流形上Laplace算子局部表达式

一般张量场的散度

$\newcommand{\ppt}[1]{\frac{\pt}{\pt{#1}}}$

$$\div(S)=\tr(X\to\nabla_X S),$$

\begin{align*} [\div(S)]_{j_1j_2\ldots j_s}^{i_1i_2\ldots i_{r-1}}& =[\div(S)]\bigg(\rd x^{i_1},\rd x^{i_2},\cdots,\rd x^{i_{r-1}},\ppt{x^{j_1}},\ppt{x^{j_2}},\cdots,\ppt{x^{j_s}}\bigg)\\ &=\tr\left(X\mapsto(\nabla_X S)\bigg(\rd x^{i_1},\cdots,\rd x^{i_{r-1}},\ppt{x^{j_1}},\cdots,\ppt{x^{j_s}}\bigg)\right)\\ &=g^{ij}\bigg\langle\ppt{x^{i}}, \left(\nabla_{\ppt{x^j}}S\right) \bigg(\rd x^{i_1},\cdots,\rd x^{i_{r-1}},\ppt{x^{j_1}},\cdots,\ppt{x^{j_s}}\bigg)\bigg\rangle\\ &=g^{ij}\bigg\langle \ppt{x^{i}},S^{i_1i_2\ldots i_{r-1}i_r}_{j_1j_2\ldots j_s;j}\ppt{x^{i_r}} \bigg\rangle\\ &=S^{i_1i_2\ldots i_{r-1}j}_{j_1j_2\ldots j_s;j}. \end{align*}

切向量场的散度

$$\div(S)=S^i_{;i}=\frac{\pt S^i}{\pt x^i}+S^k\Gamma_{ki}^i,$$

$$\Gamma_{ki}^i=\frac{1}{2}g^{il}\left\{ \pt_kg_{li}+\pt_{i}g_{kl}-\pt_l g_{ki} \right\}=\frac{1}{2}g^{il}\pt_kg_{li}.$$

$$\pt_k G=(\pt_k g_{li})\cdot(g^{il}G),$$

$$\pt_k\log G=g^{il}\pt_k g_{li}.$$

$$\Gamma_{ki}^i=\frac{1}{2}\pt_k\log G=\pt_k\log\sqrt{G}=\frac{1}{\sqrt G}\pt_k\sqrt{G}.$$

函数的梯度场的散度

\begin{align*} \div S&=\div(\mathrm{grad} f) =f^j_{;j}\\ &=\pt_j f^j+f^j\frac{1}{\sqrt G}\pt_j\sqrt{G}\\ &=\frac{1}{\sqrt G}\pt_j(f^j\sqrt{G})\\ &=\frac{1}{\sqrt G}\pt_j\bigg(\sqrt{G}g^{ij}\ppt{x^i} f\bigg). \end{align*}

Definition .假设$f\in C^\infty(M)$, 则Laplac-Beltrami算子定义为:

$$\Delta f:=-\div(\mathrm{grad}f).$$

分部积分公式

$$\int_M\Delta f h \;\rd v_g=-\int_M\div(\mathrm{grad}f)h\rd v_g =\int_M\langle\mathrm{grad}f,\mathrm{grad}h\rangle\rd v_g,$$