习题 An Introduction to Manifold - Loring W. Tu, Problem 14.10

给定光滑流形 $M$
给定 $M$ 上的两个光滑切向量场 $X,Y$
给定 $M$ 上的两个光滑函数 $f,g$
验证

$$ \bbox[5pt]{ \mathtip{[fX,gY]}{Lie 括号: \langle{\mathfrak{X}(M),\mathfrak{X}(M)}\rangle\to\mathfrak{X}(M)} = fg\mathtip{[X,Y]}{Lie 括号: \langle{\mathfrak{X}(M),\mathfrak{X}(M)}\rangle\to\mathfrak{X}(M)} + f\mathtip{\langle{X,g}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}Y - g\mathtip{\langle{Y,f}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}X } $$

Proof.

$\forall h \in \mathtip{C^\infty(M)}{M 上的全体光滑函数}$ $$ \bbox[5pt]{ \begin{aligned} & \mathtip{\langle{\mathtip{[fX,gY]}{Lie 括号: \langle{\mathfrak{X}(M),\mathfrak{X}(M)}\rangle\to\mathfrak{X}(M)},h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} \\ \mathtip{\color{skyblue}{=}}{定义: Lie 括号} & \mathtip{\langle{fX,\mathtip{\langle{gY,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - \mathtip{\langle{gY,\mathtip{\langle{fX,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} \\ = & \mathtip{\langle{fX,g\mathtip{\langle{Y,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - \mathtip{\langle{gY,f\mathtip{\langle{X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}\\ = & fg\mathtip{\langle{X,\mathtip{\langle{Y,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} + f\mathtip{\langle{X,g}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}\mathtip{\langle{Y,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - fg\mathtip{\langle{Y,\mathtip{\langle{X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - g\mathtip{\langle{Y,f}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}\mathtip{\langle{X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} \\ = & fg\mathtip{\langle{X,\mathtip{\langle{Y,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} + \mathtip{\langle{f\mathtip{\langle{X,g}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}Y,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - fg\mathtip{\langle{Y,\mathtip{\langle{X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - \mathtip{\langle{g\mathtip{\langle{Y,f}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} \\ = & fg\mathtip{\langle{X,\mathtip{\langle{Y,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - fg\mathtip{\langle{Y,\mathtip{\langle{X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} + \mathtip{\langle{f\mathtip{\langle{X,g}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}Y,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - \mathtip{\langle{g\mathtip{\langle{Y,f}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} \\ \mathtip{\color{skyblue}{=}}{定义: Lie 括号} & fg\mathtip{\langle{\mathtip{[X,Y]}{Lie 括号: \langle{\mathfrak{X}(M),\mathfrak{X}(M)}\rangle\to\mathfrak{X}(M)},h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} + \mathtip{\langle{f\mathtip{\langle{X,g}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}Y,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - \mathtip{\langle{g\mathtip{\langle{Y,f}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} \\ = & \mathtip{\langle{fg\mathtip{[X,Y]}{Lie 括号: \langle{\mathfrak{X}(M),\mathfrak{X}(M)}\rangle\to\mathfrak{X}(M)},h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} + \mathtip{\langle{f\mathtip{\langle{X,g}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}Y,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} - \mathtip{\langle{g\mathtip{\langle{Y,f}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} \\ = & \mathtip{\langle{fg\mathtip{[X,Y]}{Lie 括号: \langle{\mathfrak{X}(M),\mathfrak{X}(M)}\rangle\to\mathfrak{X}(M)} + f\mathtip{\langle{X,g}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}Y - g\mathtip{\langle{Y,f}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}X,h}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)} \end{aligned} } $$

综上

$$ \bbox[5pt]{ \mathtip{[fX,gY]}{Lie 括号: \langle{\mathfrak{X}(M),\mathfrak{X}(M)}\rangle\to\mathfrak{X}(M)} = fg\mathtip{[X,Y]}{Lie 括号: \langle{\mathfrak{X}(M),\mathfrak{X}(M)}\rangle\to\mathfrak{X}(M)} + f\mathtip{\langle{X,g}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}Y - g\mathtip{\langle{Y,f}\rangle}{\langle\mathfrak{X}(M),C^\infty(M)\rangle\to C^\infty(M)}X } $$