这篇文章是微分形式，主要目标是为广义相对论的学习服务。参考的教材为梁灿彬的《微分几何与广义相对论》、陈斌的《广义相对论》梅加强的《流形与几何初步》，分别是物理系和数学系的教材。最近 A-SOUL 给我的打击很大，博客停更了一段时间。希望广义相对论的学习能帮助调整状态吧。
微分形式 设 ω ∈ Γ ( ⊗ 0 , p T M ) \omega \in \varGamma \left(\otimes^{0,p} TM\right) ω ∈ Γ ( ⊗ 0 , p T M ) p p p X 1 , ⋯ X p ∈ X ( M ) X_1, \cdots X_p \in \mathfrak{X} (M) X 1 , ⋯ X p ∈ X ( M ) ( 1 , 2 , ⋯ , p ) (1,2,\cdots,p) ( 1 , 2 , ⋯ , p ) π \pi π
ω ( X π ( 1 ) , X π ( 2 ) , ⋯ , X π ( p ) ) = ( − 1 ) sgn ( π ) ω ( X 1 , ⋯ X p ) \omega \left( X_{\pi(1)}, X_{\pi(2)}, \cdots, X_{\pi(p)} \right) = (-1)^{\operatorname{sgn}(\pi)} \omega(X_1, \cdots X_p)
ω ( X π ( 1 ) , X π ( 2 ) , ⋯ , X π ( p ) ) = ( − 1 ) s g n ( π ) ω ( X 1 , ⋯ X p )
其中 sgn ( π ) \operatorname{sgn}(\pi) s g n ( π ) 1 1 1 − 1 -1 − 1 ω \omega ω p p p 反称协变张量 或 p p p 微分形式 。p p p ⋀ p T ∗ M \bigwedge^p T^*M ⋀ p T ∗ M ⋀ p ( M ) \bigwedge^p (M) ⋀ p ( M )
外形式在局部坐标系下的分量也具有反对称性质，设 { x μ } \{x^\mu\} { x μ } M M M X μ = ∂ ∂ x μ X_\mu = \frac{\partial}{\partial x^\mu} X μ = ∂ x μ ∂
ω ( ∂ ∂ x μ π ( 1 ) , ∂ ∂ x μ π ( 2 ) , ⋯ , ∂ ∂ x μ π ( p ) ) = ( − 1 ) sgn ( π ) ω ( ∂ ∂ x μ 1 , ∂ ∂ x μ 2 , ⋯ ∂ ∂ x μ p ) \omega \left( \frac{\partial}{\partial x^{\mu_{\pi(1)}}}, \frac{\partial}{\partial x^{\mu_{\pi(2)}}}, \cdots, \frac{\partial}{\partial x^{\mu_{\pi(p)}}} \right) = (-1)^{\operatorname{sgn}(\pi)} \omega \left(\frac{\partial}{\partial x^{\mu_1}}, \frac{\partial}{\partial x^{\mu_2}},\cdots \frac{\partial}{\partial x^{\mu_p}} \right)
ω ( ∂ x μ π ( 1 ) ∂ , ∂ x μ π ( 2 ) ∂ , ⋯ , ∂ x μ π ( p ) ∂ ) = ( − 1 ) s g n ( π ) ω ( ∂ x μ 1 ∂ , ∂ x μ 2 ∂ , ⋯ ∂ x μ p ∂ )
ω μ π ( 1 ) μ π ( 2 ) ⋯ μ π ( p ) = ( − 1 ) sgn ( π ) ω μ 1 μ 2 ⋯ μ p \omega_{\mu_{\pi(1)}\mu_{\pi(2)}\cdots\mu_{\pi(p)}} = (-1)^{\operatorname{sgn}(\pi)} \omega_{\mu_1 \mu_2 \cdots \mu_p}
ω μ π ( 1 ) μ π ( 2 ) ⋯ μ π ( p ) = ( − 1 ) s g n ( π ) ω μ 1 μ 2 ⋯ μ p
张量的反称化变换 设 ω ∈ Γ ( ⊗ 0 , p T M ) \omega \in \varGamma \left(\otimes^{0,p} TM\right) ω ∈ Γ ( ⊗ 0 , p T M ) p p p ω \omega ω Alt : Γ ( ⊗ 0 , p T M ) → Γ ( ⋀ p T ∗ M ) \operatorname{Alt}: \varGamma \left(\otimes^{0,p} TM\right) \to \varGamma \left(\bigwedge^p T^*M\right) A l t : Γ ( ⊗ 0 , p T M ) → Γ ( ⋀ p T ∗ M ) X 1 , ⋯ X p ∈ X ( M ) X_1, \cdots X_p \in \mathfrak{X} (M) X 1 , ⋯ X p ∈ X ( M )
Alt ( ω ) ( X 1 , ⋯ X p ) = 1 p ! ∑ π ( − 1 ) sgn ( π ) ω ( X π ( 1 ) , ⋯ X π ( p ) ) \operatorname{Alt} (\omega) (X_1, \cdots X_p) = \frac{1}{p!} \sum_\pi (-1)^{\operatorname{sgn}(\pi)} \omega (X_{\pi(1)}, \cdots X_{\pi(p)})
A l t ( ω ) ( X 1 , ⋯ X p ) = p ! 1 π ∑ ( − 1 ) s g n ( π ) ω ( X π ( 1 ) , ⋯ X π ( p ) )
其中 π \pi π ( 1 , 2 , ⋯ , p ) (1,2,\cdots,p) ( 1 , 2 , ⋯ , p )
反称化变换 Alt \operatorname{Alt} A l t
反称化变换 Alt \operatorname{Alt} A l t
Proof
利用置换群的重排定理，某一置换 σ \sigma σ π \pi π
ω \omega ω Alt ( ω ) = ω \operatorname{Alt} (\omega) = \omega A l t ( ω ) = ω
Proof
充分性：由性质1立刻得到。
必要性：利用 π \pi π π − 1 \pi^{-1} π − 1 Alt ω \operatorname{Alt} \omega A l t ω ω ( X π ( 1 ) , ⋯ X π ( p ) ) \omega (X_{\pi(1)}, \cdots X_{\pi(p)}) ω ( X π ( 1 ) , ⋯ X π ( p ) ) ω ( X 1 , ⋯ X p ) \omega (X_1, \cdots X_p) ω ( X 1 , ⋯ X p ) ( − 1 ) sgn ( π − 1 ) (-1)^{\operatorname{sgn}(\pi^{-1})} ( − 1 ) s g n ( π − 1 ) sgn ( π − 1 ) = sgn ( π ) \operatorname{sgn}(\pi^{-1}) = \operatorname{sgn}(\pi) s g n ( π − 1 ) = s g n ( π ) ω \omega ω p ! p! p ! p ! p! p ! ω \omega ω
Alt \operatorname{Alt} A l t Alt 2 = Alt \operatorname{Alt}^2 = \operatorname{Alt} A l t 2 = A l t
Alt \operatorname{Alt} A l t M M M p p p I m ( Alt ) = ⋀ p ( M ) \mathrm{Im}(\operatorname{Alt}) = \bigwedge^p (M) I m ( A l t ) = ⋀ p ( M )
张量分量的反称化变换 张量反称化变换后的分量用中括号 [ ] [ \ ] [ ]
Alt ( ω ) μ 1 μ 2 ⋯ μ p = ω [ μ 1 μ 2 ⋯ μ p ] = 1 p ! ∑ π ( − 1 ) sgn ( π ) ω μ π ( 1 ) μ π ( 2 ) ⋯ μ π ( p ) \operatorname{Alt} (\omega)_{\mu_1 \mu_2 \cdots \mu_p} = \omega_{[\mu_1 \mu_2 \cdots \mu_p]} = \frac{1}{p!} \sum_\pi (-1)^{\operatorname{sgn}(\pi)} \omega_{\mu_{\pi(1)}\mu_{\pi(2)}\cdots\mu_{\pi(p)}}
A l t ( ω ) μ 1 μ 2 ⋯ μ p = ω [ μ 1 μ 2 ⋯ μ p ] = p ! 1 π ∑ ( − 1 ) s g n ( π ) ω μ π ( 1 ) μ π ( 2 ) ⋯ μ π ( p )
因此 ω \omega ω
ω μ 1 μ 2 ⋯ μ p = ω [ μ 1 μ 2 ⋯ μ p ] \omega_{\mu_1 \mu_2 \cdots \mu_p} = \omega_{[\mu_1 \mu_2 \cdots \mu_p]}
ω μ 1 μ 2 ⋯ μ p = ω [ μ 1 μ 2 ⋯ μ p ]
张量的对称化变换 设 ω ∈ Γ ( ⊗ 0 , p T M ) \omega \in \varGamma \left(\otimes^{0,p} TM\right) ω ∈ Γ ( ⊗ 0 , p T M ) p p p ω \omega ω Sym : Γ ( ⊗ 0 , p T M ) → Γ ( ⋀ p T ∗ M ) \operatorname{Sym}: \varGamma \left(\otimes^{0,p} TM\right) \to \varGamma \left(\bigwedge^p T^*M\right) S y m : Γ ( ⊗ 0 , p T M ) → Γ ( ⋀ p T ∗ M ) X 1 , ⋯ X p ∈ X ( M ) X_1, \cdots X_p \in \mathfrak{X} (M) X 1 , ⋯ X p ∈ X ( M )
Sym ( ω ) ( X 1 , ⋯ X p ) = 1 p ! ∑ π ω ( X π ( 1 ) , ⋯ X π ( p ) ) \operatorname{Sym} (\omega) (X_1, \cdots X_p) = \frac{1}{p!} \sum_\pi \omega (X_{\pi(1)}, \cdots X_{\pi(p)})
S y m ( ω ) ( X 1 , ⋯ X p ) = p ! 1 π ∑ ω ( X π ( 1 ) , ⋯ X π ( p ) )
其中 π \pi π ( 1 , 2 , ⋯ , p ) (1,2,\cdots,p) ( 1 , 2 , ⋯ , p )
与反称化变换类似，对称化变换具有性质：
对称化变换 Sym \operatorname{Sym} S y m
ω \omega ω Sym ( ω ) = ω \operatorname{Sym} (\omega) = \omega S y m ( ω ) = ω Sym \operatorname{Sym} S y m Sym 2 = Sym \operatorname{Sym}^2 = \operatorname{Sym} S y m 2 = S y m Sym \operatorname{Sym} S y m M M M p p p
张量分量的对称化变换 张量反称化变换后的分量用小括号 ( ) (\ ) ( )
Sym ( ω ) μ 1 μ 2 ⋯ μ p = ω ( μ 1 μ 2 ⋯ μ p ) = 1 p ! ∑ π ω μ π ( 1 ) μ π ( 2 ) ⋯ μ π ( p ) \operatorname{Sym} (\omega)_{\mu_1 \mu_2 \cdots \mu_p} = \omega_{(\mu_1 \mu_2 \cdots \mu_p)} = \frac{1}{p!} \sum_\pi \omega_{\mu_{\pi(1)}\mu_{\pi(2)}\cdots\mu_{\pi(p)}}
S y m ( ω ) μ 1 μ 2 ⋯ μ p = ω ( μ 1 μ 2 ⋯ μ p ) = p ! 1 π ∑ ω μ π ( 1 ) μ π ( 2 ) ⋯ μ π ( p )
因此 ω \omega ω
ω μ 1 μ 2 ⋯ μ p = ω ( μ 1 μ 2 ⋯ μ p ) \omega_{\mu_1 \mu_2 \cdots \mu_p} = \omega_{(\mu_1 \mu_2 \cdots \mu_p)}
ω μ 1 μ 2 ⋯ μ p = ω ( μ 1 μ 2 ⋯ μ p )
下面给出一些关于小括号和中括号的性质：
缩并时括号具有传染性：T [ μ 1 ⋯ μ p ] S μ 1 ⋯ μ p = T [ μ 1 ⋯ μ p ] S [ μ 1 ⋯ μ p ] = T μ 1 ⋯ μ p S [ a 1 ⋯ a l ] T_{\left[\mu_1 \cdots \mu_p\right]} S^{\mu_1 \cdots \mu_p}=T_{\left[\mu_1 \cdots \mu_p\right]} S^{\left[\mu_1 \cdots \mu_p\right]}=T_{\mu_1 \cdots \mu_p} S^{\left[a_{1} \cdots a_{l}\right]} T [ μ 1 ⋯ μ p ] S μ 1 ⋯ μ p = T [ μ 1 ⋯ μ p ] S [ μ 1 ⋯ μ p ] = T μ 1 ⋯ μ p S [ a 1 ⋯ a l ]
括号内的同种子括号可以随意增删：T ( μ ( ν ρ ) σ ) = T ( μ ν ρ σ ) T_{(\mu (\nu \rho) \sigma)} = T_{(\mu \nu \rho\sigma)} T ( μ ( ν ρ ) σ ) = T ( μ ν ρ σ )
括号内夹异种子括号得零：T [ μ ( ν ρ ) σ ] = 0 T_{[\mu (\nu \rho) \sigma]} = 0 T [ μ ( ν ρ ) σ ] = 0
异种括号缩并得零：T [ μ 1 ⋯ μ p ] S ( μ 1 ⋯ μ p ) = 0 T_{\left[\mu_1 \cdots \mu_p\right]} S^{\left(\mu_1 \cdots \mu_p\right)}=0 T [ μ 1 ⋯ μ p ] S ( μ 1 ⋯ μ p ) = 0
外积 设 ω , η \omega,\eta ω , η p , q p,q p , q p + q p+q p + q
ω ∧ η = ( p + q ) ! p ! q ! Alt ( ω ⊗ η ) \omega \wedge \eta = \frac{(p+q)!}{p!q!} \operatorname{Alt} (\omega \otimes \eta)
ω ∧ η = p ! q ! ( p + q ) ! A l t ( ω ⊗ η )
映射 ∧ : ⋀ p ( M ) × ⋀ q ( M ) → ⋀ p + q ( M ) \wedge:\bigwedge^p (M) \times \bigwedge^q (M) \to \bigwedge^{p+q} (M) ∧ : ⋀ p ( M ) × ⋀ q ( M ) → ⋀ p + q ( M ) 外积 运算。
外积 ω ∧ η \omega \wedge \eta ω ∧ η
( ω ∧ η ) μ 1 ⋯ μ p ν 1 ⋯ ν q = ( p + q ) ! p ! q ! ω [ μ 1 ⋯ μ p η ν 1 ⋯ ν q ] (\omega \wedge \eta)_{\mu_1 \cdots \mu_p \nu_1 \cdots \nu_q} =\frac{(p+q)!}{p!q!} \omega_{\left[\mu_1 \cdots \mu_p \right.} \eta_{\left. \nu_1 \cdots \nu_q \right]}
( ω ∧ η ) μ 1 ⋯ μ p ν 1 ⋯ ν q = p ! q ! ( p + q ) ! ω [ μ 1 ⋯ μ p η ν 1 ⋯ ν q ]
外积具有性质：
双线性：ω ∧ ( η + θ ) = ω ∧ η + ω ∧ θ \omega \wedge (\eta + \theta) = \omega \wedge \eta + \omega \wedge \theta ω ∧ ( η + θ ) = ω ∧ η + ω ∧ θ ( ω + η ) ∧ θ = ω ∧ θ + η ∧ θ (\omega + \eta) \wedge \theta = \omega \wedge \theta + \eta \wedge \theta ( ω + η ) ∧ θ = ω ∧ θ + η ∧ θ ω ∧ ( λ η ) = ( λ ω ) ∧ η = λ ( ω ∧ η ) \omega \wedge (\lambda \eta) = (\lambda \omega) \wedge \eta = \lambda (\omega \wedge \eta) ω ∧ ( λ η ) = ( λ ω ) ∧ η = λ ( ω ∧ η )
非交换律：ω ∧ η = ( − 1 ) p q η ∧ ω \omega \wedge \eta = (-1)^{pq} \eta \wedge \omega ω ∧ η = ( − 1 ) p q η ∧ ω ω , η \omega,\eta ω , η p , q p,q p , q
结合律：ω ∧ ( η ∧ θ ) = ( ω ∧ η ) ∧ θ \omega \wedge (\eta \wedge \theta) = (\omega \wedge \eta) \wedge \theta ω ∧ ( η ∧ θ ) = ( ω ∧ η ) ∧ θ
拉回映射分配律：ϕ ∗ ( ω ∧ η ) = ϕ ∗ ω ∧ ϕ ∗ η \phi^* (\omega \wedge \eta) = \phi^*\omega \wedge \phi^* \eta ϕ ∗ ( ω ∧ η ) = ϕ ∗ ω ∧ ϕ ∗ η ϕ : M → N \phi:M \to N ϕ : M → N ω , η \omega,\eta ω , η N N N
由此可以得到三个推论：
推论一：ω 1 ∧ ⋯ ∧ ω n = ( p 1 + ⋯ + p n ) p 1 ! ⋯ p n ! Alt ( ω 1 ⊗ ⋯ ⊗ ω n ) \omega_1 \wedge \cdots \wedge \omega_n = \dfrac{\left( p_1 + \cdots + p_n \right)}{p_1 ! \cdots p_n! }\operatorname{Alt} (\omega_1 \otimes \cdots \otimes \omega_n) ω 1 ∧ ⋯ ∧ ω n = p 1 ! ⋯ p n ! ( p 1 + ⋯ + p n ) A l t ( ω 1 ⊗ ⋯ ⊗ ω n ) ω i \omega_i ω i p i p_i p i
推论二：d x μ 1 ∧ ⋯ ∧ d x μ p = ∑ π ( − 1 ) sgn ( π ) d x μ π ( 1 ) ⊗ ⋯ ⊗ d x μ π ( p ) \mathrm{d} x^{\mu_1} \wedge \cdots \wedge \mathrm{d} x^{\mu_p} = \sum\limits_\pi (-1)^{\operatorname{sgn}(\pi)} \mathrm{d} x^{\mu_{\pi(1)}} \otimes \cdots \otimes \mathrm{d} x^{\mu_{\pi(p)}} d x μ 1 ∧ ⋯ ∧ d x μ p = π ∑ ( − 1 ) s g n ( π ) d x μ π ( 1 ) ⊗ ⋯ ⊗ d x μ π ( p )
Proof
根据推论一以及反称化变换的定义，立刻得到
d x μ 1 ∧ ⋯ ∧ d x μ p = p ! Alt ( d x μ 1 ⊗ ⋯ ⊗ d x μ p ) = p ! ⋅ 1 p ! ∑ π ( − 1 ) sgn ( π ) d x μ π ( 1 ) ⊗ ⋯ ⊗ d x μ π ( p ) = ∑ π ( − 1 ) sgn ( π ) d x μ π ( 1 ) ⊗ ⋯ ⊗ d x μ π ( p ) \begin{aligned}
\mathrm{d} x^{\mu_1} \wedge \cdots \wedge \mathrm{d} x^{\mu_p} &= p!\operatorname{Alt} (\mathrm{d} x^{\mu_1} \otimes \cdots \otimes \mathrm{d} x^{\mu_p}) \\
&= p! \cdot \frac{1}{p!} \sum_\pi (-1)^{\operatorname{sgn}(\pi)} \mathrm{d} x^{\mu_{\pi(1)}} \otimes \cdots \otimes \mathrm{d} x^{\mu_{\pi(p)}} \\
&= \sum_\pi (-1)^{\operatorname{sgn}(\pi)} \mathrm{d} x^{\mu_{\pi(1)}} \otimes \cdots \otimes \mathrm{d} x^{\mu_{\pi(p)}}
\end{aligned}
d x μ 1 ∧ ⋯ ∧ d x μ p = p ! A l t ( d x μ 1 ⊗ ⋯ ⊗ d x μ p ) = p ! ⋅ p ! 1 π ∑ ( − 1 ) s g n ( π ) d x μ π ( 1 ) ⊗ ⋯ ⊗ d x μ π ( p ) = π ∑ ( − 1 ) s g n ( π ) d x μ π ( 1 ) ⊗ ⋯ ⊗ d x μ π ( p )
推论三：设 ω \omega ω p p p ω \omega ω
ω = ω μ 1 ⋯ μ p d x μ 1 ⊗ ⋯ ⊗ d x μ p \omega = \omega_{\mu_1\cdots \mu_p} \, \mathrm{d} x^{\mu_1} \otimes \cdots \otimes \mathrm{d} x^{\mu_p}
ω = ω μ 1 ⋯ μ p d x μ 1 ⊗ ⋯ ⊗ d x μ p
ω = ∑ μ 1 < ⋯ < μ p ω μ 1 ⋯ μ p d x μ 1 ∧ ⋯ ∧ d x μ p \omega = \sum_{\mu_1 < \cdots<\mu_p} \omega_{\mu_1\cdots \mu_p}\, \mathrm{d} x^{\mu_1} \wedge \cdots \wedge\mathrm{d} x^{\mu_p}
ω = μ 1 < ⋯ < μ p ∑ ω μ 1 ⋯ μ p d x μ 1 ∧ ⋯ ∧ d x μ p
ω = 1 p ! ω μ 1 ⋯ μ p d x μ 1 ∧ ⋯ ∧ d x μ p \omega = \frac{1}{p!}\omega_{\mu_1\cdots \mu_p} \, \mathrm{d} x^{\mu_1} \wedge \cdots \wedge\mathrm{d} x^{\mu_p}
ω = p ! 1 ω μ 1 ⋯ μ p d x μ 1 ∧ ⋯ ∧ d x μ p
其中重复指标代表每个指标都在 [ 0 , dim M ] [0,\dim M] [ 0 , dim M ]
例：考虑 4 维流形上的 2-形式场 ω = ω μ ν d x μ ⊗ d x ν \omega = \omega_{\mu\nu} \, \mathrm{d} x^{\mu} \otimes \mathrm{d} x^\nu ω = ω μ ν d x μ ⊗ d x ν ( ω μ ν ) = ( O 2 × 2 − I 2 × 2 I 2 × 2 O 2 × 2 ) (\omega_{\mu\nu}) = \begin{pmatrix}
O_{2\times2} & -I_{2\times2}\\
I_{2\times2} & O_{2\times2}
\end{pmatrix} ( ω μ ν ) = ( O 2 × 2 I 2 × 2 − I 2 × 2 O 2 × 2 ) ∧ \wedge ∧ ω \omega ω
首先将之展开，得到
ω = − d x 0 ⊗ d x 2 − d x 1 ⊗ d x 3 + d x 2 ⊗ d x 0 + d x 3 ⊗ d x 1 \omega = -\mathrm{d} x^0 \otimes \mathrm{d} x^2 -\mathrm{d} x^1 \otimes \mathrm{d} x^3 + \mathrm{d} x^2 \otimes \mathrm{d} x^0 + \mathrm{d} x^3 \otimes \mathrm{d} x^1
ω = − d x 0 ⊗ d x 2 − d x 1 ⊗ d x 3 + d x 2 ⊗ d x 0 + d x 3 ⊗ d x 1
另一方面，由于 ( ω μ ν ) (\omega_{\mu\nu}) ( ω μ ν ) ω \omega ω
根据推论二，可得 d x μ ∧ d x ν = d x μ ⊗ d x ν − d x ν ⊗ d x μ \mathrm{d} x^\mu \wedge \mathrm{d} x^\nu = \mathrm{d} x^\mu \otimes\mathrm{d} x^\nu - \mathrm{d} x^\nu \otimes\mathrm{d} x^\mu d x μ ∧ d x ν = d x μ ⊗ d x ν − d x ν ⊗ d x μ μ , ν \mu,\nu μ , ν ω \omega ω
ω = d x 2 ∧ d x 0 + d x 3 ∧ d x 1 = ∑ μ < ν ω μ ν d x μ ∧ d x ν = 1 2 ω μ ν d x μ ∧ d x ν \omega = \mathrm{d} x^2 \wedge \mathrm{d} x^0 + \mathrm{d} x^3 \wedge \mathrm{d} x^1 = \sum_{\mu<\nu} \omega_{\mu\nu} \mathrm{d} x^\mu \wedge \mathrm{d} x^\nu = \frac{1}{2}\omega_{\mu\nu} \mathrm{d} x^\mu \wedge \mathrm{d} x^\nu
ω = d x 2 ∧ d x 0 + d x 3 ∧ d x 1 = μ < ν ∑ ω μ ν d x μ ∧ d x ν = 2 1 ω μ ν d x μ ∧ d x ν
外微分 设 ω \omega ω p p p d : ⋀ p ( M ) → ⋀ p + 1 ( M ) \mathrm{d}: \bigwedge^p (M) \to \bigwedge^{p+1} (M) d : ⋀ p ( M ) → ⋀ p + 1 ( M ) X 1 , ⋯ , X p + 1 ∈ X ( M ) X_1, \cdots, X_{p+1} \in \mathfrak{X} (M) X 1 , ⋯ , X p + 1 ∈ X ( M )
d ω ( X 1 , ⋯ X p + 1 ) = ∑ i = 1 p + 1 ( − 1 ) i − 1 X i ω ( X 1 , ⋯ , X i , ⋯ , X p + 1 ) + ∑ i < j ( − 1 ) i − 1 ω ( X 1 , ⋯ , X i , ⋯ , [ X i , X j ] , ⋯ , X p + 1 ) = ∑ i = 1 p + 1 ( − 1 ) i − 1 ∇ X i ω ( X 1 , ⋯ , X i , ⋯ , X p + 1 ) \begin{aligned}
\mathrm{d} \omega \left(X_1, \cdots X_{p+1} \right) & = \sum_{i=1}^{p+1} (-1)^{i-1} X_i \omega \left(X_1, \cdots,\cancel{X_i}, \cdots,X_{p+1} \right) + \sum_{i<j} (-1)^{i-1} \omega \left( X_1, \cdots,\cancel{X_i},\cdots,\left[X_i,X_j\right],\cdots,X_{p+1}\right) \\
& = \sum_{i=1}^{p+1}(-1)^{i-1} \nabla_{X_i} \omega\left(X_{1}, \cdots, \cancel{X_i}, \cdots, X_{p+1}\right)
\end{aligned}
d ω ( X 1 , ⋯ X p + 1 ) = i = 1 ∑ p + 1 ( − 1 ) i − 1 X i ω ( X 1 , ⋯ , X i , ⋯ , X p + 1 ) + i < j ∑ ( − 1 ) i − 1 ω ( X 1 , ⋯ , X i , ⋯ , [ X i , X j ] , ⋯ , X p + 1 ) = i = 1 ∑ p + 1 ( − 1 ) i − 1 ∇ X i ω ( X 1 , ⋯ , X i , ⋯ , X p + 1 )
其中 X \cancel{X} X ∇ \nabla ∇ d ω \mathrm{d} \omega d ω ω \omega ω 外微分 。
设 { x μ } \{x^\mu\} { x μ } M M M d ω \mathrm{d} \omega d ω
d ω = ∑ μ 1 < ⋯ < μ p ∂ ω μ 1 ⋯ μ p ∂ x ν d x ν ∧ d x μ 1 ∧ ⋯ ∧ d x μ p \mathrm{d} \omega = \sum_{\mu_1 < \cdots<\mu_p}\frac{\partial\omega_{\mu_1\cdots\mu_p}}{\partial x^\nu} \mathrm{d} x^\nu \wedge \mathrm{d} x^{\mu_1} \wedge \cdots \wedge\mathrm{d} x^{\mu_p}
d ω = μ 1 < ⋯ < μ p ∑ ∂ x ν ∂ ω μ 1 ⋯ μ p d x ν ∧ d x μ 1 ∧ ⋯ ∧ d x μ p
( d ω ) μ 1 ⋯ μ p + 1 = ∑ i = 1 p + 1 ( − 1 ) i − 1 ∂ μ i ω μ 1 ⋯ μ i − 1 μ i + 1 ⋯ μ p + 1 (\mathrm{d} \omega)_{\mu_1\cdots\mu_{p+1}} = \sum_{i=1}^{p+1} (-1)^{i-1}\partial_{\mu_i} \omega_{\mu_1 \cdots\mu_{i-1} \mu_{i+1} \cdots \mu_{p+1}}
( d ω ) μ 1 ⋯ μ p + 1 = i = 1 ∑ p + 1 ( − 1 ) i − 1 ∂ μ i ω μ 1 ⋯ μ i − 1 μ i + 1 ⋯ μ p + 1
( d ω ) μ 1 ⋯ μ p + 1 = ( p + 1 ) ∇ [ μ 1 ω μ 2 ⋯ μ p + 1 ] (\mathrm{d} \omega)_{\mu_1\cdots\mu_{p+1}} = (p+1) \nabla_{[\mu_1} \omega_{\mu_2\cdots\mu_{p+1}]}
( d ω ) μ 1 ⋯ μ p + 1 = ( p + 1 ) ∇ [ μ 1 ω μ 2 ⋯ μ p + 1 ]
如果 d ω = 0 \mathrm{d} \omega = 0 d ω = 0 ω \omega ω 闭形式 ；如果 ω = d η \omega = \mathrm{d}\eta ω = d η ω \omega ω 恰当形式 。恰当形式必为闭形式。所有 M M M p p p M M M p p p H d R p ( M ) H^p_{\mathrm{d} R} (M) H d R p ( M )
外微分具有性质：
d ( ω ∧ η ) = d ω ∧ η + ( − 1 ) p ω ∧ d η \mathrm{d} (\omega \wedge \eta)=\mathrm{d} \omega \wedge \eta+(-1)^{p} \omega \wedge \mathrm{d} \eta d ( ω ∧ η ) = d ω ∧ η + ( − 1 ) p ω ∧ d η ω \omega ω p p p Poincare 引理：d 2 ≡ 0 \mathrm{d}^2 \equiv 0 d 2 ≡ 0
外微分和拉回映射可交换：d ( ϕ ∗ ω ) = ϕ ∗ ( d ω ) \mathrm{d}(\phi^* \omega) = \phi^* (\mathrm{d} \omega) d ( ϕ ∗ ω ) = ϕ ∗ ( d ω ) ϕ : M → N \phi:M \to N ϕ : M → N ω \omega ω N N N
外微分和李导数可以交换：d ∘ L X = L X ∘ d \mathrm{d} \circ \mathcal{L}_X = \mathcal{L}_X \circ \mathrm{d} d ∘ L X = L X ∘ d X X X
内乘 设 ω \omega ω p p p X X X p − 1 p-1 p − 1 ι X ω \iota_X \omega ι X ω Y 1 , ⋯ , Y p − 1 ∈ X ( M ) Y_1,\cdots,Y_{p-1} \in \mathfrak{X} (M) Y 1 , ⋯ , Y p − 1 ∈ X ( M )
ι X ω ( Y 1 , ⋯ , Y p − 1 ) = ω ( X , Y 1 , ⋯ , Y p − 1 ) \iota_X \omega \left( Y_1,\cdots,Y_{p-1}\right) = \omega \left(X,Y_1,\cdots,Y_{p-1}\right)
ι X ω ( Y 1 , ⋯ , Y p − 1 ) = ω ( X , Y 1 , ⋯ , Y p − 1 )
称 ι : Γ ( T M ) × ⋀ p ( M ) → ⋀ p − 1 ( M ) \iota: \varGamma (TM) \times \bigwedge^p (M) \to \bigwedge^{p-1} (M) ι : Γ ( T M ) × ⋀ p ( M ) → ⋀ p − 1 ( M )
内乘具有以下性质：
ι X ∘ ι X = 0 \iota_X \circ \iota_X = 0 ι X ∘ ι X = 0
ι f X ( ω ) = ι X ( f ω ) = f ι X ( ω ) \iota_{fX} (\omega) = \iota_X (f\omega) = f \iota_X (\omega) ι f X ( ω ) = ι X ( f ω ) = f ι X ( ω ) f f f
ι X ( ω ∧ η ) = ι X ω ∧ η + ( − 1 ) p ω ∧ ι X η \iota_{X}(\omega \wedge \eta)=\iota_{X}\omega \wedge \eta+(-1)^{p} \omega \wedge \iota_{X} \eta ι X ( ω ∧ η ) = ι X ω ∧ η + ( − 1 ) p ω ∧ ι X η ω \omega ω p p p
设 { x μ } \{x^\mu\} { x μ } M M M
ι ∂ ∂ x μ ( d x μ 1 ∧ ⋯ ∧ d x μ p ) = { 0 , μ ∉ { μ i } i = 1 p , ( − 1 ) i − 1 d x μ 1 ∧ ⋯ ∧ d x μ i ∧ ⋯ ∧ d x μ p , μ = μ i \iota_{\frac{\partial}{\partial x^\mu}}\left(\mathrm{d} x^{\mu_1} \wedge \cdots \wedge \mathrm{d} x^{\mu_p}\right)= \begin{cases} 0, & \mu \notin\left\{\mu_i\right\}_{i=1}^{p}, \\ (-1)^{i-1} \mathrm{d} x^{\mu_1} \wedge \cdots \wedge \cancel{\mathrm{d}x^{\mu_i}}\wedge \cdots \wedge \mathrm{d} x^{\mu_p}, & \mu=\mu_{i}\end{cases}
ι ∂ x μ ∂ ( d x μ 1 ∧ ⋯ ∧ d x μ p ) = { 0 , ( − 1 ) i − 1 d x μ 1 ∧ ⋯ ∧ d x μ i ∧ ⋯ ∧ d x μ p , μ ∈ / { μ i } i = 1 p , μ = μ i
L X = d ∘ ι X + ι X ∘ d \mathcal{L}_X = \mathrm{d} \circ \iota_X + \iota_X \circ \mathrm{d} L X = d ∘ ι X + ι X ∘ d
同向 设 M M M ( U α , φ α ) (U_\alpha,\varphi_\alpha) ( U α , φ α ) ( U β , φ β ) (U_\beta,\varphi_\beta) ( U β , φ β ) U α ∩ U β U_\alpha \cap U_\beta U α ∩ U β
det J ( φ β ∘ φ α − 1 ) > 0 \operatorname{det} J\left(\varphi_{\beta} \circ \varphi_{\alpha}^{-1}\right) >0
d e t J ( φ β ∘ φ α − 1 ) > 0
则称这两个局部坐标系是同向 的。
可定向流形 设 M M M M M M { ( U α , φ α ) } \{(U_\alpha, \varphi_\alpha)\} { ( U α , φ α ) } U α ∩ U β ≠ ∅ U_\alpha \cap U_\beta \neq \varnothing U α ∩ U β = ∅ det J ( φ β ∘ φ α − 1 ) > 0 \operatorname{det} J\left(\varphi_{\beta} \circ \varphi_{\alpha}^{-1}\right) >0 d e t J ( φ β ∘ φ α − 1 ) > 0 M M M { ( U α , φ α ) } \{(U_\alpha, \varphi_\alpha)\} { ( U α , φ α ) } M M M
定向 设 M M M D \mathscr{D} D D \mathscr{D} D D \mathscr{D} D D \mathscr{D} D M M M M M M
定理：连通的可定向微分流形恰好有两个定向。
体积形式 n n n M M M n n n Ω \Omega Ω 体积形式 。
选定体积形式 Ω \Omega Ω M M M 已定向 的，每个局部坐标系 ( U α , φ α ) (U_\alpha,\varphi_\alpha) ( U α , φ α ) Ω ( ∂ ∂ x α 1 , ∂ ∂ x α 2 , ⋯ ∂ ∂ x α n ) \Omega \left(\frac{\partial}{\partial x^1_\alpha}, \frac{\partial}{\partial x^2_\alpha},\cdots \frac{\partial}{\partial x^n_\alpha} \right) Ω ( ∂ x α 1 ∂ , ∂ x α 2 ∂ , ⋯ ∂ x α n ∂ ) Ω \Omega Ω M M M Ω ( ∂ ∂ x α 1 , ∂ ∂ x α 2 , ⋯ ∂ ∂ x α n ) > 0 \Omega \left(\frac{\partial}{\partial x^1_\alpha}, \frac{\partial}{\partial x^2_\alpha},\cdots \frac{\partial}{\partial x^n_\alpha} \right)>0 Ω ( ∂ x α 1 ∂ , ∂ x α 2 ∂ , ⋯ ∂ x α n ∂ ) > 0 ( U α , φ α ) (U_\alpha,\varphi_\alpha) ( U α , φ α ) 右手 的，否则称为左手 的。
两个体积形式 Ω 1 \Omega_1 Ω 1 Ω 2 \Omega_2 Ω 2
sgn Ω 1 ( ∂ ∂ x α 1 , ∂ ∂ x α 2 , ⋯ ∂ ∂ x α n ) = sgn Ω 2 ( ∂ ∂ x α 1 , ∂ ∂ x α 2 , ⋯ ∂ ∂ x α n ) ∀ α ∈ Γ \operatorname{sgn} \Omega_1 \left(\frac{\partial}{\partial x^1_\alpha}, \frac{\partial}{\partial x^2_\alpha},\cdots \frac{\partial}{\partial x^n_\alpha} \right) = \operatorname{sgn} \Omega_2 \left(\frac{\partial}{\partial x^1_\alpha}, \frac{\partial}{\partial x^2_\alpha},\cdots \frac{\partial}{\partial x^n_\alpha} \right) \qquad \forall \alpha\in \Gamma
s g n Ω 1 ( ∂ x α 1 ∂ , ∂ x α 2 ∂ , ⋯ ∂ x α n ∂ ) = s g n Ω 2 ( ∂ x α 1 ∂ , ∂ x α 2 ∂ , ⋯ ∂ x α n ∂ ) ∀ α ∈ Γ
相容的体积形式给出流形的同一个定向。
定理：n n n M M M n n n
Proof
在 U α ∩ U β U_\alpha \cap U_\beta U α ∩ U β
Ω ( ∂ ∂ x β ′ 1 , ⋯ ∂ ∂ x β ′ n ) = det J ( φ β ∘ φ α − 1 ) Ω ( ∂ ∂ x α 1 , ∂ ∂ x α 2 , ⋯ ∂ ∂ x α n ) \Omega \left(\frac{\partial}{\partial x^{\prime 1}_\beta}, \cdots \frac{\partial}{\partial x^{\prime n}_\beta} \right) = \operatorname{det} J\left(\varphi_{\beta} \circ \varphi_{\alpha}^{-1}\right) \Omega \left(\frac{\partial}{\partial x^1_\alpha}, \frac{\partial}{\partial x^2_\alpha},\cdots \frac{\partial}{\partial x^n_\alpha} \right)
Ω ( ∂ x β ′ 1 ∂ , ⋯ ∂ x β ′ n ∂ ) = d e t J ( φ β ∘ φ α − 1 ) Ω ( ∂ x α 1 ∂ , ∂ x α 2 ∂ , ⋯ ∂ x α n ∂ )
因此 det J ( φ β ∘ φ α − 1 ) > 0 \operatorname{det} J\left(\varphi_{\beta} \circ \varphi_{\alpha}^{-1}\right) >0 d e t J ( φ β ∘ φ α − 1 ) > 0
与度规适配的体积形式 设 ( M , g ) (M, g) ( M , g )
ϵ = G d x 1 ∧ ⋯ ∧ d x n \epsilon = \sqrt{G} \mathrm{d} x^1 \wedge \cdots \wedge \mathrm{d} x^n
ϵ = G d x 1 ∧ ⋯ ∧ d x n
其中 { x μ } \{x^\mu\} { x μ } M M M g = g μ ν d x μ ⊗ d x ν g = g_{\mu\nu} \mathrm{d} x^\mu \otimes \mathrm{d} x^\nu g = g μ ν d x μ ⊗ d x ν G = det ( g μ ν ) G = \det (g_{\mu\nu}) G = det ( g μ ν ) ϵ \epsilon ϵ 与度规适配的体积形式 ，也称为 Levi-Civita 张量 。
/* 注：不能混淆 Levi-Civita 张量和 Levi-Civita 符号！后者仅仅是置换逆序数的简记而已 */
当流形 ( M , g ) (M, g) ( M , g )
ϵ = ∣ G ∣ d x 1 ∧ ⋯ ∧ d x n \epsilon = \sqrt{|G|} \,\mathrm{d} x^1 \wedge \cdots \wedge \mathrm{d} x^n
ϵ = ∣ G ∣ d x 1 ∧ ⋯ ∧ d x n
Levi- Civita 张量 ϵ \epsilon ϵ
体积形式 ϵ \epsilon ϵ
Proof
设 ( U α , φ α ) (U_\alpha,\varphi_\alpha) ( U α , φ α ) ( U β , φ β ) (U_\beta,\varphi_\beta) ( U β , φ β ) M M M U α ∩ U β ≠ ∅ U_\alpha \cap U_\beta \neq \varnothing U α ∩ U β = ∅ ϵ \epsilon ϵ
ϵ = det ( g μ ν α ) d x α 1 ∧ ⋯ ∧ d x α n = det ( g μ ν β ) d x β 1 ∧ ⋯ ∧ d x β n \epsilon = \sqrt{\det (g_{\mu\nu}^\alpha)} \mathrm{d} x^1_\alpha \wedge \cdots \wedge \mathrm{d} x^n_\alpha = \sqrt{\det \left(g_{\mu\nu}^\beta \right)} \mathrm{d} x^1_\beta \wedge \cdots \wedge \mathrm{d} x^n_\beta
ϵ = det ( g μ ν α ) d x α 1 ∧ ⋯ ∧ d x α n = det ( g μ ν β ) d x β 1 ∧ ⋯ ∧ d x β n
在 U α ∩ U β U_\alpha \cap U_\beta U α ∩ U β
d x β 1 ∧ ⋯ ∧ d x β n = det J ( φ β ∘ φ α − 1 ) d x α 1 ∧ ⋯ ∧ d x α n \mathrm{d} x^1_\beta \wedge \cdots \wedge \mathrm{d} x^n_\beta = \operatorname{det} J\left(\varphi_{\beta} \circ \varphi_{\alpha}^{-1}\right) \mathrm{d} x^1_\alpha \wedge \cdots \wedge \mathrm{d} x^n_\alpha
d x β 1 ∧ ⋯ ∧ d x β n = d e t J ( φ β ∘ φ α − 1 ) d x α 1 ∧ ⋯ ∧ d x α n
det ( g i j β ) n × n = [ det J ( φ α ∘ φ β − 1 ) ] 2 det ( g i j α ) n × n \operatorname{det}\left(g_{i j}^{\beta}\right)_{n \times n}=\left[\operatorname{det} J\left(\varphi_{\alpha} \circ \varphi_{\beta}^{-1}\right)\right]^{2} \operatorname{det}\left(g_{i j}^{\alpha}\right)_{n \times n}
d e t ( g i j β ) n × n = [ d e t J ( φ α ∘ φ β − 1 ) ] 2 d e t ( g i j α ) n × n
det J ( φ β ∘ φ α − 1 ) ⋅ det J ( φ α ∘ φ β − 1 ) = 1 \operatorname{det} J\left(\varphi_{\beta} \circ \varphi_{\alpha}^{-1}\right) \cdot\operatorname{det} J\left(\varphi_{\alpha} \circ \varphi_{\beta}^{-1}\right)=1
d e t J ( φ β ∘ φ α − 1 ) ⋅ d e t J ( φ α ∘ φ β − 1 ) = 1
因此二者是相容的，故体积形式 ϵ \epsilon ϵ
若局部坐标基是标准正交的，则 ϵ \epsilon ϵ
ϵ μ 1 ⋯ μ n ϵ μ 1 ⋯ μ n = ( − 1 ) s n ! \epsilon^{\mu_1\cdots\mu_n}\epsilon_{\mu_1\cdots\mu_n} = (-1)^s n! ϵ μ 1 ⋯ μ n ϵ μ 1 ⋯ μ n = ( − 1 ) s n ! s s s g μ ν g_{\mu\nu} g μ ν − 1 -1 − 1 s = 0 s=0 s = 0
带边流形 令 H + n = { x = ( x 1 , x 2 , ⋯ , x n ) ∈ R n ∣ x n ≥ 0 } \mathbb{H}_{+}^{n}=\left\{x=\left(x^{1}, x^{2}, \cdots, x^{n}\right) \in \mathbb{R}^{n} \mid x^{n} \ge 0\right\} H + n = { x = ( x 1 , x 2 , ⋯ , x n ) ∈ R n ∣ x n ≥ 0 } R n \R^n R n ∂ H + n = { x ∈ R n ∣ x n = 0 } \partial\mathbb{H}_+^n = \left\{x \in \mathbb{R}^{n} \mid x^{n} = 0\right\} ∂ H + n = { x ∈ R n ∣ x n = 0 } n − 1 n-1 n − 1 M M M A 2 A_2 A 2 T 2 T_2 T 2 M M M { U α } α ∈ Γ \{U_\alpha\}_{\alpha \in \Gamma} { U α } α ∈ Γ φ α : U α → H + n \varphi_\alpha : U_\alpha \to \mathbb{H}_+^n φ α : U α → H + n
映射 φ α : U α → φ ( U α ) ⊂ H + n \varphi_\alpha : U_\alpha \to \varphi\left(U_\alpha\right) \subset\mathbb{H}_+^n φ α : U α → φ ( U α ) ⊂ H + n U α U_\alpha U α H + n \mathbb{H}_{+}^{n} H + n
当 U α ∩ U β ≠ ∅ U_\alpha \cap U_\beta \neq \varnothing U α ∩ U β = ∅ φ β ∘ φ α − 1 \varphi_{\beta} \circ \varphi_{\alpha}^{-1} φ β ∘ φ α − 1 C r C^r C r
则称 M M M C r C^r C r 带边流形 ，其边界 ∂ M \partial M ∂ M
∂ M = { p ∈ M ∣ ∃ α ∈ Γ , φ α ( p ) ∈ ∂ H + n } \partial M = \left\{ p \in M \mid \exist \alpha \in \Gamma , \varphi_\alpha(p) \in \partial\mathbb{H}_+^n \right\}
∂ M = { p ∈ M ∣ ∃ α ∈ Γ , φ α ( p ) ∈ ∂ H + n }
其上的局部坐标转换映射为 φ α ∣ U α ∩ ∂ M \varphi_\alpha|_{U_\alpha \cap \partial M} φ α ∣ U α ∩ ∂ M x n x^n x n U α ∩ ∂ M U_\alpha \cap \partial M U α ∩ ∂ M M M M n n n ∂ M \partial M ∂ M n − 1 n-1 n − 1
诱导定向 若体积形式 Ω = Ω ( U α ) d x α 1 ∧ ⋯ ∧ d x α n \Omega= \Omega(U_\alpha) \, \mathrm{d} x^1_\alpha \wedge \cdots \wedge \mathrm{d} x^n_\alpha Ω = Ω ( U α ) d x α 1 ∧ ⋯ ∧ d x α n n n n M M M ∂ M \partial M ∂ M
Ω ′ = Ω ( U α ) ( − 1 ) n d x α 1 ∧ ⋯ ∧ d x α n − 1 \Omega^\prime = \Omega(U_\alpha) (-1)^n \, \mathrm{d} x^1_\alpha \wedge \cdots \wedge \mathrm{d} x^{n-1}_\alpha
Ω ′ = Ω ( U α ) ( − 1 ) n d x α 1 ∧ ⋯ ∧ d x α n − 1
这样选取的定向满足
− d x α n ∧ Ω ′ = Ω ( U α ) ( − 1 ) n − 1 d x α n ∧ d x α 1 ∧ ⋯ ∧ d x α n − 1 = Ω - \,\mathrm{d} x^{n}_\alpha \wedge \Omega^\prime = \Omega(U_\alpha) (-1)^{n-1} \, \mathrm{d} x^{n}_\alpha \wedge \mathrm{d} x^1_\alpha \wedge \cdots \wedge \mathrm{d} x^{n-1}_\alpha = \Omega
− d x α n ∧ Ω ′ = Ω ( U α ) ( − 1 ) n − 1 d x α n ∧ d x α 1 ∧ ⋯ ∧ d x α n − 1 = Ω
映射的秩 设 f : M → N f : M → N f : M → N C r C^r C r p ∈ M p \in M p ∈ M q = f ( p ) ∈ N
q = f(p) \in N q = f ( p ) ∈ N p p p ( U , ϕ ) (U, \phi) ( U , ϕ ) q q q ( V , ψ ) (V, \psi) ( V , ψ )
rank p f = rank J ( ψ ∘ f ∘ φ − 1 ) ( φ ( p ) ) \operatorname{rank}_{p} f=\operatorname{rank} J\left(\psi \circ f \circ \varphi^{-1}\right)(\varphi(p))
r a n k p f = r a n k J ( ψ ∘ f ∘ φ − 1 ) ( φ ( p ) )
称为 f f f p p p 秩 。
浸入、嵌入和淹没 设 f : M m → N n f : M^m → N^n f : M m → N n C r C^r C r rank p f ≡ m , ∀ p ∈ M \operatorname{rank}_{p} f \equiv m, \forall p \in M r a n k p f ≡ m , ∀ p ∈ M f f f C r C^r C r 浸入(immersion) ；如果 f f f C r C^r C r f f f M M M f ( M ) f(M) f ( M ) f f f C r C^r C r 嵌入(embedding) ；如果 rank p f ≡ n , ∀ p ∈ M \operatorname{rank}_{p} f \equiv n, \forall p \in M r a n k p f ≡ n , ∀ p ∈ M f f f C r C^r C r 淹没(submersion) 。
子流形 设 M , N M, N M , N M ⊂ N M \subset N M ⊂ N i : M → N i : M \to N i : M → N M M M N N N 子流形 ；如果包含映射 i : M → N i : M → N i : M → N M M M N N N
超曲面 设 i : Σ → M i:\Sigma \to M i : Σ → M i i i dim Σ = dim M − 1 \dim \Sigma = \dim M -1 dim Σ = dim M − 1 i ( Σ ) ⊂ M i(\Sigma) \subset M i ( Σ ) ⊂ M dim M − 1 \dim M-1 dim M − 1 维超曲面 ，也称为余 1 1 1 维超曲面 ，简称超曲面 (hypersurface) 。
总可以找到一个光滑函数 S ( x ) S(x) S ( x )
S ( x ) = σ S(x) = \sigma
S ( x ) = σ
是一个常数。因此，余 1 维超曲面 Σ \Sigma Σ M M M S S S σ \sigma σ 余 1 维超曲面族 。事实上，在流形上某点的邻域上选定一个局域坐标系后，使得某个给定的坐标分量的值为常数 的点的集合就是流形上的余 1 维超曲面，称为坐标超曲面 。而整个邻域可以看成是由上述余 1 维超曲面族"扫出来"的。这个图像在几何学里叫做分层 (foliation) 。
切矢量 超曲面的切丛 W W W
法余矢 对于由 S ( x ) = σ S(x) = \sigma S ( x ) = σ d S \mathrm{d}S d S
n μ = ∇ μ S n_\mu = \nabla_\mu S
n μ = ∇ μ S
对于超曲面上任意一条曲线的切矢量 T T T
d S ( T ) = ∇ T S = n μ T μ = 0 \mathrm{d}S(T) = \nabla_T S = n_\mu T^\mu = 0
d S ( T ) = ∇ T S = n μ T μ = 0
这是因为 S S S S ( x ) = σ S(x) = \sigma S ( x ) = σ
法矢量 设 n n n M M M g g g n μ = g μ ν n ν n^\mu = g_{\mu\nu} n^\nu n μ = g μ ν n ν
n μ = ∇ μ S n^\mu = \nabla^\mu S
n μ = ∇ μ S
需要注意的是，法矢量并不一定就不含有任何切于超曲面的成分，甚至有可能发生法矢量切于超曲面的情况，即 n μ ∈ W n^\mu \in W n μ ∈ W
矢量 T μ T^\mu T μ n μ T μ n_\mu T^\mu n μ T μ 0 0 0 n μ n^\mu n μ n μ n μ = g μ ν n μ n ν = 0 n_\mu n^\mu = g_{\mu\nu} n^\mu n^\nu =0 n μ n μ = g μ ν n μ n ν = 0 n n n g μ ν n μ n ν g_{\mu\nu} n^\mu n^\nu g μ ν n μ n ν n μ n^\mu n μ
类时超曲面：超曲面上的法矢量 n μ n^\mu n μ g μ ν n μ n ν < 0 g_{\mu\nu} n^\mu n^\nu<0 g μ ν n μ n ν < 0
类空超曲面：超曲面上的法矢量 n μ n^\mu n μ g μ ν n μ n ν > 0 g_{\mu\nu} n^\mu n^\nu>0 g μ ν n μ n ν > 0
类光超曲面：超曲面上的法矢量 n μ n^\mu n μ g μ ν n μ n ν = 0 g_{\mu\nu} n^\mu n^\nu=0 g μ ν n μ n ν = 0
Frobenius 定理：超曲面上的法矢量满足
n [ μ ∇ ν n ρ ] = 0 n_{[\mu} \nabla_\nu n_{\rho ]} = 0
n [ μ ∇ ν n ρ ] = 0
设流形 M M M ( U , φ ) (U,\varphi) ( U , φ ) { x μ } \{x^\mu\} { x μ } Σ \Sigma Σ ( V α , ψ α ) (V_\alpha,\psi_\alpha) ( V α , ψ α ) { y a } \{y^a\} { y a }
e a μ = ∂ x μ ∂ y a e μ a = ∂ y a ∂ x μ e^\mu_a = \frac{\partial x^\mu}{\partial y^a} \qquad e_\mu^a = \frac{\partial y^a}{\partial x^\mu}
e a μ = ∂ y a ∂ x μ e μ a = ∂ x μ ∂ y a
满足 e a μ e μ b = δ a b e^\mu_a e_\mu^b = \delta_a^b e a μ e μ b = δ a b e a μ e^\mu_a e a μ e a = ∂ ∂ y a e_a = \frac{\partial}{\partial y^a} e a = ∂ y a ∂ μ \mu μ μ \mu μ μ \mu μ
由于 e a = ∂ ∂ y a e_a = \frac{\partial}{\partial y^a} e a = ∂ y a ∂
n μ e a μ = 0 n_\mu e_a^\mu = 0
n μ e a μ = 0
除非特殊声明，一般认为法向量是归一化的。记 g μ ν n μ n ν = ϵ g_{\mu\nu} n^\mu n^\nu = \epsilon g μ ν n μ n ν = ϵ ϵ \epsilon ϵ + 1 , − 1 +1,-1 + 1 , − 1 0 0 0
诱导度规 超曲面上的诱导度规定义为
h a b = g a b − ϵ n a n b h_{ab} = g_{ab} - \epsilon n_a n_b
h a b = g a b − ϵ n a n b
这使得对于任何超曲面的切矢 T a T^a T a
h a b T a T b = g a b T a T b − ϵ ( n a T a ) ( n b T b ) = g a b T a T b h_{ab} T^a T^b = g_{ab} T^a T^b -\epsilon (n_a T^a)(n_b T^b) = g_{ab} T^a T^b
h a b T a T b = g a b T a T b − ϵ ( n a T a ) ( n b T b ) = g a b T a T b
即诱导度规作用的结果等价于将流形 M M M g μ ν g_{\mu\nu} g μ ν g a b g_{ab} g a b
g a b = e a μ e b ν g μ ν h μ ν = e μ a e ν b h a b g_{ab} = e_a^\mu e^\nu_b g_{\mu\nu} \qquad h_{\mu\nu} = e_\mu^a e_\nu^b h_{ab}
g a b = e a μ e b ν g μ ν h μ ν = e μ a e ν b h a b
例：在 R 3 \mathbb{R}^3 R 3 S 2 : x 2 + y 2 + z 2 = R 2 S^2: x^2 + y^2 + z^2 = R^2 S 2 : x 2 + y 2 + z 2 = R 2 ( θ , ϕ ) (\theta,\phi) ( θ , ϕ )
x = R sin θ cos ϕ , y = R sin θ sin ϕ , z = R cos θ x = R \sin \theta \cos \phi,\ y=R\sin\theta\sin\phi, \ z=R\cos \theta
x = R sin θ cos ϕ , y = R sin θ sin ϕ , z = R cos θ
求 S 2 S^2 S 2 S 2 S^2 S 2 h a b h_{ab} h a b
解：直接对 S 2 S^2 S 2
2 x d x + 2 y d y + 2 z d z = 0 2x \mathrm{d}x + 2y\mathrm{d}y + 2z\mathrm{d}z = 0
2 x d x + 2 y d y + 2 z d z = 0
即 n μ = 2 x μ n_\mu = 2x_\mu n μ = 2 x μ n μ = x μ R n_\mu = \frac{x_\mu}{R} n μ = R x μ n μ = x μ R n^\mu = \frac{x^\mu}{R} n μ = R x μ n μ n μ = 1 n_\mu n^\mu =1 n μ n μ = 1
R 3 \mathbb{R}^3 R 3
g x x = 1 , g y y = 1 , g z z = 1 g x y = g y x = g z x = g x z = g y z = g z y = 0 \begin{gathered}
g_{xx} = 1, \ g_{yy}=1,\ g_{zz}=1 \\
g_{xy}= g_{yx} = g_{zx} = g_{xz} = g_{yz} = g_{zy} = 0
\end{gathered}
g x x = 1 , g y y = 1 , g z z = 1 g x y = g y x = g z x = g x z = g y z = g z y = 0
以及坐标转换系数
e θ x = R cos θ cos ϕ e ϕ x = − R sin θ sin ϕ e θ y = R cos θ sin ϕ e ϕ y = R sin θ cos ϕ e θ z = − R sin θ e ϕ z = 0 \begin{aligned}
e_\theta^x &= R \cos\theta\cos\phi \qquad e_\phi^x = - R \sin\theta\sin\phi \\
e_\theta^y &= R \cos\theta\sin\phi \qquad\, e_\phi^y = R \sin\theta\cos\phi \\
e_\theta^z &= - R\sin\theta \qquad \quad \ \ \, e_\phi^z = 0
\end{aligned}
e θ x e θ y e θ z = R cos θ cos ϕ e ϕ x = − R sin θ sin ϕ = R cos θ sin ϕ e ϕ y = R sin θ cos ϕ = − R sin θ e ϕ z = 0
因此有
g θ θ = e θ μ e θ ν g μ ν = R 2 g ϕ ϕ = e ϕ μ e ϕ ν g μ ν = R 2 sin 2 θ g θ ϕ = g ϕ θ = 0 n θ = e θ μ n μ = 0 n ϕ = e ϕ μ n μ = 0 \begin{gathered}
g_{\theta\theta} = e_\theta^\mu e_\theta^\nu g_{\mu\nu} = R^2 \qquad g_{\phi\phi} = e_\phi^\mu e_\phi^\nu g_{\mu\nu} = R^2 \sin^2 \theta \qquad g_{\theta\phi} = g_{\phi\theta} = 0 \\
n_\theta = e_\theta^\mu n_\mu = 0 \qquad n_\phi = e_\phi^\mu n_\mu = 0
\end{gathered}
g θ θ = e θ μ e θ ν g μ ν = R 2 g ϕ ϕ = e ϕ μ e ϕ ν g μ ν = R 2 sin 2 θ g θ ϕ = g ϕ θ = 0 n θ = e θ μ n μ = 0 n ϕ = e ϕ μ n μ = 0
以及诱导度规
h a b = g a b − ϵ n a n b = g a b h_{ab} = g_{ab}-\epsilon n_a n_b =g_{ab}
h a b = g a b − ϵ n a n b = g a b
法余矢量和法矢量在 S 2 S^2 S 2 0 0 0
积分 设 M M M n n n M M M Ω \Omega Ω ω \omega ω M M M n n n ( U α , φ α ) (U_\alpha,\varphi_\alpha) ( U α , φ α ) ω = ω α d x α 1 ∧ ⋯ ∧ d x α n \omega = \omega_\alpha \mathrm{d} x^1_\alpha \wedge \cdots \wedge \mathrm{d} x^n_\alpha ω = ω α d x α 1 ∧ ⋯ ∧ d x α n ω \omega ω U α U_\alpha U α
∫ U α ω = sgn ( ω ) ∣ U α ∫ φ α ( U α ) ω α ∘ φ α − 1 ( x 1 , ⋯ , x n ) d x 1 ⋯ d x n \int_{U_\alpha} \omega = \operatorname{sgn}(\omega)|_{U_\alpha} \int_{\varphi_\alpha (U_\alpha)} \omega_\alpha \circ\varphi^{-1}_\alpha (x_1,\cdots,x_n)\, \mathrm{d}x_1 \cdots\mathrm{d}x_n
∫ U α ω = s g n ( ω ) ∣ U α ∫ φ α ( U α ) ω α ∘ φ α − 1 ( x 1 , ⋯ , x n ) d x 1 ⋯ d x n
其中 sgn ( ω ) ∣ U α \operatorname{sgn}(\omega)|_{U_\alpha} s g n ( ω ) ∣ U α ω \omega ω ω ( ∂ ∂ x α 1 , ∂ ∂ x α 2 , ⋯ ∂ ∂ x α n ) \omega \left(\frac{\partial}{\partial x^1_\alpha}, \frac{\partial}{\partial x^2_\alpha},\cdots \frac{\partial}{\partial x^n_\alpha} \right) ω ( ∂ x α 1 ∂ , ∂ x α 2 ∂ , ⋯ ∂ x α n ∂ ) Ω ( ∂ ∂ x α 1 , ∂ ∂ x α 2 , ⋯ ∂ ∂ x α n ) \Omega \left(\frac{\partial}{\partial x^1_\alpha}, \frac{\partial}{\partial x^2_\alpha},\cdots \frac{\partial}{\partial x^n_\alpha} \right) Ω ( ∂ x α 1 ∂ , ∂ x α 2 ∂ , ⋯ ∂ x α n ∂ ) sgn ( ω ) ∣ U α \operatorname{sgn}(\omega)|_{U_\alpha} s g n ( ω ) ∣ U α + 1 +1 + 1 − 1 -1 − 1
推论：上述积分的定义和局部坐标的选取无关。即若 supp ω \operatorname{supp} \omega s u p p ω ( U β , φ β ) (U_\beta,\varphi_\beta) ( U β , φ β ) U α ∩ U β U_\alpha \cap U_\beta U α ∩ U β
Proof
由于 ( U β , φ β ) (U_\beta,\varphi_\beta) ( U β , φ β ) ( U α , φ α ) (U_\alpha,\varphi_\alpha) ( U α , φ α ) ∣ det J ( φ β ∘ φ α − 1 ) ∣ = det J ( φ β ∘ φ α − 1 ) > 0 \left| \operatorname{det} J\left(\varphi_{\beta} \circ \varphi_{\alpha}^{-1}\right) \right| = \operatorname{det} J\left(\varphi_{\beta} \circ \varphi_{\alpha}^{-1}\right) >0 ∣ ∣ ∣ d e t J ( φ β ∘ φ α − 1 ) ∣ ∣ ∣ = d e t J ( φ β ∘ φ α − 1 ) > 0
∫ φ β ( U α ∩ U β ) ω β d x β 1 ⋯ d x β n = ∫ φ α ( U β ∩ U α ) ∣ det J ( φ β ∘ φ α − 1 ) ∣ ⋅ ω β d x α 1 ⋯ d x α n = ∫ φ α ( U α ) ω α d x α 1 ⋯ d x α n \int_{\varphi_\beta\left(U_\alpha \cap U_\beta\right)} \omega_\beta \, \mathrm{d} x_\beta^1 \cdots \mathrm{d} x_\beta^n =\int_{\varphi_{\alpha}\left(U_{\beta} \cap U_{\alpha}\right)} \left| \operatorname{det} J\left(\varphi_{\beta} \circ \varphi_{\alpha}^{-1}\right) \right| \cdot \omega_{\beta} \, \mathrm{d} x_{\alpha}^{1} \cdots \mathrm{d} x_{\alpha}^{n} =\int_{\varphi_{\alpha}\left(U_{\alpha}\right)} \omega_{\alpha} \, \mathrm{d} x_{\alpha}^{1} \cdots \mathrm{d} x_{\alpha}^{n}
∫ φ β ( U α ∩ U β ) ω β d x β 1 ⋯ d x β n = ∫ φ α ( U β ∩ U α ) ∣ ∣ ∣ d e t J ( φ β ∘ φ α − 1 ) ∣ ∣ ∣ ⋅ ω β d x α 1 ⋯ d x α n = ∫ φ α ( U α ) ω α d x α 1 ⋯ d x α n
下面将积分推广到流形 M M M supp ω \operatorname{supp} \omega s u p p ω { φ α } \left\{ \varphi_\alpha \right\} { φ α }
∫ M ω = ∑ α ∫ M φ α ⋅ ω \int_{M} \omega=\sum_{\alpha} \int_{M} \varphi_{\alpha} \cdot \omega
∫ M ω = α ∑ ∫ M φ α ⋅ ω
称为 ω \omega ω M M M
推论：上述积分的定义和开覆盖及单位分解的选取无关。
无边流形上的积分 设 M M M n n n ω \omega ω n − 1 n-1 n − 1
∫ M d ω = 0 \int_M \mathrm{d} \omega = 0
∫ M d ω = 0
Stokes 积分公式 设 M M M n n n ω \omega ω n − 1 n-1 n − 1
∫ M d ω = ∫ ∂ M i ∗ ω = ∫ ∂ M ω \int_{M} \mathrm{d} \omega=\int_{\partial M} i^{*} \omega=\int_{\partial M} \omega
∫ M d ω = ∫ ∂ M i ∗ ω = ∫ ∂ M ω
其中 i : ∂ M → M i : \partial M \to M i : ∂ M → M ∂ M \partial M ∂ M
散度 设 M M M n n n Ω \Omega Ω X X X M M M L X Ω \mathcal{L}_X \Omega L X Ω
L X Ω = div ( X ) Ω \mathcal{L}_X \Omega = \operatorname{div} (X) \, \Omega
L X Ω = d i v ( X ) Ω
系数 div X \operatorname{div} X d i v X X X X 散度 ，映射 div : X ( M ) → C ∞ ( M ) \operatorname{div} : \mathfrak{X}(M) \to C^\infty (M) d i v : X ( M ) → C ∞ ( M )
高斯定理 设 M M M n n n Ω \Omega Ω X X X M M M
∫ M div ( X ) Ω = ∫ ∂ M ι X Ω \int_{M} \operatorname{div}(X) \, \Omega=\int_{\partial M} \iota_{X} \Omega
∫ M d i v ( X ) Ω = ∫ ∂ M ι X Ω
Proof
利用内乘的性质 L X = d ∘ ι X + ι X ∘ d \mathcal{L}_X = \mathrm{d} \circ \iota_X + \iota_X \circ \mathrm{d} L X = d ∘ ι X + ι X ∘ d
逐点内积 在黎曼流形 ( M , g ) (M,g) ( M , g ) { e 1 , ⋯ , e n } \{e_1,\cdots,e_n\} { e 1 , ⋯ , e n } { e 1 , ⋯ , e n } \{e^1,\cdots,e^n\} { e 1 , ⋯ , e n } T , S T,S T , S ( p , q ) (p,q) ( p , q )
T = T μ 1 ⋯ μ n ν 1 ⋯ ν n e 1 ⊗ ⋯ ⊗ e n ⊗ e 1 ⊗ ⋯ ⊗ e n T = T^{\mu_1 \cdots \mu_n}{_{\nu_1 \cdots \nu_n}} e_1 \otimes \cdots \otimes e_n \otimes e^1 \otimes \cdots \otimes e^n
T = T μ 1 ⋯ μ n ν 1 ⋯ ν n e 1 ⊗ ⋯ ⊗ e n ⊗ e 1 ⊗ ⋯ ⊗ e n
S = S μ 1 ⋯ μ n ν 1 ⋯ ν n e 1 ⊗ ⋯ ⊗ e n ⊗ e 1 ⊗ ⋯ ⊗ e n S = S^{\mu_1 \cdots \mu_n}{_{\nu_1 \cdots \nu_n}} e_1 \otimes \cdots \otimes e_n \otimes e^1 \otimes \cdots \otimes e^n
S = S μ 1 ⋯ μ n ν 1 ⋯ ν n e 1 ⊗ ⋯ ⊗ e n ⊗ e 1 ⊗ ⋯ ⊗ e n
则它们之间的逐点内积定义为
⟨ T , S ⟩ = 1 ( p + q ) ! T μ 1 ⋯ μ n ν 1 ⋯ ν n S μ 1 ⋯ μ n ν 1 ⋯ ν n \langle T,S \rangle = \frac{1}{(p+q)!} T^{\mu_1 \cdots \mu_n}{_{\nu_1 \cdots \nu_n}} S^{\mu_1 \cdots \mu_n}{_{\nu_1 \cdots \nu_n}}
⟨ T , S ⟩ = ( p + q ) ! 1 T μ 1 ⋯ μ n ν 1 ⋯ ν n S μ 1 ⋯ μ n ν 1 ⋯ ν n
两个向量场 X , Y X,Y X , Y
⟨ X , Y ⟩ = g μ ν X μ Y ν \langle X,Y \rangle = g_{\mu\nu} X^\mu Y^\nu
⟨ X , Y ⟩ = g μ ν X μ Y ν
两个 p p p ω , η \omega,\eta ω , η
ω = ∑ μ 1 < μ p ω μ 1 ⋯ μ p d x μ 1 ∧ ⋯ ∧ d x μ p η = ∑ μ 1 < μ p η μ 1 ⋯ μ p d x μ 1 ∧ ⋯ ∧ d x μ p \omega = \sum_{\mu_1 < \mu_p} \omega_{\mu_1\cdots\mu_p}\, \mathrm{d} x^{\mu_1} \wedge \cdots \wedge\mathrm{d} x^{\mu_p} \qquad \eta= \sum_{\mu_1 < \mu_p} \eta_{\mu_1\cdots\mu_p}\, \mathrm{d} x^{\mu_1} \wedge \cdots \wedge\mathrm{d} x^{\mu_p}
ω = μ 1 < μ p ∑ ω μ 1 ⋯ μ p d x μ 1 ∧ ⋯ ∧ d x μ p η = μ 1 < μ p ∑ η μ 1 ⋯ μ p d x μ 1 ∧ ⋯ ∧ d x μ p
⟨ ω , η ⟩ = ∑ μ 1 < μ p ω μ 1 ⋯ μ p η μ 1 ⋯ μ p \langle \omega,\eta \rangle = \sum_{\mu_1 < \mu_p} \omega_{\mu_1\cdots\mu_p} \eta_{\mu_1\cdots\mu_p}
⟨ ω , η ⟩ = μ 1 < μ p ∑ ω μ 1 ⋯ μ p η μ 1 ⋯ μ p
Hodge内积/整体内积 设 ( M , g ) (M,g) ( M , g ) ϵ \epsilon ϵ ⋀ ( M ) \bigwedge(M) ⋀ ( M ) ( ⋅ , ⋅ ) (\cdot,\cdot) ( ⋅ , ⋅ ) ω , η \omega,\eta ω , η
( ω , η ) = ∫ M ⟨ ω , η ⟩ ϵ (\omega,\eta) = \int_M \langle \omega,\eta \rangle \epsilon
( ω , η ) = ∫ M ⟨ ω , η ⟩ ϵ
对于异次数的微分形式，规定其内积为 0 0 0 ( ⋅ , ⋅ ) (\cdot,\cdot) ( ⋅ , ⋅ ) Hodge内积 或整体内积 。
Hodge 星算子 在 n n n ( M , g ) (M,g) ( M , g ) ∗ : ⋀ p ( M ) → ⋀ n − p ( M ) *: \bigwedge^p (M) \to \bigwedge^{n-p} (M) ∗ : ⋀ p ( M ) → ⋀ n − p ( M )
∗ ω μ 1 ⋯ μ n − p = 1 p ! ω ν 1 ⋯ ν p g ν 1 ρ 1 ⋯ g ν p ρ p ϵ ρ 1 ⋯ ρ p μ 1 ⋯ μ n − p {^*}\omega_{\mu_1\cdots\mu_{n-p}} = \frac{1}{p!} \omega_{\nu_1\cdots\nu_p}g^{\nu_1 \rho_1}\cdots g^{\nu_p \rho_p} \epsilon_{\rho_1\cdots\rho_p \mu_1\cdots\mu_{n-p}}
∗ ω μ 1 ⋯ μ n − p = p ! 1 ω ν 1 ⋯ ν p g ν 1 ρ 1 ⋯ g ν p ρ p ϵ ρ 1 ⋯ ρ p μ 1 ⋯ μ n − p
∗ ω = 1 ( n − p ) ! ( ∗ ω ) μ 1 ⋯ μ n − p d x μ 1 ∧ ⋯ ∧ d x μ n − p ^* \omega = \frac{1}{(n-p)!} ({^*}\omega)_{\mu_1\cdots\mu_{n-p}} \, \mathrm{d} x^{\mu_1} \wedge \cdots \wedge\mathrm{d} x^{\mu_{n-p}}
∗ ω = ( n − p ) ! 1 ( ∗ ω ) μ 1 ⋯ μ n − p d x μ 1 ∧ ⋯ ∧ d x μ n − p
则 ∗ * ∗ Hodge 星算子 ，∗ ω ^* \omega ∗ ω ω \omega ω 对偶微分形式 。
Hodge 星算子 ∗ * ∗
∗ 1 = ϵ *1 = \epsilon ∗ 1 = ϵ ∗ ϵ = 1 * \epsilon =1 ∗ ϵ = 1 ∗ ∗ ω = ( − 1 ) p ( n − p ) ω ** \omega = (-1)^{p(n-p)} \omega ∗ ∗ ω = ( − 1 ) p ( n − p ) ω ⟨ ∗ ω , ∗ η ⟩ = ⟨ ω , η ⟩ \langle * \omega, * \eta\rangle=\langle\omega, \eta\rangle ⟨ ∗ ω , ∗ η ⟩ = ⟨ ω , η ⟩ ω ∧ ∗ η = ⟨ ω , η ⟩ ϵ \omega \wedge * \eta=\langle\omega, \eta\rangle \epsilon ω ∧ ∗ η = ⟨ ω , η ⟩ ϵ
三维Euclid空间中，度规张量是最基本的 δ μ ν \delta_{\mu\nu} δ μ ν { d x , d y , d z } \{\mathrm{d} x,\mathrm{d} y,\mathrm{d}z \} { d x , d y , d z } ϵ = d x ∧ d y ∧ d z \epsilon = \mathrm{d} x \wedge \mathrm{d} y \wedge \mathrm{d}z ϵ = d x ∧ d y ∧ d z
∗ d x = d y ∧ d z *\mathrm{d} x = \mathrm{d}y \wedge \mathrm{d} z
∗ d x = d y ∧ d z
∗ d y = d z ∧ d x *\mathrm{d}y= \mathrm{d}z \wedge \mathrm{d} x
∗ d y = d z ∧ d x
∗ d z = d x ∧ d y *\mathrm{d} z = \mathrm{d} x\wedge \mathrm{d} y
∗ d z = d x ∧ d y
Minkowski 空间是伪黎曼流形，我们采用号差为 − 2 -2 − 2 ( + , − , − , − ) (+,-,-,-) ( + , − , − , − ) ( t , x , y , z ) (t,x,y,z) ( t , x , y , z ) ϵ = d t ∧ d x ∧ d y ∧ d z \epsilon = \mathrm{d} t \wedge \mathrm{d} x \wedge \mathrm{d} y \wedge \mathrm{d}z ϵ = d t ∧ d x ∧ d y ∧ d z
∗ ( d x μ ) = η μ λ ε λ ν ρ σ 1 3 ! d x ν ∧ d x ρ ∧ d x σ * \left(\mathrm{d}x^{\mu}\right)=\eta^{\mu \lambda} \varepsilon_{\lambda \nu \rho \sigma} \frac{1}{3 !} \mathrm{d}x^{\nu} \wedge \mathrm{d}x^{\rho} \wedge \mathrm{d}x^{\sigma}
∗ ( d x μ ) = η μ λ ε λ ν ρ σ 3 ! 1 d x ν ∧ d x ρ ∧ d x σ
∗ ( d x μ ∧ d x ν ) = η μ κ η ν λ ε κ λ ρ σ 1 2 ! d x ρ ∧ d x σ * \left(\mathrm{d}x^{\mu} \wedge \mathrm{d}x^{\nu}\right)=\eta^{\mu \kappa} \eta^{\nu \lambda} \varepsilon_{\kappa \lambda \rho \sigma} \frac{1}{2 !} \mathrm{d}x^{\rho} \wedge \mathrm{d}x^{\sigma}
∗ ( d x μ ∧ d x ν ) = η μ κ η ν λ ε κ λ ρ σ 2 ! 1 d x ρ ∧ d x σ
余微分算子 定义余微分算子 δ : ⋀ p ( M ) → ⋀ p − 1 ( M ) \delta: \bigwedge^p (M) \to \bigwedge^{p-1} (M) δ : ⋀ p ( M ) → ⋀ p − 1 ( M )
δ = ( − 1 ) n ( p − 1 ) + 1 ⋆ d ⋆ \delta = (-1)^{n(p-1)+1} \star \mathrm{d} \, \star
δ = ( − 1 ) n ( p − 1 ) + 1 ⋆ d ⋆
将 p p p p − 1 p-1 p − 1
定理：余微分算子 δ \delta δ d \mathrm{d} d
( d ω , η ) = ( ω , δ η ) (\mathrm{d}\omega,\eta) = (\omega,\delta\eta)
( d ω , η ) = ( ω , δ η )
de Rham上同调群 p p p H d R p ( M ) H^p_{\mathrm{d} R} (M) H d R p ( M ) M M M p p p
H d R p ( M ) = { ω ∈ Ω q ( M ) ∣ d ω = 0 } / { d η ∣ η ∈ Ω q − 1 ( M ) } H^p_{\mathrm{dR}} (M) = \left\{ \omega \in \Omega^q(M) \mid \mathrm{d}\omega = 0 \right\} / \left\{ \mathrm{d}\eta \mid \eta \in \Omega^{q-1}(M)\right\}
H d R p ( M ) = { ω ∈ Ω q ( M ) ∣ d ω = 0 } / { d η ∣ η ∈ Ω q − 1 ( M ) }
Poincaré引理 当 p = 0 p=0 p = 0 H d R 0 ( R n ) = R H^0_{\mathrm{dR}} (\mathbb{R}^n) = \mathbb{R} H d R 0 ( R n ) = R 1 ≤ p ≤ n 1\le p \le n 1 ≤ p ≤ n H d R p ( R n ) = { 0 } H^p_{\mathrm{dR}} (\mathbb{R}^n)=\{0\} H d R p ( R n ) = { 0 }
推论：设 M M M H d R p ( R × M ) = H d R p ( M ) , ∀ p ≥ 0 H^p_{\mathrm{dR}} ( \mathbb{R} \times M) = H^p_{\mathrm{dR}} (M),\, \forall p \ge 0 H d R p ( R × M ) = H d R p ( M ) , ∀ p ≥ 0
