updates

chapters/chap2.tex

@@ -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.