习题 An Introduction to Manifold - Loring W. Tu, Problem 14.13

给定光滑流形 $M,N$
给定光滑同胚映射 $F : M \to N$
给定 $M$ 上的光滑函数 $g$ 和光滑切向量场 $X$
证明

$$ \bbox[5pt]{ F_*(gX) = (g \circ F^{-1})F_*(X) } $$

Proof.

$\forall f \in \mathtip{C^\infty(N)}{M 上的全体光滑函数} , \forall p \in \mathtip{N}{视作集合}$ $$ \bbox[5pt]{ \begin{aligned} & \mathtip{\langle{F_*(gX),f}\rangle}{\langle{\mathfrak{X}(N),C^\infty(N)}\rangle \to C^\infty(N)}(p) \\ = & \mathtip{\langle{F_*(gX)(p),f}\rangle}{\langle{T_pN,C^\infty(N)}\rangle\to\mathbb{R}} \\ = & \mathtip{\left\langle{\underset{*,F^{-1}(p)}{F}(g(F^{-1}(p))X_{F^{-1}(p)}),f}\right\rangle}{\langle{T_pN,C^\infty(N)}\rangle\to\mathbb{R}} \\ = & \mathtip{\left\langle{g(F^{-1}(p))X_{F^{-1}(p)},f \circ F}\right\rangle}{\langle{T_{F^{-1}(p)}M,C^\infty(M)}\rangle\to\mathbb{R}} \\ = & (g \circ F^{-1})(p)\mathtip{\left\langle{X_{F^{-1}(p)},f \circ F}\right\rangle}{\langle{T_{F^{-1}(p)}M,C^\infty(M)}\rangle\to\mathbb{R}} \\ = & (g \circ F^{-1})(p)\mathtip{\left\langle{\underset{*,F^{-1}(p)}{F}(X_{F^{-1}(p)}),f}\right\rangle}{\langle{T_pN,C^\infty(N)}\rangle\to\mathbb{R}} \\ = & (g \circ F^{-1})(p)\mathtip{\langle{F_*(X)(p),f}\rangle}{\langle{T_pN,C^\infty(N)}\rangle\to\mathbb{R}} \\ = & (g \circ F^{-1})(p)\mathtip{\langle{F_*(X),f}\rangle}{\langle{\mathfrak{X}(N),C^\infty(N)}\rangle \to C^\infty(N)}(p)\\ = & \mathtip{\langle{(g \circ F^{-1})F_*(X),f}\rangle}{\langle{\mathfrak{X}(N),C^\infty(N)}\rangle \to C^\infty(N)}(p) \end{aligned} } $$

综上

$$ \bbox[5pt]{ F_*(gX) = (g \circ F^{-1})F_*(X) } $$