Problem 13(1)

弧长参数：三角换元（$x’=\cos\theta(s),y’=\sin\theta(s).$）

Problem 15: curves on sphere

$\gamma(s)\subset S^2$ satisfies $<t,\gamma(s)-P_0>,$ thus $\gamma(s)-P_0 \in span{n,b}.$

Problem 16: approximation of curves at some point

$$r(s) = r(0) + (s-\frac{\kappa^2s^3}{6})t + (\frac{\kappa s^2}{2}+\frac{\kappa’s^3}{6})n +{\frac{\kappa\tau s^3}{6}}b + \mathcal{o}(s^3).$$

Setup: Frenet Framework ${r;t,n,b}$, $P_0 = r(0), P = r(s). d(P,L) = <P-P_0,n>. d(P_0,P)^2 = x^2+y^2+z^2.$

LHS = $\lim_{s\to 0} \frac{2*(\frac{\kappa}{2})s^2 + \mathcal{O}(s^3)}{s^2 + \mathcal{O}(s^3)} = \kappa(P_0). $

Here is my quick review on root system and the classification of semisimple Lie algebras: pdf

For lemma 3.2.5 and proposition 3.3.6, I viewed them in a different way to supplement the material in the textbook. Click the Bilibili Links to see my explanation.

$\textcolor{darkblue}{\textit{Setup}:}$
$$ \gamma_0, \gamma_1: [0,1] \to \Omega $$
are $\textcolor{brown}{C^1}$ $\textbf{curves}$, $\textcolor{brown}{C^1}-\textbf{homotopic}$ (with endpoints fixed),
in the fixed endpoint case:
$$\gamma_0(0)=\gamma_1(0), \gamma_0(1)=\gamma_1(1);$$
or in the closed case:
$$\gamma_0(1)=\gamma_1(0), \gamma_0(1)=\gamma_1(0).$$

$\textcolor{darkblue}{\textit{Output}:}$
$$ \int_{\gamma_0} f(z)dz = \int_{\gamma_1} f(z)dz, \forall f \in \mathcal{H}(\Omega). $$
$\textit{In particular}$, if $\gamma$ is closed in $\Omega$ which is homotopic to a point, then
$$\oint_{\gamma} f(z)dz = 0.$$
This is the case if $\gamma$ is the boundary of a subregion $\Omega_1 \subset \Omega$ such that
$\overline{\Omega_1}$ is $\textbf{diffeomorphic} (i.e. \textbf{bi-differentiable})$ to a closed disk.

$\textcolor{darkgreen}{\textit{1. Whitney Approximation Theorem}}$

Let $M$ be a $\textcolor{brown}{\textit{smooth manifold}}$ and let $F: M \to \mathbb{R}^k$ be a $\textcolor{darkblue}{\textit{continuous}}$ function. Given any positive $\textcolor{darkblue}{\textit{continuous}}$ function $\delta: M \to \mathbb{R}$, there exists a $\textcolor{brown}{\textit{smooth}}$ function $\tilde{F}: M \to \mathbb{R}^k$ that is $\delta-$close to $F$.

$\textcolor{darkgreen}{\textit{2. Whitney Embedding Theorem}}$

$\textcolor{darkgreen}{\textit{3. Whitney Approximation Theorem on Manifolds}}$

Let $N, M$ be $\textcolor{brown}{\textit{smooth manifolds}}$, and let $F: N \to M$ be a $\textcolor{darkblue}{\textit{continuous}}$ map. Then $F$ is homotopic to a $\textcolor{brown}{\textit{smooth}}$ map $\tilde{F}: N \to M$.

$\textcolor{darkgreen}{\textit{4. Corollary}}$

If $F,G: M\to N$ are homotopic smooth maps, then they are $\textcolor{brown}{\textit{smoothly homotopic.}}$ If $F$ is homotopic to $G$ relative to some closed subset $A \subset M$, then they are smoothly homotopic relative to $A$.

By virtue of these properties of homotopy, one can apply Stoke’s Theorem which demands both $M$, the oriented $n-$dimensional manifold with boundary, and $\omega$, the compactly supported $(n-1)-$form on $M$, are smooth.

$\textcolor{darkgreen}{\textit{Definition: }}$

$\textbf{Exact Differential Forms}$: $\exists f\in C^{\infty}(M)$ such that $\omega = df$.
$\textbf{Closed Differential Forms}$: $d\omega = 0$.

$\textcolor{darkgreen}{\textit{Local Exactness of Closed covector field}}$
$\omega$: a closed covector field on a smooth manifold $M$. Then every $p \in M$ has a neighborhood on which $\omega$ is exact.

Now follows naturally the proof in page 21.

For $A\in \mathcal{X}$, it is
$\textcolor{darkblue}{\textit{列紧}:}$ 任意点列有收敛子列。
$\textcolor{darkblue}{\textit{自列紧}:}$ 该收敛子列收敛回子集$A$。

$\textcolor{darkblue}{\textit{完全有界}:}$ 任意$\epsilon$都有有穷$\epsilon$网。
$\textcolor{darkblue}{\textit{可分}:}$ 含有可数的稠密子集。

For $C(M)$ and $F\subset C(M)$ where $(M,\rho)$ is acompact normed space, F is
$\textcolor{darkblue}{\textit{一致有界（函数）}:}$ 存在$M_1$, for any $\phi \in F, x \in M, |\phi(x)|\leq M_1$. i.e. subset $F$ is 有界（函数空间） in the normed space $(C(M),d)$, here $d(u,v) = max|u-v|$.
$\textcolor{darkblue}{\textit{等度连续}:}$ For any $\epsilon, \phi \in F$, we have $$\delta(\epsilon)>0,|\phi(x_1)-\phi(x_2)|<\epsilon$$ once $\rho(x_1,x_2)<\delta$.