Machine-translated from Chinese.
第一变分公式
我们来研究$\gamma$曲线的长度极短性.
取单参数曲线族$\{\gamma_u\}_{-\varepsilon\le u\le \varepsilon},$
满足$\gamma_0=\gamma,$ $\gamma_u(a)=\gamma(a),$ $\gamma_u(b)=\gamma(b).$
记$L(u):[-\varepsilon,\varepsilon]\rightarrow \mathbb{R}$为曲线$\gamma_u$的长度.
那么当$L'(0)=0,$ $L''(0)>0$时, $L(0)$处有极小值;
若这对所有满足条件的单参数曲线族$\{\gamma_u\}$都成立,
则局部上$\gamma$取得极短长度.
我们先取消$\gamma_u(a)=\gamma(a),$ $\gamma_u(b)=\gamma(b)$的假设,
令$U$为$\{\gamma_u\}$的横截向量场, $T$切于所有曲线$\gamma_u(t).$
不妨设$t$是基曲线$\gamma$的弧长参数, $|\dot\gamma|=1.$ 那么,
$$
L(u)=\int_a^b |\dot\gamma_u(t)|dt,\quad L'(u)=\int_a^b\frac{d {} }{d {}u}\sqrt{\left<{}\dot\gamma_u(t),\dot\gamma_u(t)\right>}dt,
$$
其中积分项可以写为:
$$
U\sqrt{\left<{}T,T\right>}=\frac{1}{|T|}\left<{}T,D_UT\right>=\frac{1}{|T|}\left<{}T,D_TU\right>=\frac{1}{|T|}\left(T\left<{}T,U\right>-\left<{}D_TT,U\right>\right),
$$
这样我们就得到了弧长第一变分公式:
$$
L'(0)=\int_a^b\frac{d {} }{d {}t}\left<{}\dot\gamma,U\right>-\left<{}D_{\dot\gamma}\dot\gamma,U\right>dt=\left<{}\dot\gamma(t),U(t)\right>|_a^b-\int_a^b\left<{}\ddot\gamma(t),U(t)\right>.
$$
称$\{\gamma_u\}$为$\gamma$的正常变分, 若$\gamma_u(a)=\gamma(a),$
$\gamma_u(b)=\gamma(b),$ $\gamma_0=\gamma.$ 对于正常变分, $U(a)=U(b)=0,$
因此第一变分公式中第一项消失. 现在我们可以看到测地线的另一等价描述:
引理 1. 曲线$\gamma:[a,b]\rightarrow M$是测地线的充要条件是, 它是长度函数关于所有正常变分的临界点.
设$N$为$M$中闭子流形, $x\notin N,$
那么$\,\exists\,y\in N$使得$d(x,y)=d(x,N).$
取最短的正规测地线$\gamma:[a,b]\rightarrow M$连接$x,y.$
引理 2. $\gamma\perp N,$ 即$\dot\gamma(b)\perp N_y.$
证: 设$Y\in N_y,$ 取$\xi:[-\varepsilon,\varepsilon]\rightarrow N,$
$\xi(0)=y,$ $\dot\xi(0)=Y.$ 那么可以取到单参数曲线族$\{\gamma_u\},$
$\gamma_0=\gamma,$ $\gamma_u(a)=\gamma(a)=x,$ $\gamma_u(b)=\xi(u).$
这样横截向量场$U$满足$U(x)=0,$ $U(y)=Y,$ 从而:
$$
L'(0)=\left<{}\dot\gamma(b),U(b)\right>=\left<{}\dot\gamma(b),Y\right>=0 \quad \Rightarrow \quad \gamma\perp N.
$$
第二变分公式
一般当$L'(0)=0$时, 计算$L''(0)$才有意义. 故现设$\gamma$为正规测地线, 则:
$$
L''(u)=\int_a^b\frac{d {} }{d {}u}\left(\frac{1}{|T|}\left<{}T,D_TU\right>\right)dt,
$$
其中积分项可写为:
$$
\begin{aligned}
U\left(\frac{1}{|T|}\left<{}T,D_TU\right>\right)&=-\frac{1}{|T|^3}\left<{}T,D_TU\right>^2+\frac{1}{|T|}\left<{}D_TU,D_TU\right>+\frac{1}{|T|}\left<{}T,D_UD_TU\right>\\
&=-\frac{1}{|T|^3}\left(T\left<{}T,U\right>-\left<{}D_TT,U\right>\right)^2+\frac{1}{|T|}|D_TU|^2+\frac{1}{|T|}\left(-\left<{}T,R_{UT}U\right>+\left<{}D_TD_UU,T\right>\right).
\end{aligned}
$$
带入到$u=0$时, 注意此时$\gamma$为正规测地线,
因此:
$$
\begin{aligned}
L''(0)&=\int_a^b -\left(\frac{d {} }{d {}t}\left<{}\dot\gamma,U\right>\right)^2+|\dot U(t)|^2-\left<{}\dot \gamma,R_{U\dot\gamma}U\right>+\frac{d {} }{d {}t}\left<{}D_UU,\dot\gamma\right>dt\\
&=\left<{}D_UU,\dot\gamma\right>|^b_a+\int_a^b\left(|\dot U(t)|^2-\left<{}R_{\dot\gamma U}\dot\gamma,U\right>-\left(\left<{}\dot\gamma,U\right>'\right)^2\right)dt.
\end{aligned}
$$
这就是弧长第二变分公式, 决定项为曲率项,
称$\left<{}D_UU,\dot\gamma\right>|^b_a$为边界项,
当$\{\gamma_u\}$为正常变分时边界项消失.
由于$U(t)$沿$\dot\gamma$的切向分量并不改变$\gamma$的长度,
我们只需考虑正交分量$U^\perp(t),$
即设$\left<{}U^\perp,\dot\gamma\right>=0,$ $U=U^\perp+f\dot\gamma.$ 那么
$$
|\dot U(t)|^2-\left(\left<{}\dot\gamma(t),U(t)\right>'\right)^2=|\dot U^\perp (t)|^2+f'(t)^2-f'(t)^2=|\dot U^\perp(t)|^2,
$$
$$
\left<{}R_{\dot\gamma U}\dot\gamma,U\right>=\left<{}R_{\dot\gamma U^\perp}\dot\gamma, U^\perp\right>.
$$
因此弧长第二变分公式也可写为:
$$
L''(0)=\left<{}D_UU,\dot\gamma\right>|^b_a+\int_a^b \left(|\dot U^\perp(t)|^2-\left<{}R_{\dot\gamma U^\perp}\dot\gamma,U^\perp\right>\right)dt.
$$
后面的应用日后补充.
文章最后更新于 2022-04-22 21:06:49
first variation formula
Let's study the extremely short length of the $\gamma$ curve.
Get the single parameter curve family $\{\gamma_u\}_{-\varepsilon\le u\le \varepsilon},$
Meet $\gamma_0=\gamma,$ $\gamma_u(a)=\gamma(a),$ $\gamma_u(b)=\gamma(b).$
Let $L(u):[-\varepsilon,\varepsilon]\rightarrow \mathbb{R}$ be the length of curve $\gamma_u$.
Then when $L'(0)=0,$ $L''(0)>0$, there is a minimum value at $L(0)$;
If this holds true for all single-parameter curve families $\{\gamma_u\}$ that satisfy the conditions,
Then $\gamma$ locally obtains an extremely short length.
Let’s first cancel the assumption of $\gamma_u(a)=\gamma(a),$ $\gamma_u(b)=\gamma(b)$,
Let $U$ be the transverse vector field of $\{\gamma_u\}$, $T$ is tangent to all curves $\gamma_u(t).$
Suppose $t$ is the arc length parameter of the base curve $\gamma$, $|\dot\gamma|=1.$ then,
$$
L(u)=\int_a^b |\dot\gamma_u(t)|dt,\quad L'(u)=\int_a^b\frac{d {} }{d {}u}\sqrt{\left<{}\dot\gamma_u(t),\dot\gamma_u(t)\right>}dt,
$$
The integral term can be written as:
$$
U\sqrt{\left<{}T,T\right>}=\frac{1}{|T|}\left<{}T,D_UT\right>=\frac{1}{|T|}\left<{}T,D_TU\right>=\frac{1}{|T|}\left(T\left<{}T,U\right>-\left<{}D_TT,U\right>\right),
$$
This way we get First variation formula of arc length:
$$
L'(0)=\int_a^b\frac{d {} }{d {}t}\left<{}\dot\gamma,U\right>-\left<{}D_{\dot\gamma}\dot\gamma,U\right>dt=\left<{}\dot\gamma(t),U(t)\right>|_a^b-\int_a^b\left<{}\ddot\gamma(t),U(t)\right>.
$$
Call $\{\gamma_u\}$ $\gamma$ normal variation, if$\gamma_u(a)=\gamma(a),$
$\gamma_u(b)=\gamma(b),$ $\gamma_0=\gamma.$ For normal variation, $U(a)=U(b)=0,$
Therefore the first term in the first variation formula disappears. Now we can see another equivalent description of the geodesic:
Lemma 1. The necessary and sufficient condition for curve $\gamma:[a,b]\rightarrow M$ to be a geodesic is that it is the critical point of the length function with respect to all normal variations.
Let $N$ be the closed submanifold in $M$, $x\notin N,$
Then $\,\exists\,y\in N$ makes $d(x,y)=d(x,N).$
Take the shortest regular geodesic line $\gamma:[a,b]\rightarrow M$ and connect it to $x,y.$
Lemma 2. $\gamma\perp N,$ is $\dot\gamma(b)\perp N_y.$
Certificate: Let $Y\in N_y,$ take $\xi:[-\varepsilon,\varepsilon]\rightarrow N,$
$\xi(0)=y,$ $\dot\xi(0)=Y.$ Then you can get the single parameter curve family $\{\gamma_u\},$
$\gamma_0=\gamma,$ $\gamma_u(a)=\gamma(a)=x,$ $\gamma_u(b)=\xi(u).$
In this way, the transverse vector field $U$ satisfies $U(x)=0,$ $U(y)=Y,$ and thus:
$$
L'(0)=\left<{}\dot\gamma(b),U(b)\right>=\left<{}\dot\gamma(b),Y\right>=0 \quad \Rightarrow \quad \gamma\perp N.
$$
second variation formula
Generally, it is meaningful to calculate $L''(0)$ when $L'(0)=0$ is used. Therefore, assuming $\gamma$ is a regular geodesic, then:
$$
L''(u)=\int_a^b\frac{d {} }{d {}u}\left(\frac{1}{|T|}\left<{}T,D_TU\right>\right)dt,
$$
The integral term can be written as:
$$
\begin{aligned}
U\left(\frac{1}{|T|}\left<{}T,D_TU\right>\right)&=-\frac{1}{|T|^3}\left<{}T,D_TU\right>^2+\frac{1}{|T|}\left<{}D_TU,D_TU\right>+\frac{1}{|T|}\left<{}T,D_UD_TU\right>\\
&=-\frac{1}{|T|^3}\left(T\left<{}T,U\right>-\left<{}D_TT,U\right>\right)^2+\frac{1}{|T|}|D_TU|^2+\frac{1}{|T|}\left(-\left<{}T,R_{UT}U\right>+\left<{}D_TD_UU,T\right>\right).
\end{aligned}
$$
When brought into $u=0$, note that $\gamma$ is a regular geodesic at this time.
Therefore:
$$
\begin{aligned}
L''(0)&=\int_a^b -\left(\frac{d {} }{d {}t}\left<{}\dot\gamma,U\right>\right)^2+|\dot U(t)|^2-\left<{}\dot \gamma,R_{U\dot\gamma}U\right>+\frac{d {} }{d {}t}\left<{}D_UU,\dot\gamma\right>dt\\
&=\left<{}D_UU,\dot\gamma\right>|^b_a+\int_a^b\left(|\dot U(t)|^2-\left<{}R_{\dot\gamma U}\dot\gamma,U\right>-\left(\left<{}\dot\gamma,U\right>'\right)^2\right)dt.
\end{aligned}
$$
this is Second variation formula of arc length, the determining term is the curvature term,
Call $\left<{}D_UU,\dot\gamma\right>|^b_a$ a boundary term,
The boundary term disappears when $\{\gamma_u\}$ is a normal variation.
Since the tangential component of $U(t)$ along $\dot\gamma$ does not change the length of $\gamma$,
We only need to consider the orthogonal component $U^\perp(t),$
That is, suppose $\left<{}U^\perp,\dot\gamma\right>=0,$ $U=U^\perp+f\dot\gamma.$ then
$$
|\dot U(t)|^2-\left(\left<{}\dot\gamma(t),U(t)\right>'\right)^2=|\dot U^\perp (t)|^2+f'(t)^2-f'(t)^2=|\dot U^\perp(t)|^2,
$$
$$
\left<{}R_{\dot\gamma U}\dot\gamma,U\right>=\left<{}R_{\dot\gamma U^\perp}\dot\gamma, U^\perp\right>.
$$
Therefore, the second variation formula of arc length can also be written as:
$$
L''(0)=\left<{}D_UU,\dot\gamma\right>|^b_a+\int_a^b \left(|\dot U^\perp(t)|^2-\left<{}R_{\dot\gamma U^\perp}\dot\gamma,U^\perp\right>\right)dt.
$$
Later applications will be added in the future.
The article was last updated on 2022-04-22 21:06:49
