### 数学代写|黎曼几何代写Riemannian geometry代考|MTH 3022

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|黎曼几何代写Riemannian geometry代考|Frobenius’ Theorem

In this section we prove Frobenius’ theorem about vector distributions.
Definition 2.33 Let $M$ be a smooth manifold. A vector distribution $D$ of rank $m$ on $M$ is a family of vector subspaces $D_{q} \subset T_{q} M$, where $\operatorname{dim} D_{q}=m$ for every $q$.

A vector distribution $D$ is said to be smooth if, for every point $q_{0} \in M$, there exists a neighborhood $O_{q_{0}}$ of $q_{0}$ and a family of smooth vector fields $X_{1}, \ldots, X_{m}$ such that
$$D_{q}=\operatorname{span}\left{X_{1}(q), \ldots, X_{m}(q)\right}, \quad \forall q \in O_{q_{0}}$$
Definition 2.34 A smooth vector distribution $D$ (or rank $m$ ) on $M$ is said to be involutive if there exists a local basis of vector fields $X_{1}, \ldots, X_{m}$ satisfying (2.38), and smooth functions $a_{i j}^{k}$ on $M$, such that
$$\left[X_{i}, X_{k}\right]=\sum_{j=1}^{m} a_{i j}^{k} X_{j}, \quad \forall i, k=1, \ldots, m$$
Exercise 2.35 Prove that a smooth vector distribution $D$ is involutive if and only if for every local basis of vector fields $X_{1}, \ldots, X_{m}$ satisfying (2.38) there exist smooth functions $a_{i j}^{k}$ such that (2.39) holds.

Definition 2.36 A smooth vector distribution $D$ on $M$ is said to be flat if for every point $q_{0} \in M$ there exists a local diffeomorphism $\phi: O_{q_{0}} \rightarrow \mathbb{R}^{n}$ such that $\phi_{*, q}\left(D_{q}\right)=\mathbb{R}^{m} \times{0}$ for all $q \in O_{q_{0}}$.

Theorem 2.37 (Frobenius Theorem) A smooth distribution is involutive if and only if it is flat.

Proof The statement is local, hence it is sufficient to prove the statement on a neighborhood of every point $q_{0} \in M$.

## 数学代写|黎曼几何代写Riemannian geometry代考|An Application of Frobenius’ Theorem

Let $M$ and $N$ be two smooth manifolds. Given vector fields $X \in \operatorname{Vec}(M)$ and $Y \in \operatorname{Vec}(N)$ we define the vector field $X \times Y \in \operatorname{Vec}(M \times N)$ as the derivation
$$(X \times Y) a=X a_{y}^{1}+Y a_{x}^{2},$$
where, given $a \in C^{\infty}(M \times N)$, we define $a_{y}^{1} \in C^{\infty}(M)$ and $a_{x}^{2} \in C^{\infty}(N)$ as follows:
$$a_{y}^{1}(x):=a(x, y), \quad a_{x}^{2}(y):=a(x, y), \quad x \in M, y \in N .$$
Notice that, if we denote by $p_{1}: M \times N \rightarrow M$ and $p_{2}: M \times N \rightarrow N$ the two projections, we have
$$\left(p_{1}\right){}(X \times Y)=X, \quad\left(p{2}\right){}(X \times Y)=Y .$$
Exercise 2.40 Let $X{1}, X_{2} \in \operatorname{Vec}(M)$ and $Y_{1}, Y_{2} \in \operatorname{Vec}(N)$. Prove that
$$\left[X_{1} \times Y_{1}, X_{2} \times Y_{2}\right]=\left[X_{1}, X_{2}\right] \times\left[Y_{1}, Y_{2}\right]$$
We can now prove the following result, which is important when dealing with Lie groups (see Chapter 7 and Section 17.5).

## 数学代写|黎曼几何代写Riemannian geometry代考|Cotangent Space

In this section we introduce covectors, which are linear functionals on the tangent space. The space of all covectors at a point $q \in M$, called cotangent space, is in algebraic terms simply the dual space to the tangent space.

Definition 2.42 Let $M$ be an $n$-dimensional smooth manifold. The cotangent space at a point $q \in M$ is the set
$$T_{q}^{} M:=\left(T_{q} M\right)^{}=\left{\lambda: T_{q} M \rightarrow \mathbb{R}, \lambda \text { linear }\right}$$
For $\lambda \in T_{q}^{*} M$ and $v \in T_{q} M$, we will denote by $\langle\lambda, v\rangle:=\lambda(v)$ the evaluation of the covector $\lambda$ on the vector $v$.

As we have seen, the differential of a smooth map yields a linear map between tangent spaces. The dual of the differential gives a linear map between cotangent spaces.

Definition 2.43 Let $\varphi: M \rightarrow N$ be a smooth map and $q \in M$. The pullback of $\varphi$ at point $\varphi(q)$, where $q \in M$, is the map
$$\varphi^{}: T_{\varphi(q)}^{} N \rightarrow T_{q}^{} M, \quad \lambda \mapsto \varphi^{} \lambda,$$
defined by duality in the following way:
$$\left\langle\varphi^{} \lambda, v\right\rangle:=\left\langle\lambda, \varphi_{} v\right\rangle, \quad \forall v \in T_{q} M, \forall \lambda \in T_{\varphi(q)}^{} N .$$ Example 2.44 Let $a: M \rightarrow \mathbb{R}$ be a smooth function and $q \in M$. The differential $d_{q} a$ of the function $a$ at the point $q \in M$, defined through the formula $$\left\langle d_{q} a, v\right):=\left.\frac{d}{d t}\right|{t=0} a(\gamma(t)), \quad v \in T{q} M,$$
where $\gamma$ is any smooth curve such that $\gamma(0)=q$ and $\gamma(0)=v$, is an element of $T_{q}^{} M$. Indeed, the right-hand side of $(2.43)$ is linear with respect to $v$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Frobenius’ Theorem

D_{q}=\operatorname{span}\left{X_{1}(q), \ldots, X_{m}(q)\right}, \quad \forall q \in O_{q_{0}}D_{q}=\operatorname{span}\left{X_{1}(q), \ldots, X_{m}(q)\right}, \quad \forall q \in O_{q_{0}}

[X一世,Xķ]=∑j=1米一个一世jķXj,∀一世,ķ=1,…,米

## 数学代写|黎曼几何代写Riemannian geometry代考|An Application of Frobenius’ Theorem

(X×是)一个=X一个是1+是一个X2,

(p1)(X×是)=X,(p2)(X×是)=是.

[X1×是1,X2×是2]=[X1,X2]×[是1,是2]

## 数学代写|黎曼几何代写Riemannian geometry代考|Cotangent Space

T_{q}^{} M:=\left(T_{q} M\right)^{}=\left{\lambda: T_{q} M \rightarrow \mathbb{R}, \lambda \text { 线性}\正确的}T_{q}^{} M:=\left(T_{q} M\right)^{}=\left{\lambda: T_{q} M \rightarrow \mathbb{R}, \lambda \text { 线性}\正确的}

⟨披λ,在⟩:=⟨λ,披在⟩,∀在∈吨q米,∀λ∈吨披(q)ñ.例 2.44 让一个:米→R是一个平滑的函数并且q∈米. 差速器dq一个功能的一个在这一点上q∈米, 通过公式定义

⟨dq一个,在):=dd吨|吨=0一个(C(吨)),在∈吨q米,

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