这篇文章是纤维丛理论的笔记。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。
左作用/右作用 李群 G G G M M M 左作用 (left action) 指的是一个 C ∞ C^\infty C ∞ L : G × M → M L: G \times M \to M L : G × M → M
L g : M → M L_g : M \to M L g : M → M ∀ g ∈ G \forall g \in G ∀ g ∈ G L g h = L g ∘ L h , ∀ g , h ∈ G L_{gh} = L_g \circ L_h ,\, \forall g,h \in G L g h = L g ∘ L h , ∀ g , h ∈ G
相应地，李群 G G G M M M 右作用 (left action) 指的是一个 C ∞ C^\infty C ∞ R : M × G → M R: M \times G \to M R : M × G → M
R g : M → M R_g : M \to M R g : M → M ∀ g ∈ G \forall g \in G ∀ g ∈ G R g h = R g ∘ R h , ∀ g , h ∈ G R_{gh} = R_g \circ R_h ,\, \forall g,h \in G R g h = R g ∘ R h , ∀ g , h ∈ G
左作用/右作用的结果 L g ( p ) L_g (p) L g ( p ) R g ( p ) R_g (p) R g ( p ) g p gp g p p g pg p g M M M
/* 注：谨记左作用/右作用的定义不要求流形 M M M M M M L e L_e L e e e e G G G
左作用 L L L 自由 (free) 的，若 g ≠ e ⟹ g p ≠ p , ∀ p ∈ M g \neq e \implies gp \neq p,\forall p \in M g = e ⟹ g p = p , ∀ p ∈ M
轨道 集合 { g p ∣ g ∈ G } \left\{gp \mid g \in G\right\} { g p ∣ g ∈ G } L g L_g L g p p p L g L_g L g p p p R R R
纤维丛 设 π : E → M \pi : E \to M π : E → M F F F G G G F F F M M M { U α } α ∈ Γ \{U_\alpha\}_{\alpha \in \Gamma} { U α } α ∈ Γ ψ α : π − 1 ( U α ) → U α × F \psi_{\alpha}: \pi^{-1}\left(U_{\alpha}\right) \rightarrow U_{\alpha} \times F ψ α : π − 1 ( U α ) → U α × F
ψ ( π − 1 ( p ) ) = { p } × F , ∀ p ∈ U α \psi\left(\pi^{-1}(p)\right)=\{p\} \times F, \quad \forall p \in U_{\alpha} ψ ( π − 1 ( p ) ) = { p } × F , ∀ p ∈ U α
当 U α ∩ U β ≠ ∅ U_\alpha \cap U_\beta \neq \varnothing U α ∩ U β = ∅ g β α : U α ⋂ U β → G g_{\beta \alpha}: U_{\alpha} \bigcap U_{\beta} \rightarrow G g β α : U α ⋂ U β → G
ψ β ∘ ψ α − 1 ( p , f ) = ( p , g β α ( p ) f ) , ∀ p ∈ U α ∩ U β , f ∈ F \psi_{\beta} \circ \psi_{\alpha}^{-1}(p, f)=\left(p, g_{\beta \alpha}(p) f\right), \quad \forall p \in U_{\alpha} \cap U_{\beta}, f \in F
ψ β ∘ ψ α − 1 ( p , f ) = ( p , g β α ( p ) f ) , ∀ p ∈ U α ∩ U β , f ∈ F
则称 E E E M M M 纤维丛 ，π \pi π 丛投影 ，F F F 纤维/标准纤维 ，E E E M M M 总空间 和底空间 ，G G G 结构群 ，ψ α \psi_\alpha ψ α π − 1 ( p ) = E p \pi^{-1}(p)=E_{p} π − 1 ( p ) = E p p p p 处的纤维 ，并与 F F F g β α g_{\beta \alpha} g β α
/* 注：纤维丛即局部上看上去像乘积流形 M × F M \times F M × F F F F E p E_p E p E p E_p E p π \pi π p p p E E E F F F E p E_p E p F F F E p E_p E p
向量丛 设 E , M E,M E , M π : E → M \pi : E \to M π : E → M M M M { U α } \{U_\alpha\} { U α } ψ α : π − 1 ( U α ) → U α × R n \psi_{\alpha}: \pi^{-1}\left(U_{\alpha}\right) \rightarrow U_{\alpha} \times \mathbb{R}^{n} ψ α : π − 1 ( U α ) → U α × R n
ψ ( π − 1 ( p ) ) = { p } × R n , ∀ p ∈ U α \psi\left(\pi^{-1}(p)\right)=\{p\} \times \mathbb{R}^{n}, \quad \forall p \in U_{\alpha} ψ ( π − 1 ( p ) ) = { p } × R n , ∀ p ∈ U α
当 U α ∩ U β ≠ ∅ U_\alpha \cap U_\beta \neq \varnothing U α ∩ U β = ∅ g β α : U α ⋂ U β → G L ( n , R ) g_{\beta \alpha}: U_{\alpha} \bigcap U_{\beta} \rightarrow \mathrm{GL}(n,\mathbb{R}) g β α : U α ⋂ U β → G L ( n , R )
ψ β ∘ ψ α − 1 ( p , v ) = ( p , g β α ( p ) v ) , ∀ p ∈ U α ∩ U β , v ∈ R n \psi_{\beta} \circ \psi_{\alpha}^{-1}(p, v)=\left(p, g_{\beta \alpha}(p) v\right), \quad \forall p \in U_{\alpha} \cap U_{\beta}, v \in \mathbb{R}^{n}
ψ β ∘ ψ α − 1 ( p , v ) = ( p , g β α ( p ) v ) , ∀ p ∈ U α ∩ U β , v ∈ R n
则称 E E E M M M n n n π \pi π E E E M M M ψ α \psi_\alpha ψ α π − 1 ( p ) = E p ≅ R n \pi^{-1}(p)=E_{p} \cong \mathbb{R}^n π − 1 ( p ) = E p ≅ R n p p p g β α g_{\beta \alpha} g β α g β α g_{\beta \alpha} g β α
/* 注：向量丛就是纤维丛定义中 F F F R n \mathbb{R}^n R n G L ( n , R ) \mathrm{GL}(n,\mathbb{R}) G L ( n , R ) M × R n M \times \mathbb{R}^n M × R n
切丛 切丛是总空间为 T M TM T M M M M T p M T_p M T p M
主丛 设 π : P → M \pi:P \to M π : P → M G G G p ∈ P , x ∈ M p \in P,x\in M p ∈ P , x ∈ M
G G G P P P R : P × G → P R:P \times G \to P R : P × G → P 点 x ∈ M x\in M x ∈ M P x P_x P x R R R p ∈ π − 1 ( x ) p \in \pi^{-1}(x) p ∈ π − 1 ( x ) P x = { p g ∣ g ∈ G } P_x = \left\{pg \mid g \in G\right\} P x = { p g ∣ g ∈ G }
存在 M M M { U α } \{U_\alpha\} { U α } ψ α : π − 1 ( U α ) → U α × G \psi_{\alpha}: \pi^{-1}\left(U_{\alpha}\right) \rightarrow U_{\alpha} \times G ψ α : π − 1 ( U α ) → U α × G ψ ( p ) = ( π ( p ) , ϕ U ( p ) ) , ∀ p ∈ P \psi(p) = (\pi(p),\phi_U (p)),\forall p \in P ψ ( p ) = ( π ( p ) , ϕ U ( p ) ) , ∀ p ∈ P ϕ U : π − 1 ( U ) → G \phi_U:\pi^{-1}\left(U\right) \to G ϕ U : π − 1 ( U ) → G ϕ U ( p g ) = ϕ U ( p ) g \phi_U(pg)= \phi_U(p)g ϕ U ( p g ) = ϕ U ( p ) g
则称 P P P M M M 主丛 (principal fiber bundle) 。
注：主丛就是纤维丛定义中标准纤维 F F F G G G R R R P x = { p ∈ P ∣ π ( p ) = x } P_x = \{ p \in P \mid \pi(p) = x\} P x = { p ∈ P ∣ π ( p ) = x } G G G P x P_x P x ϕ U \phi_U ϕ U p ∈ U ⊆ P p\in U\subseteq P p ∈ U ⊆ P ϕ p : π − 1 ( π ( p ) ) = P π ( p ) → G \phi_p :\pi^{-1}(\pi(p)) = P_{\pi(p)} \to G ϕ p : π − 1 ( π ( p ) ) = P π ( p ) → G R p : G → P π ( p ) R_p : G \to P_{\pi(p)} R p : G → P π ( p ) R p R_p R p P π ( p ) P_{\pi(p)} P π ( p ) p p p R : P × G → P R:P \times G \to P R : P × G → P P π ( p ) P_{\pi(p)} P π ( p ) p p p
( p g ) ( p h ) = p ( g h ) , ∀ g , h ∈ G (pg) (ph) = p(gh), \, \forall g,h\in G
( p g ) ( p h ) = p ( g h ) , ∀ g , h ∈ G
则纤维 P π ( p ) P_{\pi(p)} P π ( p ) P π ( p ) P_{\pi(p)} P π ( p ) p p p π − 1 ( x ) \pi^{-1}(x) π − 1 ( x ) P x P_x P x R R R ϕ U : π − 1 ( U ) → G \phi_U:\pi^{-1}\left(U\right) \to G ϕ U : π − 1 ( U ) → G p p p ϕ U ( p g ) = ϕ U ( p ) g \phi_U(pg)= \phi_U(p)g ϕ U ( p g ) = ϕ U ( p ) g
标架丛 设 π : E → M \pi:E \to M π : E → M x ∈ M x \in M x ∈ M x x x E x E_x E x { e 1 , ⋯ , e n } \{e_1,\cdots,e_n\} { e 1 , ⋯ , e n } x x x x x x F x F_x F x π F : F x ↦ x \pi_F:F_x \mapsto x π F : F x ↦ x F ( E ) F(E) F ( E )
F ( E ) = ⨆ x ∈ M F x F(E) = \bigsqcup_{x\in M} F_x
F ( E ) = x ∈ M ⨆ F x
称为与向量丛 E E E 标架丛 (frame bundle) 。标架丛 F ( E ) F(E) F ( E ) E E E ( U α , φ α ) (U_\alpha,\varphi_\alpha) ( U α , φ α ) E E E x ∈ M x \in M x ∈ M φ : E x → R n \varphi: E_x \to \mathbb{R}^n φ : E x → R n F ( E ) F(E) F ( E )
ψ : π − 1 ( U α ) → U α × G L ( n , R ) ( x , F x ) ↦ ( x , φ ( F x ) ) \begin{aligned}
\psi: \pi^{-1} (U_\alpha) &\to U_\alpha \times \mathrm{GL}(n,\mathbb{R}) \\
(x,F_x) &\mapsto (x,\varphi(F_x))
\end{aligned}
ψ : π − 1 ( U α ) ( x , F x ) → U α × G L ( n , R ) ↦ ( x , φ ( F x ) )
其中 φ ( F x ) \varphi(F_x) φ ( F x ) φ \varphi φ E x E_x E x n n n E x E_x E x
当 U α ∩ U β ≠ ∅ U_\alpha \cap U_\beta \neq \varnothing U α ∩ U β = ∅ E E E x ∈ U α ∩ U β x \in U_\alpha \cap U_\beta x ∈ U α ∩ U β E x E_x E x { e 1 , ⋯ , e n } \{e_1,\cdots,e_n\} { e 1 , ⋯ , e n } { e 1 ′ , ⋯ , e n ′ } \{e^\prime_1,\cdots,e^\prime_n\} { e 1 ′ , ⋯ , e n ′ } E x E_x E x P ∈ G L ( n , R ) P \in \mathrm{GL}(n,\mathbb{R}) P ∈ G L ( n , R )
[ e 1 ′ , ⋯ , e n ′ ] = [ e 1 , ⋯ , e n ] P [e^\prime_1,\cdots,e^\prime_n] = [e_1,\cdots,e_n] P
[ e 1 ′ , ⋯ , e n ′ ] = [ e 1 , ⋯ , e n ] P
则 F ( E ) F(E) F ( E )
ψ : π − 1 ( U α ) → U α × G L ( n , R ) ( x , F x ) ↦ ( x , P − 1 φ ( F x ) ) \begin{aligned}
\psi: \pi^{-1} (U_\alpha) &\to U_\alpha \times \mathrm{GL}(n,\mathbb{R}) \\
(x,F_x) &\mapsto (x, P^{-1} \varphi (F_x))
\end{aligned}
ψ : π − 1 ( U α ) ( x , F x ) → U α × G L ( n , R ) ↦ ( x , P − 1 φ ( F x ) )
其中 P − 1 ∈ G L ( n , R ) P^{-1} \in \mathrm{GL}(n,\mathbb{R}) P − 1 ∈ G L ( n , R )
更进一步地，标架丛是一个主丛。易见其标准纤维和结构群均为一般线性群 G L ( n , R ) \mathrm{GL}(n,\mathbb{R}) G L ( n , R ) G L ( n , R ) \mathrm{GL}(n,\mathbb{R}) G L ( n , R )
标准正交标架丛 如果向量丛 E E E E p E_p E p E p E_p E p G G G n n n O ( n ) \mathrm{O}(n) O ( n ) E E E E E E G G G n n n S O ( n ) \mathrm{SO}(n) S O ( n )
切标架丛 取 E E E T M TM T M F ( M ) F(M) F ( M )
铅直丛 设 E E E M M M π : E → M \pi : E \to M π : E → M V E VE V E T E TE T E
V E = ker π ∗ = { ξ ∈ T E ∣ π ∗ ξ = 0 } VE = \operatorname{ker} \pi_* = \left\{ \xi \in TE \mid\pi_* \xi =0\right\}
V E = k e r π ∗ = { ξ ∈ T E ∣ π ∗ ξ = 0 }
其纤维 V p E = ( V E ) p = T p E p ⊂ ( T E ) p V_p E = (VE)_p = T_p E_p\subset (TE)_p V p E = ( V E ) p = T p E p ⊂ ( T E ) p 铅锤子空间 ，铅直子空间中的元素 ξ \xi ξ
主丛上的联络 设 π : P → M \pi:P \to M π : P → M G G G { p ↦ H p } \{p \mapsto H_p\} { p ↦ H p }
T p P = H p ⊕ V p T_p P = H_p \oplus V_p T p P = H p ⊕ V p H p g = ( R g ) ∗ H p , ∀ p ∈ P , g ∈ G H_{pg} = (R_g)_* H_p,\,\forall p \in P ,g \in G H p g = ( R g ) ∗ H p , ∀ p ∈ P , g ∈ G H p H_p H p p p p
