# 空间曲面

## 定义

$\displaystyle{ \textbf{x}: D\rightarrow {\rm I\!R}^3 \qquad (u,v)\mapsto (x^1(u,v),x^2(u,v),x^3(u,v)) }$

\displaystyle{ \begin{align} &x_u=\left( \frac{\partial{x^1}}{\partial{u}}, \frac{\partial{x^2}}{\partial{u}}, \frac{\partial{x^3}}{\partial{u}}\right)\\ &x_v=\left( \frac{\partial{x^1}}{\partial{v}}, \frac{\partial{x^2}}{\partial{v}}, \frac{\partial{x^3}}{\partial{v}}\right) \end{align} }

## 曲面的几何

### 切平面

• 定理：一个向量$\displaystyle{ \textbf{v} }$在切平面$\displaystyle{ T_p(M) }$上的充要条件是它可以写成如下形式：

$\displaystyle{ v=\lambda_1 x_u+\lambda_2 x_v }$

$\displaystyle{ \textbf{v}=\alpha'(0)=\textbf{x}_u(u_0,v_0)\frac{du}{dt}(0)+\textbf{x}_v(u_0,v_0)\frac{dv}{dt}(0) }$

(2) 其次，证明，对于任意一组非零常数$\displaystyle{ \lambda_1,\lambda_2 }$$\displaystyle{ \lambda_1\textbf{x}_u+\lambda_2\textbf{x}_v }$都是某一条曲线$\displaystyle{ \alpha(t) }$的切线。

$\displaystyle{ \alpha(t)=\textbf{x}(u_0+t\lambda_1,v_0+t\lambda_2) }$

$\displaystyle{ \alpha'(0)=\lambda_1 \textbf{x}_u+\lambda_2 \textbf{x}_v }$

### 法向量

$\displaystyle{ N=\textbf{x}_u(u_0,v_0) \times \textbf{x}_v(u_0,v_0) }$

### 几何属性

#### 标量场沿曲面的运动

\displaystyle{ \begin{align} \frac{d}{dt}(g(\alpha(t)))&=\frac{\partial g}{\partial x} \frac{dx}{dt}+\frac{\partial g}{\partial y} \frac{dy}{dt}+\frac{\partial g}{\partial z} \frac{dz}{dt}\\ &=\frac{\partial g}{\partial x} \frac{d\alpha^1}{dt}+\frac{\partial g}{\partial y} \frac{d\alpha^2}{dt}+\frac{\partial g}{\partial z} \frac{d\alpha^3}{dt}\\ &=\left(\frac{\partial g}{\partial x},\frac{\partial g}{\partial y},\frac{\partial g}{\partial z}\right)\cdot \left(\frac{d\alpha^1}{dt},\frac{d\alpha^1}{dt},\frac{d\alpha^3}{dt}\right)\\ &=\nabla g(\alpha(t))\cdot \alpha'(t) \end{align} }

$\displaystyle{ \textbf{v}[g](p)\equiv \frac{d}{dt}(g(\alpha(t)))|_{t=0}=\nabla g(p)\cdot \textbf{v} }$

### 向量场沿曲面运动

$\displaystyle{ W=\sum_{i=1}^{3}w^i\textbf{e}_i }$

$\displaystyle{ \nabla_{\textbf{v}}W\equiv (\textbf{v}[w^1],\textbf{v}[w^2],\textbf{v}[w^3])=\sum_{i=1}^{3}\textbf{v}[w^i]\textbf{e}_i }$

### 形状算符

\displaystyle{ \begin{align} &S(\textbf{x}_u)\cdot \textbf{x}_u=\textbf{x}_{uu}\cdot U,\\ &S(\textbf{x}_u)\cdot \textbf{x}_v=\textbf{x}_{uv}\cdot U, \\ &S (\textbf{x}_v)\cdot \textbf{x}_v=\textbf{x}_{vv}\cdot U. \end{align} }

#### 关于形状算符的一种矩阵表示

$\displaystyle{ \left( \begin{matrix} -\frac{Fm-Gl}{EG-F^2} & \frac{-Fl+Em}{EG-F^2}\\ -\frac{-mG+Fn}{EG-F^2}&\frac{En-Fm}{EG-F^2}\\ \end{matrix} \right) }$

$\displaystyle{ l=S(\textbf{x}_u)\cdot \textbf{x}_u, m=S(\textbf{x}_u)\cdot \textbf{x}_v=S(\textbf{x}_v)\cdot \textbf{x}_u, n=S(\textbf{x}_v)\cdot \textbf{x}_v }$

### 曲面的曲率

$\displaystyle{ \kappa(\textbf{u})=S_p(\textbf{u})\cdot \textbf{u} }$

$\displaystyle{ \kappa(\textbf{u})=\kappa(0)\cos(\theta) }$

#### 高斯曲率

$\displaystyle{ \Kappa(p)=\det(S_p) }$

$\displaystyle{ H(p)=\frac{1}{2}Tr(S_p) }$

## 测地线

$\displaystyle{ \alpha''=AT+B(U\times T)+CU }$

$\displaystyle{ B=\alpha''\cdot (U\times T) }$

$\displaystyle{ C=\alpha''\cdot U }$

$\displaystyle{ B=U\cdot \alpha'\times \alpha''=|U||\alpha'\times \alpha''|\cos\theta=|T\times T'|\cos\theta=\kappa_{\alpha}\cos\theta }$

$\displaystyle{ \alpha''_{tan}=\kappa_g U\times T }$

$\displaystyle{ \alpha''_{normal}=(\alpha''\cdot U)U }$

### 测地线方程

$\displaystyle{ \alpha''=\textbf{x}_{uu} u'^2=\textbf{x}_{vv}v'u'+\textbf{x}_u u''+\textbf{x}_{vu}u'v'+\textbf{x}_{vv}v'^2+\textbf{x}_v v'' }$

$\displaystyle{ \alpha''=\textbf{x}_u\left [{u''+\frac{E_u}{2E}u'^2+\frac{E_v}{E}u'v'-\frac{G_v}{2E}v'^2}\right ] + \textbf{x}_v\left [v''-\frac{E_v}{2G}u'^2+\frac{G_u}{G}u'v'+\frac{G_v}{2G}v'^2\right ]+U\left [lu'^2+2mu'v'+nv'^2\right ] }$

\displaystyle{ \begin{align} &u''+\frac{E_u}{2E}u'^2+\frac{E_v}{E}u'v'-\frac{G_v}{2E}v'^2=0\\ &v''-\frac{E_v}{2G}u'^2+\frac{G_u}{G}u'v'+\frac{G_v}{2G}v'^2=0 \end{align} }

$\displaystyle{ l=S(\textbf{x}_u)\cdot \textbf{x}_u, m=S(\textbf{x}_u)\cdot \textbf{x}_v=S(\textbf{x}_v)\cdot \textbf{x}_u, n=S(\textbf{x}_v)\cdot \textbf{x}_v }$

## 非欧几里德空间的情况

### 度规

1. $\displaystyle{ g(a\textbf{u}+b\textbf{v},\textbf{w})=ag(\textbf{u},\textbf{w})+bg(\textbf{v},\textbf{w}) }$
2. $\displaystyle{ g(\textbf{w},a\textbf{u}+b\textbf{v})=ag(\textbf{w},\textbf{u})+bg(\textbf{w},\textbf{v}) }$

$\displaystyle{ g(\textbf{u},\textbf{v})=\textbf{u}\cdot \textbf{v} }$

$\displaystyle{ g(\textbf{w}_1,\textbf{w}_2)=\frac{\textbf{w}_1\cdot \textbf{w}_2}{v^2} }$

$\displaystyle{ g(\textbf{x},\textbf{y})=\textbf{x}^TA\textbf{y} }$

$\displaystyle{ \left( \begin{matrix} \frac{1}{v^2} & 0\\ 0&\frac{1}{v^2}\\ \end{matrix} \right) }$

## 在度规下的曲率和测地线

$\displaystyle{ K=-\frac{1}{2\sqrt{EG}}\left( \frac{\partial}{\partial v}\left(\frac{E_v}{\sqrt{EG}}\right)+\frac{\partial}{\partial u}\left(\frac{G_u}{\sqrt{EG}}\right)\right) }$

### 举例：彭加莱平面

$\displaystyle{ E=g(\textbf{x}_u,\textbf{x}_u)=\frac{1}{v^2}, F=g(\textbf{x}_u,\textbf{x}_v)=0, G=g(\textbf{x}_v,\textbf{x}_v)=\frac{1}{v^2} }$

$\displaystyle{ K=-\frac{1}{2\sqrt{1/v^4}}\left( \frac{\partial}{\partial v}\left( \frac{-2/v^3}{\sqrt{1/v^4}}\right)\right)=-1 }$

$\displaystyle{ u''-\frac{2}{v}u'v'=0, v''+\frac{1}{v}u'^2-\frac{1}{v}v'^2=0 }$

$\displaystyle{ (u-d)^2+v^2=1/c^2 }$

## 参考文献

1. John Oprea: Differential Geometry and Its Applications,Pearson Education, 2004