updates

chapters/chap0.tex

@@ -833,6 +833,122 @@ Here is Table~\ref{tab:analog_of_classical_probability_theory_and_non_commutativ
     \vspace{0.5cm}
 \end{table}
 
+\section{Manifolds}
+
+In this section, we will introduce some basic definitions and theorems used in manifold theory that are relevant to our study. Assuming no prior knowledge of manifold theory but basic topology understanding. We will provide brief definitions and explanations for each term. From the most abstract Manifold definition to the Riemannian manifolds and related theorems.
+
+\subsection{Manifolds}
+
+\begin{defn}
+    \label{defn:m-manifold}
+
+An $m$-manifold is a Topological space $X$ that is
+
+\begin{enumerate}
+    \item Hausdroff: every distinct two points in $X$ can be separated by two disjoint open sets.
+    \item Second countable: $X$ has countable basis.
+    \item Every point $p$ has an open neighborhood $p\in U$ that is homeomorphic to an open subset of $\mathbb{R}^m$.
+\end{enumerate}
+\end{defn}
+
+
+\begin{examples}
+    \label{example:second_countable_space}
+    Let $X=\mathbb{R}$ and $\mathcal{B}=\{(a,b)|a,b\in \mathbb{R},a<b\}$ (collection of all open intervals with rational endpoints).
+
+    Since the rational numbers are countable, so $\mathcal{B}$ is countable.
+
+    So $\mathbb{R}$ is second countable.
+
+    Likewise, $\mathbb{R}^n$ is also second countable.
+\end{examples}
+
+\begin{examples}
+    \label{example:manifold}
+1-manifold is a curve and 2-manifold is a surface.
+\end{examples}
+
+\begin{theorem}
+    \label{Theorem of imbedded space}
+    If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^n$ for some $n$.
+\end{theorem}
+
+This theorem might save you from imagining abstract structures back to real dimension. Good news, at least you stay in some real numbers.
+
+\subsection{Smooth manifolds and Lie groups}
+
+This section is adopted from \cite{lee_introduction_2012}
+
+\begin{defn}
+    \label{defn:partial_derivative}
+
+    Let $U\subseteq \mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ be a map.
+
+    For any $a=(a_1,\cdots,a_n)\in U$, $j\in \{1,\cdots,n\}$, the $j$-th partial derivative of $F$ at $a$ is defined as
+
+    $$
+    \begin{aligned}
+    \frac{\partial f}{\partial x_j}(a)&=\lim_{h\to 0}\frac{f(a_1,\cdots,a_j+h,\cdots,a_n)-f(a_1,\cdots,a_j,\cdots,a_n)}{h} \\
+    &=\lim_{h\to 0}\frac{f(a+he_j)-f(a)}{h}
+    \end{aligned}
+$$
+
+\end{defn}
+
+\begin{defn}
+    \label{defn:continuously_differentiable_map}
+    Let $U\subseteq \mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ be a map.
+
+    If for any $j\in \{1,\cdots,n\}$, the $j$-th partial derivative of $f$ is continuous at $a$, then $f$ is continuously differentiable at $a$.
+
+    If $\forall a\in U$, $\frac{\partial f}{\partial x_j}$ exists and is continuous at $a$, then $f$ is continuously differentiable on $U$. or $C^1$ map. (Note that $C^0$ map is just a continuous map.)
+\end{defn} 
+
+
+\begin{defn}
+    \label{defn:smooth_map}
+    A function $f:U\to \mathbb{R}^n$ is smooth if it is of class $C^k$ for every $k\geq 0$ on $U$. Such function is called a diffeomorphism if it is also a \textbf{\texttt{bijection}} and its \textbf{\texttt{inverse is also smooth}}.
+\end{defn}
+
+
+\begin{defn}
+    \label{defn:chart}
+
+    Let $M$ be a smooth manifold. A \textbf{\texttt{chart}} is a pair $(U,\varphi)$ where $U\subseteq M$ is an open subset and $\varphi:U\to \hat{U}\subseteq \mathbb{R}^n$ is a homeomorphism (a continuous bijection map and its inverse is also continuous).
+
+    If $p\in U$ and $\varphi(p)=0$, then we say that $p$ is the origin of the chart $(U,\varphi)$.
+
+    For $p\in U$, we note that the continuous function $\varphi(p)=(x_1(p),\cdots,x_n(p))$ gives a vector in $\mathbb{R}^n$. The $(x_1(p),\cdots,x_n(p))$ is called the \textbf{\texttt{local coordinates}} of $p$ in the chart $(U,\varphi)$.
+
+\end{defn}
+
+\begin{defn}
+    \label{defn:atlas}
+    Let $M$ be a smooth manifold. An \textbf{\texttt{atlas}} is a collection of charts $\mathcal{A}=\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$ such that $M=\bigcup_{\alpha\in I} U_\alpha$.
+
+    An atlas is said to be \textbf{\texttt{smooth}} if the transition maps $\phi_\alpha\circ \phi_\beta^{-1}:\phi_\beta(U_\alpha\cap U_\beta)\to \phi_\alpha(U_\alpha\cap U_\beta)$ are smooth for all $\alpha, \beta\in I$.
+\end{defn}
+
+
+\begin{defn}
+    \label{defn:smooth_manifold}
+    A smooth manifold is a pair $(M,\mathcal{A})$ where $M$ is a topological manifold and $\mathcal{A}$ is a smooth atlas.
+\end{defn}
+
+TODO: There is some section gaps here, from smooth manifold to smooth submersion. 
+
+Here are some additional propositions that will be helpful for our study in later sections:
+
+This one is from \cite{lee_introduction_2012} Theorem 4.26
+
+\begin{theorem}
+    \label{theorem:local_section_theorem}
+
+    Let $M$ and $N$ be smooth manifolds and $\pi:M\to N$ is a smooth map. Then $\pi$ is a smooth submersion if and only if every point of $M$ is in the image of a smooth local section of $\pi$ (a local section of $\pi$ is a map $\sigma:U\to M$ defined on some open subset $U\subseteq N$ with $\pi\circ \sigma=Id_U$).   
+\end{theorem}
+
+
+
 \section{Quantum physics and terminologies}
 
 In this section, we will introduce some terminologies and theorems used in quantum physics that are relevant to our study. Assuming no prior knowledge of quantum physics, we will provide brief definitions and explanations for each term.