519 lines
21 KiB
Markdown
519 lines
21 KiB
Markdown
# Math 401, Fall 2025: Thesis notes, R2, Levy's concentration theorem and Levy's family
|
|
|
|
> Progress: 2/5=40% (denominator and enumerator may change)
|
|
|
|
## Levy's concentration theorem
|
|
|
|
> [!TIP]
|
|
>
|
|
> 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.
|
|
|
|
Our goal is to prove the generalized version of Levy's concentration theorem used in Hayden's work for $\eta$-Lipschitz functions.
|
|
|
|
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.)
|
|
|
|
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.
|
|
|
|
$$
|
|
\operatorname{Pr}[|f(x)-M_f|>\epsilon]\leq \exp(-\frac{n\epsilon^2}{2\eta^2})
|
|
$$
|
|
$$
|
|
\operatorname{Pr}[|f(x)-\langle f\rangle|>\epsilon]\leq 2\exp(-\frac{(n-1)\epsilon^2}{2\eta^2})
|
|
$$
|
|
|
|
### Levy's concentration theorem via sub-Gaussian concentration
|
|
|
|
> [!TIP]
|
|
>
|
|
> 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.
|
|
|
|
#### Isoperimetric inequality on $\mathbb{R}^n$
|
|
|
|
Among all subsets $A\subset \mathbb{R}^n$ with a given volume, the Euclidean ball has the minimal area.
|
|
|
|
That is, for any $\epsilon>0$, Euclidean balls minimize the volume of the $\epsilon$-neighborhood of $A$.
|
|
|
|
Where the volume of the $\epsilon$-neighborhood of $A$ is defined as
|
|
|
|
$$
|
|
A_\epsilon(A)\coloneqq \{x\in \mathbb{R}^n: \exists y\in A, \|x-y\|_2\leq \epsilon\}=A+\epsilon B_2^n
|
|
$$
|
|
|
|
Here the $\|\cdot\|_2$ is the Euclidean norm. (The theorem holds for both geodesic metric on sphere and Euclidean metric on $\mathbb{R}^n$.)
|
|
|
|
#### Isoperimetric inequality on the sphere
|
|
|
|
Let $\sigma_n(A)$ denotes the normalized area of $A$ on $n$ dimensional sphere $S^n$. That is $\sigma_n(A)\coloneqq\frac{\operatorname{Area}(A)}{\operatorname{Area}(S^n)}$.
|
|
|
|
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 $A$.
|
|
|
|
> 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_.
|
|
|
|
To continue prove the theorem, we use sub-Gaussian concentration *(Chapter 3 of _High-dimensional probability_ by Roman Vershynin)* of sphere $\sqrt{n}S^n$.
|
|
|
|
This will leads to some constant $C>0$ such that the following lemma holds:
|
|
|
|
#### The "Blow-up" lemma
|
|
|
|
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 $t\geq 0$,
|
|
|
|
$$
|
|
\sigma(A_t)\geq 1-2\exp(-ct^2)
|
|
$$
|
|
|
|
where $A_t=\{x\in S^n: \operatorname{dist}(x,A)\leq t\}$ and $c$ is some positive constant.
|
|
|
|
#### Proof of the Levy's concentration theorem
|
|
|
|
Proof:
|
|
|
|
Without loss of generality, we can assume that $\eta=1$. Let $M$ denotes the median of $f(X)$.
|
|
|
|
So $\operatorname{Pr}[|f(X)\leq M|]\geq \frac{1}{2}$, and $\operatorname{Pr}[|f(X)\geq M|]\geq \frac{1}{2}$.
|
|
|
|
Consider the sub-level set $A\coloneqq \{x\in \sqrt{n}S^n: |f(x)|\leq M\}$.
|
|
|
|
Since $\operatorname{Pr}[X\in A]\geq \frac{1}{2}$, by the blow-up lemma, we have
|
|
|
|
$$
|
|
\operatorname{Pr}[X\in A_t]\geq 1-2\exp(-ct^2)
|
|
$$
|
|
|
|
And since
|
|
|
|
$$
|
|
\operatorname{Pr}[X\in A_t]\leq \operatorname{Pr}[f(X)\leq M+t]
|
|
$$
|
|
|
|
Combining the above two inequalities, we have
|
|
|
|
$$
|
|
\operatorname{Pr}[f(X)\leq M+t]\geq 1-2\exp(-ct^2)
|
|
$$
|
|
|
|
## Levy's concentration theorem via Levy family
|
|
|
|
> [!TIP]
|
|
>
|
|
> This version of Levy's concentration theorem can be found in:
|
|
> - [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)
|
|
> - [Metric Measure Geometry by Takashi Shioya](https://arxiv.org/pdf/1410.0428)
|
|
|
|
|
|
### Levy's concentration theorem (Gromov's version)
|
|
|
|
> 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.
|
|
|
|
#### Theorem $3\frac{1}{2}.19$ Levy concentration theorem:
|
|
|
|
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.
|
|
|
|
That is
|
|
|
|
$$
|
|
\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
|
|
$$
|
|
|
|
where
|
|
|
|
$$
|
|
\kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
|
|
$$
|
|
|
|
$a_0$ 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:
|
|
|
|
$$
|
|
\mu(f^{-1}(-\infty,a_0])\geq \frac{1}{2} \text{ and } \mu(f^{-1}[a_0,\infty))\geq \frac{1}{2}
|
|
$$
|
|
|
|
Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.
|
|
|
|
> Detailed proof by Takashi Shioya.
|
|
>
|
|
> The central idea is to draw the connection between the given three topological spaces, $S^{2n+1}$, $CP^n$ and $\mathbb{R}$.
|
|
|
|
First, we need to introduce the following distribution and lemmas/theorems:
|
|
|
|
**OBSERVATION**
|
|
|
|
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).
|
|
|
|
We denote the normalized Riemannian volume measure on $S^n(\sqrt{n})$ as $\sigma^n(\cdot)$, and $\sigma^n(S^n(\sqrt{n}))=1$.
|
|
|
|
#### Definition of Gaussian measure on $\mathbb{R}^k$
|
|
|
|
We denote the Gaussian measure on $\mathbb{R}^k$ as $\gamma^k$.
|
|
|
|
$$
|
|
d\gamma^k(x)\coloneqq\frac{1}{\sqrt{2\pi}^k}\exp(-\frac{1}{2}\|x\|^2)dx
|
|
$$
|
|
|
|
$x\in \mathbb{R}^k$, $\|x\|^2=\sum_{i=1}^k x_i^2$ is the Euclidean norm, and $dx$ is the Lebesgue measure on $\mathbb{R}^k$.
|
|
|
|
Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on $\mathbb{R}^k$ with standard deviation $1$.
|
|
|
|
#### Maxwell-Boltzmann distribution law
|
|
|
|
> 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 $n\to \infty$.
|
|
|
|
For any natural number $k$,
|
|
|
|
$$
|
|
\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}\to \frac{d\gamma^k(x)}{dx}
|
|
$$
|
|
|
|
where $(\pi_{n,k})_*\sigma^n$ is the push-forward measure of $\sigma^n$ by $\pi_{n,k}$.
|
|
|
|
In other words,
|
|
|
|
$$
|
|
(\pi_{n,k})_*\sigma^n\to \gamma^k\text{ weakly as }n\to \infty
|
|
$$
|
|
|
|
<details>
|
|
<summary>Proof</summary>
|
|
|
|
We denote the $n$ dimensional volume measure on $\mathbb{R}^k$ as $\operatorname{vol}_k$.
|
|
|
|
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 $\pi_{n,k}$).
|
|
|
|
So,
|
|
|
|
$$
|
|
\begin{aligned}
|
|
\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}&=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(x))}{\operatorname{vol}_k(S^n(\sqrt{n}))}\\
|
|
&=\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}\\
|
|
\end{aligned}
|
|
$$
|
|
|
|
as $n\to \infty$.
|
|
|
|
note that $\lim_{n\to \infty}{(1-\frac{a}{n})^n}=e^{-a}$ for any $a>0$.
|
|
|
|
$(n-\|x\|^2)^{\frac{n-k}{2}}=\left(n(1-\frac{\|x\|^2}{n})\right)^{\frac{n-k}{2}}\to n^{\frac{n-k}{2}}\exp(-\frac{\|x\|^2}{2})$
|
|
|
|
So
|
|
|
|
$$
|
|
\begin{aligned}
|
|
\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}&=\frac{e^{-\frac{\|x\|^2}{2}}}{\int_{x\in \mathbb{R}^k}e^{-\frac{\|x\|^2}{2}}dx}\\
|
|
&=\frac{1}{(2\pi)^{\frac{k}{2}}}e^{-\frac{\|x\|^2}{2}}\\
|
|
&=\frac{d\gamma^k(x)}{dx}
|
|
\end{aligned}
|
|
$$
|
|
|
|
QED
|
|
|
|
</details>
|
|
|
|
#### Proof of the Levy's concentration theorem via the Maxwell-Boltzmann distribution law
|
|
|
|
We use the Maxwell-Boltzmann distribution law and Levy's isoperimetric inequality to prove the Levy's concentration theorem.
|
|
|
|
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.
|
|
|
|
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.
|
|
|
|
<details>
|
|
<summary>Proof</summary>
|
|
|
|
Let $f:S^n\to \mathbb{R}$ be a 1-Lipschitz function.
|
|
|
|
Consider the two sets of points on the sphere $S^n$ with radius $\sqrt{n}$:
|
|
|
|
$$
|
|
\Omega_+=\{x\in S^n: f(x)\leq a_0-\epsilon\}, \Omega_-=\{x\in S^n: f(x)\geq a_0+\epsilon\}
|
|
$$
|
|
|
|
Note that $\Omega_+\cup \Omega_-$ is the whole sphere $S^n(\sqrt{n})$.
|
|
|
|
By the Levy's isoperimetric inequality, we have
|
|
|
|
$$
|
|
\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))\leq \operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_+))+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_-))
|
|
$$
|
|
|
|
We define $\kappa_n(\epsilon)$ as the following:
|
|
|
|
$$
|
|
\kappa_n(\epsilon)=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))}{\operatorname{vol}_k(S^n(\sqrt{n}))}=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
|
|
$$
|
|
|
|
By the Levy's isoperimetric inequality, and the Maxwell-Boltzmann distribution law, we have
|
|
|
|
$$
|
|
\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
|
|
$$
|
|
</details>
|
|
|
|
## Levy's Isoperimetric inequality
|
|
|
|
> This section is from the Appendix $C_+$ of Gromov's book _Metric Structures for Riemannian and Non-Riemannian Spaces_.
|
|
|
|
Not very edible for undergraduates.
|
|
|
|
## Differential Geometry
|
|
|
|
> This section is designed for stupids like me skipping too much essential materials in the book.
|
|
|
|
> This part might be extended to a separate note, let's check how far we can go from this part.
|
|
>
|
|
> References:
|
|
>
|
|
> - [Introduction to Smooth Manifolds by John M. Lee]
|
|
>
|
|
> - [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)
|
|
|
|
### Manifold
|
|
|
|
> Unexpectedly, a good definition of the manifold is defined in the topology I.
|
|
>
|
|
> Check section 36. This topic extends to a wonderful chapter 8 in the book where you can hardly understand chapter 2.
|
|
|
|
#### Definition of m-manifold
|
|
|
|
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 $\mathbb{R}^m$.
|
|
|
|
<details>
|
|
<summary>Example of second countable space</summary>
|
|
|
|
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.
|
|
</details>
|
|
|
|
<details>
|
|
<summary>Example of manifold</summary>
|
|
|
|
1-manifold is a curve and 2-manifold is a surface.
|
|
|
|
</details>
|
|
|
|
#### Theorem of imbedded space
|
|
|
|
If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^n$ for some $n$.
|
|
|
|
This theorem might save you from imagining abstract structures back to real dimension. Good news, at least you stay in some real numbers.
|
|
|
|
### Smooth manifolds and Lie groups
|
|
|
|
> This section is waiting for the completion of book Introduction to Smooth Manifolds by John M. Lee.
|
|
|
|
#### Partial derivatives
|
|
|
|
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
|
|
|
|
$$
|
|
\begin{aligned}
|
|
\frac{\partial f}{\partial x_j}(a)&=\lim_{h\to 0}\frac{f(a_1,\cdots,a_j+h,\cdots,a_n)-f(a_1,\cdots,a_j,\cdots,a_n)}{h} \\
|
|
&=\lim_{h\to 0}\frac{f(a+he_j)-f(a)}{h}
|
|
\end{aligned}
|
|
$$
|
|
|
|
#### Continuously differentiable maps
|
|
|
|
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.)
|
|
|
|
#### Smooth maps
|
|
|
|
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**.
|
|
|
|
#### Charts
|
|
|
|
Let $M$ be a smooth manifold. A **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 **local coordinates** of $p$ in the chart $(U,\varphi)$.
|
|
|
|
#### Atlas
|
|
|
|
Let $M$ be a smooth manifold. An **atlas** is a collection of charts $\mathcal{A}=\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$ such that $M=\bigcup_{\alpha\in I} U_\alpha$.
|
|
|
|
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 $\alpha, \beta\in I$.
|
|
|
|
#### Smooth manifold
|
|
|
|
A smooth manifold is a pair $(M,\mathcal{A})$ where $M$ is a topological manifold and $\mathcal{A}$ is a smooth atlas.
|
|
|
|
#### Fundamental group
|
|
|
|
A **fundamental group** of a point $p$ in a topological space $X$ is the group of all paths (continuous map $f:I\to X$, $I=[0,1]\subseteq \mathbb{R}$) from $p$ to $p$.
|
|
|
|
- Product defined as composition of paths.
|
|
- Identity element is the constant path from $p$ to $p$.
|
|
- Inverse is the reverse path.
|
|
|
|
#### smooth local coordinate representations
|
|
|
|
If $M$ is a smooth manifold, then any chart $(U,\varphi)$ contained in the given maximal smooth atlas is called a **smooth chart**, and the map $\varphi$ is called a **smooth coordinate map** because it gives a coordinate
|
|
|
|
#### Lie group
|
|
|
|
Lie group is a group (satisfying group axioms: closure, associativity, identity, inverses) that is also a smooth manifold. with the operator $m:G\times G\to G$, and the inverse operation $i:G\to G$ that are both smooth.
|
|
|
|
In short, a Lie group is a group that is also a smooth manifold with map $G\times G\to G$ given by $(g,h)\mapsto gh^-1$ that is smooth.
|
|
|
|
<details>
|
|
<summary>Example of Lie group</summary>
|
|
|
|
The general linear group $GL(n,\mathbb{R})$ is the group of all $n\times n$ invertible matrices over $\mathbb{R}$.
|
|
|
|
This is a Lie group since
|
|
|
|
1. Multiplication is a smooth map $GL(n,\mathbb{R})\times GL(n,\mathbb{R})\to GL(n,\mathbb{R})$ since it is a polynomial map.
|
|
2. Inverse is a smooth map $GL(n,\mathbb{R})\to GL(n,\mathbb{R})$ by cramer's rule.
|
|
|
|
---
|
|
|
|
If $G$ is a Lie group, then any open subgroup (with subgroup topology and open set in $G$) $H$ of $G$ is also a Lie group.
|
|
|
|
</details>
|
|
|
|
#### Translation map on Lie group
|
|
|
|
If $G$ is a Lie group, then the translation map $L_g:G\to G$ given by $L_g(h)=gh$ and $R_g:G\to G$ given by $R_g(h)=hg$ are both smooth and are diffeomorphisms on $G$.
|
|
|
|
#### Derivation and tangent vectors
|
|
|
|
The directional derivative of a geometric tangent vector $v_a\in \mathbb{R}^n_a$ yields a map $D_v\vert_a:C^\infty(\mathbb{R}^n)\to \mathbb{R}$ given by the formula
|
|
|
|
$$
|
|
D_v\vert_a(f)=D_v f(a)=\frac{d}{dt}\bigg\vert_{t=0}f(a+tv_a)
|
|
$$
|
|
|
|
Note that this is a linear over $\mathbb{R}$, and satisfies the product rule.
|
|
|
|
$$
|
|
D_v\vert_a(f\cdot g)=f(a)D_v\vert_a(g)+g(a)D_v\vert_a(f)
|
|
$$
|
|
|
|
We can generalize this representation to the following definition:
|
|
|
|
If $a$ is a point of $\mathbb{R}^n$, then a **derivation at $a$** is a linear map $w:C^\infty(\mathbb{R}^n)\to \mathbb{R}$ such that it is linear over $\mathbb{R}$ and satisfies the product rule.
|
|
|
|
$$
|
|
w(f\cdot g)=w(f)\cdot g(a)+f(a)\cdot w(g)
|
|
$$
|
|
|
|
Let $T_a\mathbb{R}^n$ denote the set of all derivations of $C^\infty(\mathbb{R}^n)$ at $a$. So $T_a\mathbb{R}^n$ is a vector space over $\mathbb{R}$.
|
|
|
|
$$
|
|
(w_1+w_2)(f)=w_1(f)+w_2(f),\quad (cw)(f)=c(w(f))
|
|
$$
|
|
|
|
Some key properties are given below and check the proof in the book for details.
|
|
|
|
1. If $f$ is a constant function, then $w(f)=0$.
|
|
2. If $f(a)=g(a)=0$, then $w(f\cdot g)=0$.
|
|
3. For each geometric tangent vector $v_a\in \mathbb{R}^n_a$, the map $D_v\vert_a:C^\infty(\mathbb{R}^n)\to \mathbb{R}$ is a derivation at $a$.
|
|
4. The map $v_a\mapsto D_v\vert_a$ is an isomorphism of vector spaces from $\mathbb{R}^n_a$ to $T_a\mathbb{R}^n$.
|
|
|
|
#### Tangent vector on Manifolds
|
|
|
|
Let $M$ be a smooth manifold. Let $p\in M$. A **tangent vector to $M$ at $p$** is a derivation at $p$ if it satisfies:
|
|
|
|
$$
|
|
v(f\cdot g)=f(p)vg+g(p)vf\prod \text{ for all } f,g\in C^\infty(M)
|
|
$$
|
|
|
|
The set of all derivations of $C^\infty(M)$ at $p$ is denoted by $T_pM$ is called tangent space to $M$ at $p$. An element of $T_pM$ is called a tangent vector to $M$ at $p$.
|
|
|
|
#### Tangent bundle
|
|
|
|
We define the tangent bundle of $M$ as the disjoint union of all the tangent spaces:
|
|
|
|
$$
|
|
TM=\bigsqcup_{p\in M} T_pM
|
|
$$
|
|
|
|
We write the element in $TM$ as pair $(p,v)$ where $p\in M$ and $v\in T_pM$.
|
|
|
|
The tangent bundle comes with a natural projection map $\pi:TM\to M$ given by $\pi(p,v)=p$.
|
|
|
|
#### Section of map
|
|
|
|
If $\pi:M\to N$ is any continuous map, a **section of $\pi$** is a continuous right inverse of $\pi$. For example $\sigma:N\to M$ is a section of $\pi$ if $\sigma\circ \pi=Id_N$.
|
|
|
|
#### Vector field
|
|
|
|
A vector field on $M$ is a section of the map $\pi:TM\to M$.
|
|
|
|
More concretely, a vector field is a continuous map $X:M\to TM$, usually written $p\mapsto X_p$, with property that
|
|
|
|
$$
|
|
\pi\circ X=Id_M
|
|
$$
|
|
|
|
> That is a map from element on the manifold to the tangent space of the manifold.
|
|
|
|
|
|
|
|
### Riemannian manifolds and geometry
|
|
|
|
#### Riemannian metric
|
|
|
|
A Riemannian metric is a smooth assignment of an inner product to each tangent space $T_pM$ of the manifold.
|
|
|
|
More formally, let $M$ be a smooth manifold. A **Riemannian metric** on $M$ is a smooth covariant 2-tensor field $g\in \mathcal{T}^2(M)$ whose value $g_p$ at each $p\in M$ is an inner product on $T_p M$.
|
|
|
|
Thus $g$ is a symmetric 2-tensor field that is positive definite in the sense that $g_p(v,v)\geq 0$ for each $p\in M$ and each $v\in T_p M$, with equality if and only if $v=0$.
|
|
|
|
Riemannian metric exists in great abundance.
|
|
|
|
A good news for smooth manifold is that every smooth manifold admits a Riemannian metric.
|
|
|
|
<details>
|
|
<summary> Example of Riemannian metrics</summary>
|
|
|
|
An example of Riemannian metric is the Euclidean metric, the bilinear form of $d(p,q)=\|p-q\|_2$ on $\mathbb{R}^n$.
|
|
|
|
More formally, the Riemannian metric $\overline{g}$ on $\mathbb{R}^n$ at each $x\in \mathbb{R}^n$ , for $v,w\in T_x \mathbb{R}^n$ with stardard coordinates $(x^1,\ldots,x^n)$ as $v=\sum_{i=1}^n v_i \partial_x^i$ and $w=\sum_{
|
|
|
|
</details>
|
|
|
|
#### Riemannian manifolds
|
|
|
|
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.
|
|
|
|
More formally, a **Riemannian manifold** is a pair $(M,g)$, where $M$ is a smooth manifold and $g$ is a specific choice of Riemannian metric on $M$.
|
|
|
|
|
|
An example of Riemannian manifold is the sphere $\mathbb{C}P^n$.
|
|
|
|
### Notion of Connection
|
|
|
|
A connection is a way to define the directional derivative of a vector field along a curve on a Riemannian manifold.
|
|
|
|
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
|
|
|
|
$$
|
|
D_VX=\lim_{h\to 0}\frac{X(p+h)-X(p)}{h}
|
|
$$
|
|
|
|
### Notion of Curvatures
|
|
|
|
> [!NOTE]
|
|
>
|
|
> Geometrically, the curvature of the manifold is radius of the tangent sphere of the manifold.
|
|
|
|
#### Nabla notation and Levi-Civita connection
|
|
|
|
|
|
|
|
#### Fundamental theorem of Riemannian geometry
|
|
|
|
Let $(M,g)$ be a Riemannian or pseudo-Riemannian manifold (with or without boundary). There exist sa unique connection $\nabla$ on $TM$ that is compatible with $g$ and symmetric. It is called the **Levi-Civita** connection of $g$ (or also, when $g$ is a positive definite, the Riemannian connection).
|
|
|
|
#### Ricci curvature
|