Machine-translated from Chinese.
The dimension of a cut locus on a smooth Riemannian manifold - Jin-Ichi
Itoh and Minoru Tanaka.
准备工作
我们来对第一次课中给出的定理进行证明. 首先给出推广的Sard定理:
定理 1 (Morse-Sard-Federer). 令$f:B_1^n(0)\rightarrow \mathbb{R}^m$为$C^1$映射, $E=\{x\in B_1^n(0)|r(df_x)\le k\}.$ 那么$\dim f(E)\le k.$
回忆对$M$为完备黎曼流形, $p\in M,$ 那么
$$
\operatorname{Cut}_p=Q_p\cup N_p\cup L_p, \quad Q_{p}:=\{x\in \operatorname{Cut}_p|\,x,p\text{共轭}\},
$$
还有$N_p$为其中恰有两条线段连接的部分,
$L_p$为其中有三条以上线段连接的部分.
给出下面几个定义:
$$
\lambda(v)=\inf \{t>0|r(d\exp_{p,tv})\le n-1\}\in (0,+\infty],
$$
$$
U_pM=\{v\in T_pM\,|\,|v|=1,\lambda(v)<+\infty\},
$$
$$
\phi(v):=\exp_p\lambda(v)v: U_pM\rightarrow M.
$$
我们给出如下的事实: 对$v\in U_pM,$
若$\nu(v):=n-r(d\exp_{p,\lambda(v)v})=1,$ 由Morse指标定理,
有$v$处邻域使得$\nu(v)\equiv 1.$ 由Jacobi场性质与隐函数定理,
$\lambda,\phi$在$v$附近是光滑的,
且$\,\exists\,\gamma:(-\varepsilon,\varepsilon)\rightarrow U_pM,$ 使得
$$
\gamma(0)=v,\quad \gamma'(t)\in \ker(d\exp_{p,\lambda(\gamma(t))\gamma(t)}).
$$
引理 2. 在前述事实基础上, 若$\phi(v)\in \operatorname{Cut}_p,$ 则$\left.\frac{d {} }{d {}t}\phi(\gamma(t))\right|_{t=0}=0.$
记$\widehat{\gamma}=\phi(\gamma(t)),$ 首先由定义,
$$
\frac{d {} }{d {}t}\widehat{\gamma}(t)=(\exp_{p,\lambda(v)v})_\ast \left(\frac{d {}\lambda(\gamma(t))}{d {}t}\gamma(t)+\lambda(\gamma(t))\frac{d {}\gamma(t)}{d {}t}\right)=\frac{d {}\lambda(\gamma(t))}{d {}t}(\exp_{p,\lambda(v)v})_\ast \gamma(t).
$$
两边套模长积分,
假设$\left.\frac{d {}\lambda(\gamma(t))}{d {}t}\right|_{t=0}<0,$
那么存在小区间$[0,a],$ 使得
$$
l(\widehat{\gamma})+\lambda(\gamma(a))=\lambda(\gamma(0)).
$$
注意到$\lambda(\gamma(a))\ge d(p,\phi(\gamma(a))),$
而由于$\phi(v)\in \operatorname{Cut}_p,$
$\lambda(\gamma(0))=d(p,\phi(v)).$ 因此,
$$
d(p,\phi(v))\le l(\widehat{\gamma})+d(p,\phi(\gamma(a)))\le d(p,\phi(v)).
$$
因此式子只能取等,
这说明$\phi(\gamma(a))$落在连接$p,\phi(v)$的最短测地线上,
这与$\lambda(\gamma(a))$的定义矛盾,
因此$\left.\frac{d {}\lambda(\gamma(t))}{d {}t}\right|_{t=0}=0,$
从而$\left.\frac{d {} }{d {}t}\phi(\gamma(t))\right|_{t=0}=0.$
证明
引理 3. $\dim Q_p\le n-2.$
我们将$Q_p$中的点分解为两部分来处理.
$$
\begin{aligned}
Q_p&=\{\exp_pv\in Q_p|r(d\exp_{p,v})\le n-1\}\\
&=\{\exp_pv\in Q_p|r(d\exp_{p,v})=n-1\}\cup \{\exp_pv\in Q_p|r(d\exp_{p,v})=n-2\}
\end{aligned}
$$
这两部分维数都是小于等于$n-2$的.
后一部分直接由推广的Sard定理得到, 前一部分间接由其得到:
它是$\{v\in U_pM|\phi(v)\in \operatorname{Cut}_p,\nu(v)=1\}$在$\phi$下的像,
且$r(d\phi)\le n-2.$
引理 4. $N_p$为$n-1$维嵌入子流形.
取$x\in N_p,$ 有两条从$p$引出的测地线, 记在$T_pM$上终点分别是$x_1,x_2.$
取它们的邻域$U_1,U_2,$ 使得$\exp_p|_{U_i}$为微分同胚且像一致.
令$\phi=\exp_p|_{U_2}^{-1}\circ \exp_p|_{U_1}:U_1\approx U_2.$
考虑$f(v)=|v|-|\phi(v)|\in C^\infty(U_1).$ 不妨设$f$非退化,
那么存在$(n-1)$维嵌入子流形$\Sigma_1,$ 其上有$|v|=|\phi(v)|.$
令$\Sigma_2=\phi(\Sigma_1),$ $\Sigma=\exp_p\Sigma_1=\exp_p\Sigma_2.$
分析可以得到$\Sigma=N_p\cap \exp_{p}|_{U_1},$
从而$N_p$是$n-1$维嵌入子流形.
类似地, 可以证明:
引理 5. $\dim L_p\le n-2$
所以我们可以看出, $\partial B_R^g(p)\cap N_p$中坏的部分主要集中$N_p$上.
$$
\partial B_R^g(p)=(\partial B_R^g(p)\cap S_p)\cup (\partial B_R^g(p)\cap N_p)\cup (\partial B_R^g(p)\cap (L_p\cup Q_p)).
$$
在$\widehat{C}_p\subset T_pM$上, 取法坐标系$\widehat{g}=\exp_p^\ast (g),$
$(\widehat{C}_p,\widehat{g})\cong (C_p,g).$ $r(v)=|v|$是光滑函数.
考虑$\widehat{N}_p=\exp_p^{-1}(N_p)$为$\widehat{C}_p$的$n-1$维子流形,
那么一个事实是:
命题 6. $\widehat{N}_p$有可数个连通分支, 是星型的.
由Sard定理, $\,\exists\,A\subset \mathbb{R}_+,$ $L^1(A)=0,$
$R\notin A$是$\widehat{N}_p$上$r(v)$的正则值. $\,\forall\,a.e. R,$
$$
\partial B_R^g(p)=(\partial B_R^g(p)\cap S_p)\cup \{\text{零测集}\}
$$
对不是正则值的$R,$ $\partial B_R^g(p)\cap N_p$是一个$n-1$维varifold,
为$\exp_p(\partial B_R^n(0)\cap \widehat{N}_p).$
$\widehat{N}_p$与$\partial B_R^n(0)$在坏点交集处的切空间是一致的,
有后面将会讨论的体积公式:
$$
\mathcal{H}^{n-1}(\partial B_R^g(p)\cap N_p)=\int \Theta d\mathcal{H}^{n-1}
$$
文章最后更新于 2022-09-22 22:35:04
The dimension of a cut locus on a smooth Riemannian manifold - Jin-Ichi
Itoh and Minoru Tanaka.
Preparation
Let's prove the theorem given in the first lesson. First, we give the generalized Sard's theorem:
Theorem 1 (Morse-Sard-Federer). Let $f:B_1^n(0)\rightarrow \mathbb{R}^m$ be the mapping of $C^1$, $E=\{x\in B_1^n(0)|r(df_x)\le k\}.$ then $\dim f(E)\le k.$
Recall that $M$ is a complete Riemannian manifold, $p\in M,$ then
$$
\operatorname{Cut}_p=Q_p\cup N_p\cup L_p, \quad Q_{p}:=\{x\in \operatorname{Cut}_p|\,x,p\text{共轭}\},
$$
And $N_p$ is the part where exactly two line segments are connected,
$L_p$ is the part connected by more than three line segments.
Give the following definitions:
$$
\lambda(v)=\inf \{t>0|r(d\exp_{p,tv})\le n-1\}\in (0,+\infty],
$$
$$
U_pM=\{v\in T_pM\,|\,|v|=1,\lambda(v)<+\infty\},
$$
$$
\phi(v):=\exp_p\lambda(v)v: U_pM\rightarrow M.
$$
We give the following facts: For $v\in U_pM,$
If $\nu(v):=n-r(d\exp_{p,\lambda(v)v})=1,$ is determined by Morse’s index theorem,
There is a neighborhood of $v$ such that $\nu(v)\equiv 1.$ According to Jacobi field properties and implicit function theorem,
$\lambda,\phi$ is smooth near $v$,
And $\,\exists\,\gamma:(-\varepsilon,\varepsilon)\rightarrow U_pM,$ makes
$$
\gamma(0)=v,\quad \gamma'(t)\in \ker(d\exp_{p,\lambda(\gamma(t))\gamma(t)}).
$$
Lemma 2. Based on the aforementioned facts, if $\phi(v)\in \operatorname{Cut}_p,$ then $\left.\frac{d {} }{d {}t}\phi(\gamma(t))\right|_{t=0}=0.$
Note that $\widehat{\gamma}=\phi(\gamma(t)),$ is first defined,
$$
\frac{d {} }{d {}t}\widehat{\gamma}(t)=(\exp_{p,\lambda(v)v})_\ast \left(\frac{d {}\lambda(\gamma(t))}{d {}t}\gamma(t)+\lambda(\gamma(t))\frac{d {}\gamma(t)}{d {}t}\right)=\frac{d {}\lambda(\gamma(t))}{d {}t}(\exp_{p,\lambda(v)v})_\ast \gamma(t).
$$
The mold length integral on both sides is
Assumption$\left.\frac{d {}\lambda(\gamma(t))}{d {}t}\right|_{t=0}<0,$
Then there is a small interval $[0,a],$ such that
$$
l(\widehat{\gamma})+\lambda(\gamma(a))=\lambda(\gamma(0)).
$$
Notice$\lambda(\gamma(a))\ge d(p,\phi(\gamma(a))),$
And because $\phi(v)\in \operatorname{Cut}_p,$
$\lambda(\gamma(0))=d(p,\phi(v)).$ Therefore,
$$
d(p,\phi(v))\le l(\widehat{\gamma})+d(p,\phi(\gamma(a)))\le d(p,\phi(v)).
$$
Therefore, the formula can only be equal to,
This shows that $\phi(\gamma(a))$ falls on the shortest geodesic connecting $p,\phi(v)$,
This contradicts the definition of $\lambda(\gamma(a))$,
Therefore $\left.\frac{d {}\lambda(\gamma(t))}{d {}t}\right|_{t=0}=0,$
Thus $\left.\frac{d {} }{d {}t}\phi(\gamma(t))\right|_{t=0}=0.$
prove
Lemma 3. $\dim Q_p\le n-2.$
We decompose the points in $Q_p$ into two parts for processing.
$$
\begin{aligned}
Q_p&=\{\exp_pv\in Q_p|r(d\exp_{p,v})\le n-1\}\\
&=\{\exp_pv\in Q_p|r(d\exp_{p,v})=n-1\}\cup \{\exp_pv\in Q_p|r(d\exp_{p,v})=n-2\}
\end{aligned}
$$
The dimensions of both parts are less than or equal to $n-2$.
The latter part is directly obtained from the generalized Sard's theorem, and the former part is indirectly obtained from it:
It is the image of $\{v\in U_pM|\phi(v)\in \operatorname{Cut}_p,\nu(v)=1\}$ under $\phi$,
And $r(d\phi)\le n-2.$
Lemma 4. $N_p$ is the $n-1$-dimensional embedding submanifold.
Take $x\in N_p,$ and there are two geodesic lines leading from $p$. Record them on $T_pM$ and their end points are $x_1,x_2.$.
Take their neighborhood $U_1,U_2,$ such that $\exp_p|_{U_i}$ is diffeomorphic and image-consistent.
Let$\phi=\exp_p|_{U_2}^{-1}\circ \exp_p|_{U_1}:U_1\approx U_2.$
Considering $f(v)=|v|-|\phi(v)|\in C^\infty(U_1).$, we might as well assume that $f$ is non-degenerate,
Then there is a $(n-1)$-dimensional embedding submanifold $\Sigma_1,$ on which there is $|v|=|\phi(v)|.$
Order $\Sigma_2=\phi(\Sigma_1),$ $\Sigma=\exp_p\Sigma_1=\exp_p\Sigma_2.$
Analysis can get $\Sigma=N_p\cap \exp_{p}|_{U_1},$
Thus $N_p$ is a $n-1$-dimensional embedding submanifold.
Similarly, it can be proved that:
Lemma 5. $\dim L_p\le n-2$
So we can see that the bad parts in $\partial B_R^g(p)\cap N_p$ are mainly concentrated in $N_p$.
$$
\partial B_R^g(p)=(\partial B_R^g(p)\cap S_p)\cup (\partial B_R^g(p)\cap N_p)\cup (\partial B_R^g(p)\cap (L_p\cup Q_p)).
$$
On $\widehat{C}_p\subset T_pM$, take the normal coordinate system $\widehat{g}=\exp_p^\ast (g),$
$(\widehat{C}_p,\widehat{g})\cong (C_p,g).$ $r(v)=|v|$ is a smooth function.
Consider $\widehat{N}_p=\exp_p^{-1}(N_p)$ to be the $n-1$-dimensional submanifold of $\widehat{C}_p$,
Then the fact is:
Proposition 6. $\widehat{N}_p$ has countable connected components and is star-shaped.
From Sard’s theorem, $\,\exists\,A\subset \mathbb{R}_+,$ $L^1(A)=0,$
$R\notin A$ is the regular value of $r(v)$ on $\widehat{N}_p$. $\,\forall\,a.e. R,$
$$
\partial B_R^g(p)=(\partial B_R^g(p)\cap S_p)\cup \{\text{零测集}\}
$$
For $R,$ $\partial B_R^g(p)\cap N_p$ that is not a regular value, it is a $n-1$ dimensional varifold,
for $\exp_p(\partial B_R^n(0)\cap \widehat{N}_p).$
The tangent spaces of $\widehat{N}_p$ and $\partial B_R^n(0)$ at the intersection of bad points are consistent.
There is a volume formula that will be discussed later:
$$
\mathcal{H}^{n-1}(\partial B_R^g(p)\cap N_p)=\int \Theta d\mathcal{H}^{n-1}
$$
The article was last updated on 2022-09-22 22:35:04
