j Λ 2 ( T p ∗ M ) \Lambda^{2} (T_{p}^{\ast}M) Λ 2 ( T p ∗ M )
( d x i ∧ d x j ) p = notation ( d x i ) p ∧ ( d x j ) p ∈ Λ 2 ( T p ∗ M )
(dx_{i} \wedge dx_{j})_{p} \overset{\text{notation}}{=} (dx_{i})_{p} \wedge (dx_{j})_{p} \in \Lambda^{2} (T_{p}^{\ast}M)
( d x i ∧ d x j ) p = notation ( d x i ) p ∧ ( d x j ) p ∈ Λ 2 ( T p ∗ M )
すると、{ ( d x i ∧ d x j ) p : i < j } \left\{ (dx_{i} \wedge dx_{j})_{p} : i \lt j \right\} { ( d x i ∧ d x j ) p : i < j } Λ 2 ( T p ∗ M ) \Lambda^{2} (T_{p}^{\ast}M) Λ 2 ( T p ∗ M ) 基底になり 、次が成立する。
( d x i ∧ d x j ) p = − ( d x j ∧ d x i ) p , i ≠ j ( d x i ∧ d x i ) p = 0
(dx_{i} \wedge dx_{j})_{p} = - (dx_{j} \wedge dx_{i})_{p},\quad i \ne j
\\[1em] (dx_{i} \wedge dx_{i})_{p} = 0
( d x i ∧ d x j ) p = − ( d x j ∧ d x i ) p , i = j ( d x i ∧ d x i ) p = 0
これで2次形式を定義する準備ができた。
定義 点 p ∈ M p \in M p ∈ M ω : M → Λ 2 ( T p ∗ M ) \omega : M \to \Lambda^{2} (T_{p}^{\ast}M) ω : M → Λ 2 ( T p ∗ M ) M M M 2次形式 exterior form of degree 2 と定義する。
ω ( p ) = a 12 ( p ) ( d x 1 ∧ d x 2 ) p + a 13 ( p ) ( d x 1 ∧ d x 3 ) p + a 23 ( p ) ( d x 2 ∧ d x 3 ) p
\omega (p) = a_{12}(p)(dx_{1} \wedge dx_{2})_{p} + a_{13}(p)(dx_{1} \wedge dx_{3})_{p} + a_{23}(p)(dx_{2} \wedge dx_{3})_{p}
ω ( p ) = a 12 ( p ) ( d x 1 ∧ d x 2 ) p + a 13 ( p ) ( d x 1 ∧ d x 3 ) p + a 23 ( p ) ( d x 2 ∧ d x 3 ) p
ω \omega ω
ω = a 12 d x 1 ∧ d x 2 + a 13 d x 1 ∧ d x 3 + a 23 d x 2 ∧ d x 3 = a i j d x i ∧ d x j ( i < j ) by Einstein notation
\begin{align*}
\omega =&\ a_{12}dx_{1} \wedge dx_{2} + a_{13}dx_{1} \wedge dx_{3} + a_{23}dx_{2} \wedge dx_{3}
\\ =&\ a_{ij}dx_{i}\wedge dx_{j} (i \lt j) & \text{by Einstein notation}
\end{align*}
ω = = a 12 d x 1 ∧ d x 2 + a 13 d x 1 ∧ d x 3 + a 23 d x 2 ∧ d x 3 a ij d x i ∧ d x j ( i < j ) by Einstein notation
この時、a i j : M → R a_{ij} : M \to \mathbb{R} a ij : M → R a i j a_{ij} a ij 微分可能 であれば、ω \omega ω 2次微分形式 differential form of degree 2 と呼ぶ。
参考 