376 lines
15 KiB
Markdown
376 lines
15 KiB
Markdown
# Math 401, Fall 2025: Thesis notes, R2, Levy's concentration theorem and Levy's family
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> Progress: 2/5=40% (denominator and enumerator may change)
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## Levy's concentration theorem
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> [!TIP]
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>
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> This version of Levy's concentration theorem can be found in [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) 15.84 and 15.85.
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Our goal is to prove the generalized version of Levy's concentration theorem used in Hayden's work for $\eta$-Lipschitz functions.
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Let $f:S^{n-1}\to \mathbb{R}$ be a $\eta$-Lipschitz function. Let $M_f$ denote the median of $f$ and $\langle f\rangle$ denote the mean of $f$. (Note this can be generalized to many other manifolds.)
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Select a random point $x\in S^{n-1}$ with $n>2$ according to the uniform measure (Haar measure). Then the probability of observing a value of $f$ much different from the reference value is exponentially small.
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$$
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\operatorname{Pr}[|f(x)-M_f|>\epsilon]\leq \exp(-\frac{n\epsilon^2}{2\eta^2})
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$$
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$$
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\operatorname{Pr}[|f(x)-\langle f\rangle|>\epsilon]\leq 2\exp(-\frac{(n-1)\epsilon^2}{2\eta^2})
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$$
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### Levy's concentration theorem via sub-Gaussian concentration
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> [!TIP]
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>
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> This version of Levy's concentration theorem can be found in [High-dimensional probability](https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-2.pdf) 5.1.4.
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#### Isoperimetric inequality on $\mathbb{R}^n$
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Among all subsets $A\subset \mathbb{R}^n$ with a given volume, the Euclidean ball has the minimal area.
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That is, for any $\epsilon>0$, Euclidean balls minimize the volume of the $\epsilon$-neighborhood of $A$.
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Where the volume of the $\epsilon$-neighborhood of $A$ is defined as
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$$
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A_\epsilon(A)\coloneqq \{x\in \mathbb{R}^n: \exists y\in A, \|x-y\|_2\leq \epsilon\}=A+\epsilon B_2^n
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$$
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Here the $\|\cdot\|_2$ is the Euclidean norm. (The theorem holds for both geodesic metric on sphere and Euclidean metric on $\mathbb{R}^n$.)
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#### Isoperimetric inequality on the sphere
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Let $\sigma_n(A)$ denotes the normalized area of $A$ on $n$ dimensional sphere $S^n$. That is $\sigma_n(A)\coloneqq\frac{\operatorname{Area}(A)}{\operatorname{Area}(S^n)}$.
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Let $\epsilon>0$. Then for any subset $A\subset S^n$, given the area $\sigma_n(A)$, the spherical caps minimize the volume of the $\epsilon$-neighborhood of $A$.
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> The above two inequalities is not proved in the Book _High-dimensional probability_. But you can find it in the Appendix C of Gromov's book _Metric Structures for Riemannian and Non-Riemannian Spaces_.
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To continue prove the theorem, we use sub-Gaussian concentration *(Chapter 3 of _High-dimensional probability_ by Roman Vershynin)* of sphere $\sqrt{n}S^n$.
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This will leads to some constant $C>0$ such that the following lemma holds:
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#### The "Blow-up" lemma
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Let $A$ be a subset of sphere $\sqrt{n}S^n$, and $\sigma$ denotes the normalized area of $A$. Then if $\sigma\geq \frac{1}{2}$, then for every $t\geq 0$,
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$$
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\sigma(A_t)\geq 1-2\exp(-ct^2)
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$$
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where $A_t=\{x\in S^n: \operatorname{dist}(x,A)\leq t\}$ and $c$ is some positive constant.
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#### Proof of the Levy's concentration theorem
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Proof:
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Without loss of generality, we can assume that $\eta=1$. Let $M$ denotes the median of $f(X)$.
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So $\operatorname{Pr}[|f(X)\leq M|]\geq \frac{1}{2}$, and $\operatorname{Pr}[|f(X)\geq M|]\geq \frac{1}{2}$.
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Consider the sub-level set $A\coloneqq \{x\in \sqrt{n}S^n: |f(x)|\leq M\}$.
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Since $\operatorname{Pr}[X\in A]\geq \frac{1}{2}$, by the blow-up lemma, we have
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$$
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\operatorname{Pr}[X\in A_t]\geq 1-2\exp(-ct^2)
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$$
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And since
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$$
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\operatorname{Pr}[X\in A_t]\leq \operatorname{Pr}[f(X)\leq M+t]
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$$
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Combining the above two inequalities, we have
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$$
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\operatorname{Pr}[f(X)\leq M+t]\geq 1-2\exp(-ct^2)
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$$
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## Levy's concentration theorem via Levy family
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> [!TIP]
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>
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> This version of Levy's concentration theorem can be found in:
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> - [Metric Structures for Riemannian and Non-Riemannian Spaces by M. Gromov](https://www.amazon.com/Structures-Riemannian-Non-Riemannian-Progress-Mathematics/dp/0817638989/ref=tmm_hrd_swatch_0?_encoding=UTF8&dib_tag=se&dib=eyJ2IjoiMSJ9.Tp8dXvGbTj_D53OXtGj_qOdqgCgbP8GKwz4XaA1xA5PGjHj071QN20LucGBJIEps.9xhBE0WNB0cpMfODY5Qbc3gzuqHnRmq6WZI_NnIJTvc&qid=1750973893&sr=8-1)
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> - [Metric Measure Geometry by Takashi Shioya](https://arxiv.org/pdf/1410.0428)
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### Levy's concentration theorem (Gromov's version)
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> The Levy's lemma can also be found in _Metric Structures for Riemannian and Non-Riemannian Spaces_ by M. Gromov. $3\frac{1}{2}.19$ The Levy concentration theory.
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#### Theorem $3\frac{1}{2}.19$ Levy concentration theorem:
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An arbitrary 1-Lipschitz function $f:S^n\to \mathbb{R}$ concentrates near a single value $a_0\in \mathbb{R}$ as strongly as the distance function does.
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That is
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$$
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\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
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$$
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where
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$$
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\kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
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$$
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$a_0$ is the **Levy mean** of function $f$, that is the level set of $f^{-1}:\mathbb{R}\to S^n$ divides the sphere into equal halves, characterized by the following equality:
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$$
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\mu(f^{-1}(-\infty,a_0])\geq \frac{1}{2} \text{ and } \mu(f^{-1}[a_0,\infty))\geq \frac{1}{2}
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$$
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Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.
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> Detailed proof by Takashi Shioya.
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>
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> The central idea is to draw the connection between the given three topological spaces, $S^{2n+1}$, $CP^n$ and $\mathbb{R}$.
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First, we need to introduce the following distribution and lemmas/theorems:
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**OBSERVATION**
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consider the orthogonal projection from $\mathbb{R}^{n+1}$, the space where $S^n$ is embedded, to $\mathbb{R}^k$, we denote the restriction of the projection as $\pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k$. Note that $\pi_{n,k}$ is a 1-Lipschitz function (projection will never increase the distance between two points).
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We denote the normalized Riemannian volume measure on $S^n(\sqrt{n})$ as $\sigma^n(\cdot)$, and $\sigma^n(S^n(\sqrt{n}))=1$.
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#### Definition of Gaussian measure on $\mathbb{R}^k$
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We denote the Gaussian measure on $\mathbb{R}^k$ as $\gamma^k$.
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$$
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d\gamma^k(x)\coloneqq\frac{1}{\sqrt{2\pi}^k}\exp(-\frac{1}{2}\|x\|^2)dx
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$$
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$x\in \mathbb{R}^k$, $\|x\|^2=\sum_{i=1}^k x_i^2$ is the Euclidean norm, and $dx$ is the Lebesgue measure on $\mathbb{R}^k$.
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Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on $\mathbb{R}^k$ with standard deviation $1$.
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#### Maxwell-Boltzmann distribution law
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> It is such a wonderful fact for me, that the projection of $n+1$ dimensional sphere with radius $\sqrt{n}$ to $\mathbb{R}^k$ is a Gaussian distribution as $n\to \infty$.
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For any natural number $k$,
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$$
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\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}\to \frac{d\gamma^k(x)}{dx}
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$$
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where $(\pi_{n,k})_*\sigma^n$ is the push-forward measure of $\sigma^n$ by $\pi_{n,k}$.
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In other words,
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$$
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(\pi_{n,k})_*\sigma^n\to \gamma^k\text{ weakly as }n\to \infty
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$$
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<details>
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<summary>Proof</summary>
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We denote the $n$ dimensional volume measure on $\mathbb{R}^k$ as $\operatorname{vol}_k$.
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Observe that $\pi_{n,k}^{-1}(x),x\in \mathbb{R}^k$ is isometric to $S^{n-k}(\sqrt{n-\|x\|^2})$, that is, for any $x\in \mathbb{R}^k$, $\pi_{n,k}^{-1}(x)$ is a sphere with radius $\sqrt{n-\|x\|^2}$ (by the definition of $\pi_{n,k}$).
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So,
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$$
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\begin{aligned}
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\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}&=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(x))}{\operatorname{vol}_k(S^n(\sqrt{n}))}\\
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&=\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}\\
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\end{aligned}
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$$
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as $n\to \infty$.
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note that $\lim_{n\to \infty}{(1-\frac{a}{n})^n}=e^{-a}$ for any $a>0$.
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$(n-\|x\|^2)^{\frac{n-k}{2}}=\left(n(1-\frac{\|x\|^2}{n})\right)^{\frac{n-k}{2}}\to n^{\frac{n-k}{2}}\exp(-\frac{\|x\|^2}{2})$
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So
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$$
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\begin{aligned}
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\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}&=\frac{e^{-\frac{\|x\|^2}{2}}}{\int_{x\in \mathbb{R}^k}e^{-\frac{\|x\|^2}{2}}dx}\\
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&=\frac{1}{(2\pi)^{\frac{k}{2}}}e^{-\frac{\|x\|^2}{2}}\\
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&=\frac{d\gamma^k(x)}{dx}
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\end{aligned}
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$$
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QED
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</details>
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#### Proof of the Levy's concentration theorem via the Maxwell-Boltzmann distribution law
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We use the Maxwell-Boltzmann distribution law and Levy's isoperimetric inequality to prove the Levy's concentration theorem.
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The goal is the same as the Gromov's version, first we bound the probability of the sub-level set of $f$ by the $\kappa_n(\epsilon)$ function by Levy's isoperimetric inequality. Then we claim that the $\kappa_n(\epsilon)$ function is bounded by the Gaussian distribution.
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Note, this section is not rigorous enough in sense of mathematics and the author should add sections about Levy family and observable diameter to make the proof more rigorous and understandable.
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<details>
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<summary>Proof</summary>
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Let $f:S^n\to \mathbb{R}$ be a 1-Lipschitz function.
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Consider the two sets of points on the sphere $S^n$ with radius $\sqrt{n}$:
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$$
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\Omega_+=\{x\in S^n: f(x)\leq a_0-\epsilon\}, \Omega_-=\{x\in S^n: f(x)\geq a_0+\epsilon\}
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$$
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Note that $\Omega_+\cup \Omega_-$ is the whole sphere $S^n(\sqrt{n})$.
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By the Levy's isoperimetric inequality, we have
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$$
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\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))\leq \operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_+))+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_-))
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$$
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We define $\kappa_n(\epsilon)$ as the following:
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$$
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\kappa_n(\epsilon)=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))}{\operatorname{vol}_k(S^n(\sqrt{n}))}=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
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$$
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By the Levy's isoperimetric inequality, and the Maxwell-Boltzmann distribution law, we have
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$$
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\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
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$$
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</details>
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## Levy's Isoperimetric inequality
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> This section is from the Appendix $C_+$ of Gromov's book _Metric Structures for Riemannian and Non-Riemannian Spaces_.
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Not very edible for undergraduates.
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## Riemannian manifolds and geometry
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> This section is designed for stupids like me skipping too much essential materials in the book.
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> This part might be extended to a separate note, let's check how far we can go from this part.
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>
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> References:
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>
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> - [Riemannian Geometry by John M. Lee](https://www.amazon.com/Introduction-Riemannian-Manifolds-Graduate-Mathematics/dp/3319917544?dib=eyJ2IjoiMSJ9.88u0uIXulwPpi3IjFn9EdOviJvyuse9V5K5wZxQEd6Rto5sCIowzEJSstE0JtQDW.QeajvjQEbsDmnEMfPzaKrfVR9F5BtWE8wFscYjCAR24&dib_tag=se&keywords=riemannian+manifold+by+john+m+lee&qid=1753238983&sr=8-1)
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### Manifold
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> Unexpectedly, a good definition of the manifold is defined in the topology I.
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>
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> Check section 36. This topic extends to a wonderful chapter 8 in the book where you can hardly understand chapter 2.
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#### Definition of m-manifold
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An $m$-manifold is a [Hausdorff space](../../Math4201/Math4201_L9#hausdorff-space) $X$ with a **countable basis** (second countable) such that each point of $x$ of $X$ has a neighborhood [homeomorphic](../../Math4201/Math4201_L10#definition-of-homeomorphism) to an open subset of $\mathbb{R}^m$.
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<details>
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<summary>Example of second countable space</summary>
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Let $X=\mathbb{R}$ and $\mathcal{B}=\{(a,b)|a,b\in \mathbb{R},a<b\}$ (collection of all open intervals with rational endpoints).
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Since the rational numbers are countable, so $\mathcal{B}$ is countable.
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So $\mathbb{R}$ is second countable.
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Likewise, $\mathbb{R}^n$ is also second countable.
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</details>
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<details>
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<summary>Example of manifold</summary>
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1-manifold is a curve and 2-manifold is a surface.
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</details>
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#### Theorem of imbedded space
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If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^n$ for some $n$.
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This theorem might save you from imagining abstract structures back to real dimension. Good news, at least you stay in some real numbers.
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### Smooth manifolds and Lie groups
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> This section is waiting for the completion of book Introduction to Smooth Manifolds by John M. Lee.
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#### Partial derivatives
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Let $U\subseteq \mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ be a map.
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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
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$$
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\begin{aligned}
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\frac{\partial f}{\partial x_j}(a)&=\lim_{h\to 0}\frac{f(a_1,\cdots,a_j+h,\cdots,a_n)-f(a_1,\cdots,a_j,\cdots,a_n)}{h} \\
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&=\lim_{h\to 0}\frac{f(a+he_j)-f(a)}{h}
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\end{aligned}
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$$
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#### Continuously differentiable maps
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Let $U\subseteq \mathbb{R}^n$ and $f:U\to \mathbb{R}^n$ be a map.
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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 $a$.
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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.)
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#### Smooth maps
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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 **bijection** and its **inverse is also smooth**.
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#### Charts
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Let $M$ be a smooth manifold. A **chart** is a pair $(U,\phi)$ where $U\subseteq M$ is an open subset and $\phi:U\to \hat{U}\subseteq \mathbb{R}^n$ is a homeomorphism (a continuous bijection map and its inverse is also continuous).
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If $p\in U$ and $\phi(p)=0$, then we say that $p$ is the origin of the chart $(U,\phi)$.
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#### Atlas
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Let $M$ be a smooth manifold. An **atlas** is a collection of charts $\mathcal{A}=\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$ such that $M=\bigcup_{\alpha\in I} U_\alpha$.
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An atlas is said to be **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 $\alpha, \beta\in I$.
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#### Smooth manifold
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A smooth manifold is a pair $(M,\mathcal{A})$ where $M$ is a topological manifold and $\mathcal{A}$ is a smooth atlas.
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### Riemannian manifolds
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A Riemannian manifold is a smooth manifold equipped with a **Riemannian metric**, which is a smooth assignment of an inner product to each tangent space $T_pM$ of the manifold.
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An example of Riemannian manifold is the sphere $\mathbb{C}P^n$.
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### Riemannian metric
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A Riemannian metric is a smooth assignment of an inner product to each tangent space $T_pM$ of the manifold.
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An example of Riemannian metric is the Euclidean metric, the bilinear form of $d(p,q)=\|p-q\|_2$ on $\mathbb{R}^n$.
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### Notion of Connection
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A connection is a way to define the directional derivative of a vector field along a curve on a Riemannian manifold.
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For every $p\in M$, where $M$ denote the manifold, suppose $M=\mathbb{R}^n$, then let $X=(f_1,\cdots,f_n)$ be a vector field on $M$. The directional derivative of $X$ along the point $p$ is defined as
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$$
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D_VX=\lim_{h\to 0}\frac{X(p+h)-X(p)}{h}
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$$
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### Notion of Curvatures
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> [!NOTE]
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>
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> Geometrically, the curvature of the manifold is radius of the tangent sphere of the manifold.
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#### Nabla notation and Levi-Civita connection
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#### Ricci curvature
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