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+++ b/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.
diff --git a/chapters/chap2.pdf b/chapters/chap2.pdf
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@@ -173,6 +173,69 @@ In this section, we will try to use the results from previous sections to estima
   So $\obdiam(S^n(1);-\kappa)=\sqrt{n}(2I^{-1}(\frac{1-\kappa}{2}))=O(\sqrt{n})$.
 \end{proof}
 
+From the previous discussion, we see that the only remaining for finding observable diameter of $\C P^n$ is to find the lipchitz function that is isometric with consistent push-forward measure.
+
+To find such metric, we need some additional results.
+
+\begin{defn}
+  \label{defn:riemannian-metric}
+
+  Let $M$ be a smooth manifold. A \textit{\textbf{Riemannian metric}} on $M$ is a smooth covariant tensor field $g\in \mathcal{T}^2(M)$ such that for each $p\in M$, $g_p$ is an inner product on $T_pM$.
+
+  $g_p(v,v)\geq 0$ for each $p\in M$ and each $v\in T_pM$. equality holds if and only if $v=0$.
+
+\end{defn}
+
+TODO: There is a hidden chapter on group action on manifolds, can you find that?
+
+\begin{theorem}
+  \label{theorem:riemannian-submersion}
+
+  Let $(\tilde{M},\tilde{g})$ be a Riemannian manifold, let $\pi:\tilde{M}\to M$ be a surjective smooth submersion, and let $G$ be a group acting on $\tilde{M}$. If the \textbf{action} is
+  \begin{enumerate}
+    \item isometric: the map $x\mapsto \varphi\cdot x$ is an isometry for each $\varphi\in G$.
+    \item vertical: every element $\varphi\in G$ takes each fiber to itself, that is $\pi(\varphi\cdot p)=\pi(p)$ for all $p\in \tilde{M}$.
+    \item transitive on fibers: for each $p,q\in \tilde{M}$ such that $\pi(p)=\pi(q)$, there exists $\varphi\in G$ such that $\varphi\cdot p = q$.
+  \end{enumerate}
+  Then there is a unique Riemannian metric on $M$ such that $\pi$ is a Riemannian submersion.
+
+\end{theorem}
+
+A natural measure for $\C P^n$ is the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric. \cite{lee_introduction_2018} Example 2.30
+
+\begin{defn}
+  \label{defn:fubini-study-metric}
+
+  Let $n$ be a positive integer, and consider the complex projective space $\C P^n$ defined as the quotient space of $\C^{n+1}$ by the equivalence relation $z\sim z'$ if there exists $\lambda \in \C$ such that $z=\lambda z'$. The map $\pi:\C^{n+1}\setminus\{0\}\to \C P^n$ sending each point in $\C^{n+1}\setminus\{0\}$ to its span is surjective smooth submersion.
+
+  Identifying $\C^{n+1}$ with $\R^{2n+2}$ with its Euclidean metric, we can view the unit sphere $S^{2n+1}$ with its roud metric $\mathring{g}$ as an embedded Riemannian submanifold of $\C^{n+1}\setminus\{0\}$. Let $p:S^{2n+1}\to \C P^n$ denote the restriction of the map $\pi$. Then $p$ is smooth, and its is surjective, because every 1-dimensional complex subspace contains elements of unit norm.
+
+\end{defn}
+
+There are many additional properties for such construction, we will check them just for curiosity.
+
+We need to show that it is a submersion.
+
+\begin{proof}
+   Let $z_0\in S^{2n+1}$ and set $\zeta_0=p(z_0)\in \C P^n$. Since $\pi$ is a smooth submersion, it has a smooth local section $\sigma: U\to \C^{n+1}$ defined on a neighborhood $U$ of $\zeta_0$ and satisfying $\sigma(\zeta_0)=z_0$ by the local section theorem (Theorem \ref{theorem:local_section_theorem}). Let $v:\C^{n+1}\setminus\{0\}\to S^{2n+1}$ be the radial projection on to the sphere:
+
+   $$
+   v(z)=\frac{z}{|z|}
+   $$
+
+   Since dividing an element of $\C^{n+1}$ by a nonzero scalar does not change its span, it follows that $p\circ v=\pi$. Therefore, if we set $\tilde{\sigma}=v\circ \sigma$, then $\tilde{\sigma}$ is a smooth local section of $p$. Apply the local section theorem (Theorem \ref{theorem:local_section_theorem}) again, this shows that $p$ is a submersion.
+
+   Define an action of $S^1$ on $S^{2n+1}$ by complex multiplication:
+
+   $$
+   \lambda (z^1,z^2,\ldots,z^{n+1})=(\lambda z^1,\lambda z^2,\ldots,\lambda z^{n+1})
+   $$
+
+   for $\lambda\in S^1$ (viewed as complex number of norm 1) and $z=(z^1,z^2,\ldots,z^{n+1})\in S^{2n+1}$. This is easily seen to be isometric, vertical, and transitive on fibers of $p$.
+
+   By (Theorem \ref{theorem:riemannian-submersion}). Therefore, there is a unique metric on $\C P^n$ such that the map $p:S^{2n+1}\to \C P^n$ is a Riemannian submersion. This metric is called the Fubini-study metric.
+\end{proof}
+
 \subsection{Observable diameter for complex projective spaces}
 
 Using the projection map and Hopf's fibration, we can estimate the observable diameter for complex projective spaces from the observable diameter of spheres.
@@ -192,7 +255,7 @@ Using the projection map and Hopf's fibration, we can estimate the observable di
 
 \end{proof}
 
-\subsection{More example for concentration of measure and observable diameter}
+\section{Example for concentration of measure and observable diameter}
 
 In this section, we wish to use observable diameter to estimate the statics of thermal dynamics of some classical systems.
 
diff --git a/main.bib b/main.bib
index 6c55e24..349acd9 100644
--- a/main.bib
+++ b/main.bib
@@ -95,6 +95,16 @@
   isbn = {978-3-319-91755-9}
 }
 
+@book{lee_introduction_2012,
+	address = {New York},
+	title = {Introduction to {Smooth} {Manifolds}},
+	isbn = {978-1-4899-9475-2 978-1-4419-9982-5},
+	language = {eng},
+	publisher = {Springer},
+	author = {Lee, John M. and Lee, John M.},
+	year = {2012},
+}
+
 @inproceedings{Hayden,
   title     = {Concentration of measure effects in quantum information},
   author    = {Hayden, Patrick},
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@@ -51,28 +51,28 @@ One can imagine the project as a big tree, where the root is in undergrad math a
   arrow/.style={-Latex}
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-\node[box, below=of lin] (prob) {Probability\\(expectation, variance, concentration)};
-\node[box, below=of real] (top) {Topology/Geometry\\(metrics, compactness)};
+% \node[box] (lin) {Linear Algebra\\(bases, maps, eigenvalues)};
+% \node[box, right=of lin] (real) {Real Analysis\\(limits, continuity, measure-lite)};
+% \node[box, below=of lin] (prob) {Probability\\(expectation, variance, concentration)};
+% \node[box, below=of real] (top) {Topology/Geometry\\(metrics, compactness)};
 
-\node[box, below=12mm of prob] (func) {Functional Analysis\\($L^p$, Hilbert spaces, operators)};
-\node[box, below=12mm of top] (quant) {Quantum Formalism\\(states, observables, partial trace)};
+% \node[box, below=12mm of prob] (func) {Functional Analysis\\($L^p$, Hilbert spaces, operators)};
+% \node[box, below=12mm of top] (quant) {Quantum Formalism\\(states, observables, partial trace)};
 
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-  \draw[arrow] (func) -- (quant);
-  \draw[arrow] (lin) -- (quant);
-  \draw[arrow] (top) -- (quant);
+% \node[box, below=14mm of func, xshift=25mm] (book) {This Book\\(Chapters 1--n)};
+% % draw arrows behind nodes
+% \begin{scope}[on background layer]
+%   \draw[arrow] (lin) -- (func);
+%   \draw[arrow] (real) -- (func);
+%   \draw[arrow] (prob) -- (func);
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-  \draw[arrow] (quant) -- (book);
-  \draw[arrow] (prob) -- (book);
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+%   \draw[arrow] (func) -- (book);
+%   \draw[arrow] (quant) -- (book);
+%   \draw[arrow] (prob) -- (book);
+% \end{scope}
 
 \end{tikzpicture}
 \caption{Dependency tree: prerequisites and how they feed into the main text.}