假设$N$是一个光滑流形等距地嵌入到$\mathbb{R}^K$, 我们知道存在$N$的管状邻域$N(\delta)\subset\mathbb{R}^K$, 使得定义在$N(\delta)$上的映射$\Pi$:
\[
\Pi:y\mapsto x\in N,\quad \mathrm{dist}(y,N)=|y-\Pi(y)|.
\]
关于$\Pi$, 我们有如下基本性质.
Proposition 1. 假设$\Pi$定义如上, 则
- $D\Pi|_y:\mathbb{R}^K\to T_xN$, $x=\Pi(y)$;
- 对$v_1,v_2\in T_xN$, 我们有$\mathrm{Hess}\Pi|_x(v_1,v_2)=-A(x)(v_1,v_2)$.
Proof . 假设$x=\Pi(y)$, 且令$v\in T_xN$是任一切方向而$\gamma(s)$是$N$中一弧长参数曲线(法截线), 使得$\gamma(0)=x$, $\gamma'(0)=v$. 我们始终用$\perp$来记垂直方向(单位). 为了计算$D\Pi(v+v^\perp)$, 我们选取以$v+v^\perp$为切向的一条曲线并计算其导数即可. 一个简单的选取方法是
\[
\alpha(s)=\gamma(s)+|y-x|[\gamma'(\lambda s)]^\perp,
\]
即过$y$且与$\gamma(s)$平行的曲线, 这里$\lambda$是任何常数. 注意到
\[
\Pi(\alpha(t))=\Pi(\gamma(t))=\gamma(t),
\]
两边对$s$在$s=0$求导得到
\[
v=\left.\frac{d\gamma(s)}{ds}\right|_{s=0}= D\Pi\cdot \alpha'(0).
\]
注意到, 由曲线论基本理论
\begin{align*}
\gamma(s)&=\gamma(0)+\gamma'(0)s+\gamma^\prime\prime(0)s^2/2+o(s^2)\\
&=x+sv+\kappa(0)v^\perp s^2/2+o(s^2),
\end{align*}
可见
\[
\gamma'(s)=v+\kappa(0)v^{\perp_0} s+o(s).
\]
故
\begin{align*}
\gamma'(\lambda s)&=\lambda\gamma’|_{\lambda s}=\lambda v+\kappa(0)v^{\perp_0} \lambda^2 s+o(s),\\
[\gamma'(\lambda s)]^{\perp_{\lambda s}}&=\lambda v^{\perp_{\lambda s}}+\kappa(0)(v^{\perp_0})^{\perp_{\lambda s}}\lambda^2 s+o(s)
\end{align*}
即
\[
\alpha'(0)=v+\lambda^2\kappa(0)|y-x|v^\perp.
\]
由$\lambda$的任意性知
\[
D\Pi(v+\lambda v^\perp)=v,\quad\forall \lambda.
\]
即$D\Pi|_y$是到切空间$T_{\Pi(y)}N$的正交投影.
\[
\alpha(s)=\gamma(s)+|y-x|[\gamma'(\lambda s)]^\perp,
\]
即过$y$且与$\gamma(s)$平行的曲线, 这里$\lambda$是任何常数. 注意到
\[
\Pi(\alpha(t))=\Pi(\gamma(t))=\gamma(t),
\]
两边对$s$在$s=0$求导得到
\[
v=\left.\frac{d\gamma(s)}{ds}\right|_{s=0}= D\Pi\cdot \alpha'(0).
\]
注意到, 由曲线论基本理论
\begin{align*}
\gamma(s)&=\gamma(0)+\gamma'(0)s+\gamma^\prime\prime(0)s^2/2+o(s^2)\\
&=x+sv+\kappa(0)v^\perp s^2/2+o(s^2),
\end{align*}
可见
\[
\gamma'(s)=v+\kappa(0)v^{\perp_0} s+o(s).
\]
故
\begin{align*}
\gamma'(\lambda s)&=\lambda\gamma’|_{\lambda s}=\lambda v+\kappa(0)v^{\perp_0} \lambda^2 s+o(s),\\
[\gamma'(\lambda s)]^{\perp_{\lambda s}}&=\lambda v^{\perp_{\lambda s}}+\kappa(0)(v^{\perp_0})^{\perp_{\lambda s}}\lambda^2 s+o(s)
\end{align*}
即
\[
\alpha'(0)=v+\lambda^2\kappa(0)|y-x|v^\perp.
\]
由$\lambda$的任意性知
\[
D\Pi(v+\lambda v^\perp)=v,\quad\forall \lambda.
\]
即$D\Pi|_y$是到切空间$T_{\Pi(y)}N$的正交投影.
最后, 注意到按照映照Hessian的定义(仿照函数Hessian),
\begin{align*}
\mathrm{Hess}\Pi(v_1,v_2)&=D_{v_1}D_{v_2}\Pi-D_{D_{v_1}v_2}\Pi
=D_{v_1}(D\Pi(v_2))-D\Pi(D_{v_1}v_2)\\
&=D_{v_1}v_2-[D_{v_1}v_2]^{\top}
=[D_{v_1}v_2]^\bot.
\end{align*}
而按照第二基本型的定义, 对$T_xN$处的法向量$\set{\nu_i}$,
\[
A^i(x)(v_1,v_2)=\inner{D_{v_1}\nu_i, v_2}
=-\inner{\nu_i,D_{v_1}v_2}
\]
故
\[
A(x)(v_1,v_2)=-\inner{\nu_i,D_{v_1}v_2}\nu_i=-[D_{v_1}v_2]^\perp.
\]
这表明第二条成立.