习题 An Introduction to Manifold - Loring W. Tu, Problem 14.12

给定 $\mathbb{R}^n$ 上的两个光滑切向量场 $X,Y$ 如下

$$ \bbox[5pt]{ X = \sum_{k=1}^{n}{a^k\frac{\partial}{\partial{x^k}}} \qquad Y = \sum_{k=1}^{n}{b^k\frac{\partial}{\partial{x^k}}} } $$

其中 $a^k,b^k \in \mathtip{C^\infty(\mathbb{R}^n)}{\mathbb{R}^n 上的全体光滑函数}$
计算 $\mathtip{[X,Y]}{Lie 括号: \langle{\mathfrak{X}(\mathbb{R}^n),\mathfrak{X}(\mathbb{R}^n)}\rangle\to\mathfrak{X}(\mathbb{R}^n)}$

Proof.

$\forall f \in \mathtip{C^\infty(\mathbb{R}^n)}{\mathbb{R}^n 上的全体光滑函数}$ $$ \bbox[5pt]{ \begin{aligned} & \mathtip{\langle{\mathtip{[X,Y]}{Lie 括号: \langle{\mathfrak{X}(\mathbb{R}^n),\mathfrak{X}(\mathbb{R}^n)}\rangle\to\mathfrak{X}(\mathbb{R}^n)},f}\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)} \\ \mathtip{\color{skyblue}{=}}{定义: Lie 括号} & \mathtip{\langle{X,\mathtip{\langle{Y,f}\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)}}\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)} - \mathtip{\langle{Y,\mathtip{\langle{X,f}\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)}}\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)} \\ = & \mathtip{\left\langle{\sum_{k=1}^{n}{a^k\frac{\partial}{\partial{x^k}}},\mathtip{\left\langle{\sum_{k=1}^{n}{b^k\frac{\partial}{\partial{x^k}}},f}\right\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)}}\right\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)} - \mathtip{\left\langle{\sum_{k=1}^{n}{b^k\frac{\partial}{\partial{x^k}}},\mathtip{\left\langle{\sum_{k=1}^{n}{a^k\frac{\partial}{\partial{x^k}}},f}\right\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)}}\right\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)} \\ = & \mathtip{\left\langle{\sum_{k=1}^{n}{a^k\frac{\partial}{\partial{x^k}}},\sum_{k=1}^{n}{b^k\frac{\partial{f}}{\partial{x^k}}}}\right\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)} - \mathtip{\left\langle{\sum_{k=1}^{n}{b^k\frac{\partial}{\partial{x^k}}},\sum_{k=1}^{n}{a^k\frac{\partial{f}}{\partial{x^k}}}}\right\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)} \\ = & \sum_{i,j=1}^{n}{\left(a^j\frac{\partial{b^i}}{\partial{x^j}} - b^j\frac{\partial{a^i}}{\partial{x^j}}\right)\frac{\partial{f}}{\partial{x^i}}} \\ = & \mathtip{\left\langle{\sum_{i,j=1}^{n}{\left(a^j\frac{\partial{b^i}}{\partial{x^j}} - b^j\frac{\partial{a^i}}{\partial{x^j}}\right)}\frac{\partial}{\partial{x^i}},f}\right\rangle}{\langle\mathfrak{X}(\mathbb{R}^n),C^\infty(\mathbb{R}^n)\rangle\to C^\infty(\mathbb{R}^n)} \end{aligned} } $$

综上

$$ \bbox[5pt]{ \mathtip{[X,Y]}{Lie 括号: \langle{\mathfrak{X}(\mathbb{R}^n),\mathfrak{X}(\mathbb{R}^n)}\rangle\to\mathfrak{X}(\mathbb{R}^n)} = \sum_{i,j=1}^{n}{\left(a^j\frac{\partial{b^i}}{\partial{x^j}} - b^j\frac{\partial{a^i}}{\partial{x^j}}\right)}\frac{\partial}{\partial{x^i}} } $$

即

$$ \bbox[5pt]{ \mathtip{\left[\sum_{k=1}^{n}{a^k\frac{\partial}{\partial{x^k}}},\sum_{k=1}^{n}{b^k\frac{\partial}{\partial{x^k}}}\right]}{Lie 括号: \langle{\mathfrak{X}(\mathbb{R}^n),\mathfrak{X}(\mathbb{R}^n)}\rangle\to\mathfrak{X}(\mathbb{R}^n)} = \sum_{i,j=1}^{n}{\left(a^j\frac{\partial{b^i}}{\partial{x^j}} - b^j\frac{\partial{a^i}}{\partial{x^j}}\right)}\frac{\partial}{\partial{x^i}} } $$