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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}, diff --git a/main.pdf b/main.pdf index 286a5a47b0ec71dbe1a81d99d758ec7a675c1469..f0cc2f99280af89d56866f6e5355bc4b295abafc 100644 GIT binary patch delta 204258 zcmZUaQ*%{wryvkiM?anHYRp5vF*u(JGQL}&-ee|S?A_lbg!z5?zQ@+ z*89|34SJ(~s-bO^fd}Wvyu*dWgka0>`r&RsSV%?+68-HPx&jd~Xo!A+O35`4gisWN(PXvO_N z{DSji|Dl12nU3w_x##1E$%LI*CPY?m;r(bN$u2iujBdFEoA&6HHC~G>U zlp8DJ52Y81+0xFcxa(sM z47S05Ed_|ly&$=Mz`@d1JSPJk_1j9G{RuR110MYon*59unpPxcjG{%*t;-$C=?OU) zm~@WKv$@$hme{qdOG0?wc#qZg%^&Oui2Dg5!I85MEC^gOvR+EThB>0pUSCV5MtO7- z*}ZO(*Rb_!9y8U>xwUelrGZBKFbjZ3Zo-VbYgAbj>0SRjHj0cJGc4d%q&(wa6Rk6w zYFa^zP61R0a>pZMudqG+W&3PqD`o@ak2$1vgJS#ND0Z&v`9_U=;W*uUl$_ffr5Wpv z`crLFI;KwO?k3H4JyFLb=TNd1>?Blz3v9au$$#cv*eu2;Bv=PJPDe8bVw9sX6dn`B zeS`%|HtK$!?+59^>S6*%%sx_Ei%m6ZjG(>bu=>e0@=7GfOTSv5*e_bWTCUxogL7`oL^wZgi{Jo^Qz+B 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