# 霍奇对偶本文重定向自 霍奇对偶

## 维数与代数

${\displaystyle {n \choose k},\,}$

${\displaystyle {n \choose n-k},\,}$

1, 3, 3, 1

## k-向量的霍奇星号的正式定义

${\displaystyle \star (e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}})=e_{i_{k+1}}\wedge e_{i_{k+2}}\wedge \cdots \wedge e_{i_{n}},}$

${\displaystyle *(e_{1}\wedge e_{2}\wedge \cdots \wedge e_{k})=e_{k+1}\wedge e_{k+2}\wedge \dots \wedge e_{n}.}$

## 星算子的指标记法

${\displaystyle (*\eta )_{i_{1},i_{2},\ldots ,i_{n-k}}={\frac {1}{k!}}\eta ^{j_{1},\ldots ,j_{k}}\,{\sqrt {|\det g|}}\,\epsilon _{j_{1},\ldots ,j_{k},i_{1},\ldots ,i_{n-k}},\,}$

## 例子

${\displaystyle *\mathrm {d} x=\mathrm {d} y\wedge \mathrm {d} z}$

${\displaystyle *\mathrm {d} y=\mathrm {d} z\wedge \mathrm {d} x}$

${\displaystyle *\mathrm {d} z=\mathrm {d} x\wedge \mathrm {d} y}$

n = 4 时，霍奇对偶作用在第二外幂（6 维）上是自同态。它是一个对合，从而可以分解为子对偶与反自对偶子空间，在这两个子空间上的作用分别为 +1 和 -1。

${\displaystyle *\,\mathrm {d} t=\mathrm {d} x\wedge \mathrm {d} y\wedge \mathrm {d} z}$
${\displaystyle *\,\mathrm {d} x=\mathrm {d} t\wedge \mathrm {d} y\wedge \mathrm {d} z}$
${\displaystyle *\,\mathrm {d} y=-\mathrm {d} t\wedge \mathrm {d} x\wedge \mathrm {d} z}$
${\displaystyle *\,\mathrm {d} z=\mathrm {d} t\wedge \mathrm {d} x\wedge \mathrm {d} y}$

2-形式

${\displaystyle *\,\mathrm {d} t\wedge \mathrm {d} x=-\mathrm {d} y\wedge \mathrm {d} z}$
${\displaystyle *\,\mathrm {d} t\wedge \mathrm {d} y=\mathrm {d} x\wedge \mathrm {d} z}$
${\displaystyle *\,\mathrm {d} t\wedge \mathrm {d} z=-\mathrm {d} x\wedge \mathrm {d} y}$
${\displaystyle *\,\mathrm {d} x\wedge \mathrm {d} y=\mathrm {d} t\wedge \mathrm {d} z}$
${\displaystyle *\,\mathrm {d} x\wedge \mathrm {d} z=-\mathrm {d} t\wedge \mathrm {d} y}$
${\displaystyle *\,\mathrm {d} y\wedge \mathrm {d} z=\mathrm {d} t\wedge \mathrm {d} x}$

## k-向量的内积

${\displaystyle \zeta \wedge *\eta =\langle \zeta ,\eta \rangle \;\omega ,\,}$

${\displaystyle \omega ={\sqrt {|\det g_{ij}|}}\;dx^{1}\wedge \ldots \wedge dx^{n},\,}$

## 对偶性

${\displaystyle **\eta =(-1)^{k(n-k)}s\;\eta ,\,}$

## 流形上的霍奇星号

${\displaystyle (\eta ,\zeta )=\int _{M}\eta \wedge *\zeta .\,}$

（截面的集合通常记做 ${\displaystyle \Omega ^{k}(M)=\Gamma (\Lambda ^{k}(M))}$；里面的元素称为外 k-形式。）

### 余微分

${\displaystyle \delta =(-1)^{nk+n+1}*d*\,}$

${\displaystyle d:\Omega ^{k}(M)\rightarrow \Omega ^{k+1}(M),\,}$

${\displaystyle \delta :\Omega ^{k}(M)\rightarrow \Omega ^{k-1}(M).\,}$

${\displaystyle \langle \delta \zeta ,\eta \rangle =\langle \zeta ,d\eta \rangle .\,}$

${\displaystyle \int _{M}d(\zeta \wedge *\eta )=0.\,}$

${\displaystyle \Delta =\delta d+d\delta }$

${\displaystyle \langle \Delta \zeta ,\eta \rangle =\langle \zeta ,\Delta \eta \rangle ,\,}$

${\displaystyle \langle \Delta \eta ,\eta \rangle \geq 0.\,}$

${\displaystyle \star :H_{\Delta }^{k}(M)\to H_{\Delta }^{n-k}(M),\,}$

## 三维中的导数

${\displaystyle \ast }$ 算子与外导数 ${\displaystyle d}$ 的组合推广了三维经典算子 gradcurldiv。具体做法如下：${\displaystyle d}$ 将一个 0-形式（函数）变成 1-形式，1-形式变成 2-形式，2-形式变成 3-形式（应用到 3-形式变成零）。

1. 对一个 0-形式（${\displaystyle \omega =f(x,y,z)}$），第一种情形，写成分量与 ${\displaystyle \operatorname {grad} }$ 算子等价：

${\displaystyle d\omega ={\frac {\partial f}{\partial x}}dx+{\frac {\partial f}{\partial y}}dy+{\frac {\partial f}{\partial z}}dz.}$

2. 第二种情形后面跟着 ${\displaystyle \ast }$，是 1-形式（${\displaystyle \omega =Adx+Bdy+Cdz}$）上一个算子，其分量是 ${\displaystyle \operatorname {curl} }$ 算子：

${\displaystyle d\omega =\left({\partial C \over \partial y}-{\partial B \over \partial z}\right)dy\wedge dz+\left({\partial C \over \partial x}-{\partial A \over \partial z}\right)dx\wedge dz+\left({\partial B \over \partial x}-{\partial A \over \partial y}\right)dx\wedge dy.}$

${\displaystyle \ast d\omega =\left({\partial C \over \partial y}-{\partial B \over \partial z}\right)dx-\left({\partial C \over \partial x}-{\partial A \over \partial z}\right)dy+\left({\partial B \over \partial x}-{\partial A \over \partial y}\right)dz.}$

3. 最后一种情形，前面与后面都有一个 ${\displaystyle \ast }$，将一个 1-形式（${\displaystyle \omega =Adx+Bdy+Cdz}$）变成 0-形式（函数）；写成分量是 ${\displaystyle \operatorname {div} }$ 算子：

${\displaystyle \ast \omega =Ady\wedge dz-Bdx\wedge dz+Cdx\wedge dy}$
${\displaystyle d\ast \omega =\left({\frac {\partial A}{\partial x}}+{\frac {\partial B}{\partial y}}+{\frac {\partial C}{\partial z}}\right)dx\wedge dy\wedge dz}$
${\displaystyle \ast d\ast \omega ={\frac {\partial A}{\partial x}}+{\frac {\partial B}{\partial y}}+{\frac {\partial C}{\partial z}}.}$

${\displaystyle \operatorname {curl} (\operatorname {grad} (f))=\operatorname {div} (\operatorname {curl} (\mathbf {F} ))=0}$

${\displaystyle \mathrm {d} \mathbf {F} =0,\qquad \mathrm {d} *{\mathbf {F} }=\mathbf {J} .}$

## 注释

1. ^ Darling, R. W. R. Differential forms and connections. Cambridge University Press. 1994.

## 参考文献

• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (Provides a basic review of differential geometry in the special case of four-dimensional space-time.)
• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 . (Provides a detailed exposition starting from basic principles, but does not treat the pseudo-Riemannian case).
• David Bleecker, Gauge Theory and Variational Principles, (1981) Addison-Wesley Publishing, New York' ISBN 0-201-10096-7. (Provides condensed review of non-Riemannian differential geometry in chapter 0).