伴随外微分算子

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 定义 7.6

设 $\mathsf{d}:{\Omega }^{r-1}(M)\to {\Omega }^r(M)$ 为外微分算子，则伴随外微分算子(adjoint exterior derivative operator) ${\mathrm{d}}^{†}:{\Omega }^r(M)\to {\Omega }^{r-1}(M)$ 定义为

$${\mathrm{d}}^{†}=(-1{)}^{mr+m+1}\ast \mathrm{d}\ast \text{\hspace{1em}(7.184a)}$$

若 $(M,g)$ 是黎曼的，则

$${\mathrm{d}}^{†}=(-1{)}^{mr+m}\ast \mathrm{d}\ast \text{\hspace{1em}(7.184b)}$$

若 $m=\dim M$ 为洛伦兹的，则

总之，我们有如下交换图（对黎曼流形而言）：

$$\begin{array}{ccc}{\Omega }^{m-r}(M)& \stackrel{(-1{)}^{mr+m+1}d}{\to }& {\Omega }^{m-r+1}(M)\\ \ast & & \downarrow \ast \\ {\Omega }^r(M)& \stackrel{d^{†}}{\to }& {\Omega }^{r-1}(M).\end{array}\text{\hspace{1em}(7.185)}$$

算子 ${\mathrm{d}}^{†}$ 是幂零的，因为 $\mathrm{d}$ 是幂零的：${\mathrm{d}}^{†2}=\ast \mathrm{d}\ast \ast \mathrm{d}\ast \propto \ast {\mathrm{d}}^2\ast =0$。

英文原文

Let $\mathsf{d}:{\Omega }^{r-1}(M)\to {\Omega }^r(M)$ be the exterior derivative operator. The adjoint exterior derivative operator ${\mathrm{d}}^{†}:{\Omega }^r(M)\to {\Omega }^{r-1}(M)$ is defined by

$${\mathrm{d}}^{†}=(-1{)}^{mr+m+1}\ast \mathrm{d}\ast \text{\hspace{1em}(7.184a)}$$

if $(M,g)$ is Riemannian and

$${\mathrm{d}}^{†}=(-1{)}^{mr+m}\ast \mathrm{d}\ast \text{\hspace{1em}(7.184b)}$$

if Lorentzian, where $m=\dim M$ .

In summary, we have the following diagram (for a Riemannian manifold),

$$\begin{array}{ccc}{\Omega }^{m-r}(M)& \stackrel{(-1{)}^{mr+m+1}d}{\to }& {\Omega }^{m-r+1}(M)\\ \ast & & \downarrow \ast \\ {\Omega }^r(M)& \stackrel{d^{†}}{\to }& {\Omega }^{r-1}(M).\end{array}\text{\hspace{1em}(7.185)}$$

The operator ${\mathrm{d}}^{†}$ is nilpotent since $\mathrm{d}$ is: ${\mathrm{d}}^{†2}=\ast \mathrm{d}\ast \ast \mathrm{d}\ast \propto \ast {\mathrm{d}}^2\ast =0$ .

出处：7.9 微分形式与霍奇理论 · node a-a2fac6d50192854165ff2d27 · 源节点 SHA-256 59912236c3754333…