## 数学代写|黎曼几何代写Riemannian geometry代考|МАТН6205

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|The First Dirichlet Eigenvalue Comparison Theorem

Following standard notations and setting (see, e.g., [Cha1] or in this context the seminal survey by Grigoryan in [Gri1]), for any precompact open set $\Omega$ in a Riemannian manifold $M$ we denote by $\lambda(\Omega)$ the smallest number $\lambda$ for which the following Dirichlet eigenvalue problem has a non-zero solution
\left{\begin{aligned} \Delta u+\lambda u &=0 \text { at all points } x \text { in } \Omega \ u(x) &=0 \text { at all points } x \text { in } \partial \Omega \end{aligned}\right.
We shall need the following beautiful observation due to Barta:

Theorem $7.1$ ([B], [Cha1]). Consider any smooth function $f$ on a domain $\Omega$ which satisfies $f_{\left.\right|{\Omega}}>0$ and $f{\mid \text {an }}=0$, and let $\lambda(\Omega)$ denote the first eigenvalue of the Dirichlet problem for $\Omega$. Then
$$\inf {\Omega}\left(\frac{\Delta f}{f}\right) \leq-\lambda(\Omega) \leq \sup {\Omega}\left(\frac{\Delta f}{f}\right)$$
If equality occurs in one of the inequalities, then they are both equalities, and $f$ is an eigenfunction for $\Omega$ corresponding to the eigenvalue $\lambda(\Omega)$.
Proof. Let $\phi$ be an eigenfunction for $\Omega$ corresponding to $\lambda(\Omega)$.
Then $\phi_{\Omega}>0$ and $\phi_{\left.\right|{\Omega}}=0$. If we let $h$ denote the difference $h=\phi-f$, then \begin{aligned} -\lambda(\Omega)=\frac{\Delta \phi}{\phi} &=\frac{\Delta f}{f}+\frac{f \Delta h-h \Delta f}{f(f+h)} \ &=\inf {\Omega}\left(\frac{\Delta f}{f}\right)+\sup {\Omega}\left(\frac{f \Delta h-h \Delta f}{f(f+h)}\right) \ &=\sup {\Omega}\left(\frac{\Delta f}{f}\right)+\inf {\Omega}\left(\frac{f \Delta h-h \Delta f}{f(f+h)}\right) \end{aligned} Here the supremum, $\sup {\Omega}\left(\frac{f \Delta h-h \Delta f}{f(f+h)}\right)$ is necessarily positive since
$$\left.f(f+h)\right|{\Omega}>0$$ and since by Green’s second formula $(6.8)$ in Theorem $6.4$ we have $$\int{\Omega}(f \Delta h-h \Delta f) d V=0 \text {. }$$
For the same reason, the infimum, $\inf _{\Omega}\left(\frac{f \Delta h-h \Delta f}{f(f+h)}\right)$ is necessarily negative. This gives the first part of the theorem. If equality occurs, then $(f \Delta h-h \Delta f)$ must vanish identically on $\Omega$, so that $-\lambda(\Omega)=\frac{\Delta f}{f}$, which gives the last part of the statement.

As already alluded to in the introduction, the key heuristic message of this report is that the Laplacian is a particularly ‘swift actor’ on minimal submanifolds (i.e., minimal extrinsic regular $R$-balls $D_{R}$ ) in ambient spaces with an upper bound $b$ on its sectional curvatures. This is to be understood in comparison with the ‘action’ of the Laplacian on totally geodesic $R$-balls $B_{R}^{b, m}$ in spaces of constant curvature b. In this section we will use Barta’s theorem to show that this phenomenon can indeed be ‘heard’ by ‘listening’ to the bass note of the Dirichlet spectrum of any given $D_{R}$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Isoperimetric Relations

In this and the following two sections we survey some comparison results concerning inequalities of isoperimetric type, mean exit times and capacities, respectively, for extrinsic minimal balls in ambient spaces with an upper bound on sectional curvature. This has been developed in a series of papers, see [Pa] and [MaP1][MaP4].

We will still assume a standard situation as in the previous section, i.e., $D_{R}$ denotes an extrinsic minimal ball of a minimal submanifold $P$ in an ambient space $N$ with the upper bound $b$ on the sectional curvatures.

Proposition 8.1. We define the following function of $t \in \mathbb{R}{+} \cup{0}$ for every $b \in \mathbb{R}$, for every $q \in \mathbb{R}$, and for every dimension $m \geq 2$ : $$L{q}^{b, m}(t)=q\left(\frac{\operatorname{Vol}\left(S_{t}^{b, m-1}\right)}{m h_{b}(t)}-\operatorname{Vol}\left(B_{t}^{b, m}\right)\right)$$
Then
$$L_{q}^{b, m}(0)=0 \text { for all } b, q, \text { and } m$$
and
$$\operatorname{sign}\left(\frac{d}{d t} L_{q}^{b, m}(t)\right)=\operatorname{sign}(b q) \text { for all } b, q, m, \text { and } t>0 \text {. }$$
Proof. This follows from a direct computation using the definition of $h_{b}(t)$ from equation (3.5) together with the volume formulae (cf. [Gr])
\begin{aligned} \operatorname{Vol}\left(B_{t}^{b, m}\right) &=\operatorname{Vol}\left(S_{1}^{0, m-1}\right) \cdot \int_{0}^{t}\left(Q_{b}(u)\right)^{m-1} d u \ \operatorname{Vol}\left(S_{t}^{b, m-1}\right) &=\operatorname{Vol}\left(S_{1}^{0, m-1}\right) \cdot\left(Q_{b}(t)\right)^{m-1} \end{aligned}

## 数学代写|黎曼几何代写Riemannian geometry代考|A Consequence of the Co-area Formula

The co-area equation (6.4) applied to our setting gives the following
Proposition 9.1. Let $D_{R}(p)$ denote a regular extrinsic minimal ball of $P$ with center $p$ in $N$. Then
$$\frac{d}{d u} \operatorname{Vol}\left(D_{u}\right) \geq \operatorname{Vol}\left(\partial D_{u}\right) \text { for all } u \leq R$$

Proof. We let $f: \bar{D}{R} \rightarrow \mathbb{R}$ denote the function $f(x)=R-r(x)$, which clearly vanishes on the boundary of $D{R}$ and is smooth except at $p$. Following the notation of the co-area formula we further let
\begin{aligned} \Omega(t) &=D_{(R-t)} \ V(t) &=\operatorname{Vol}\left(D_{(R-t)}\right) \text { and } \ \Sigma(t) &=\partial D_{(R-t)} \end{aligned}
Then
\begin{aligned} \operatorname{Vol}\left(D_{u}\right) &=V(R-u) \text { so that } \ \frac{d}{d u} \operatorname{Vol}\left(D_{u}\right) &=-V^{\prime}(t){\left.\right|{i=n-u}} . \end{aligned}
The co-area equation (6.4) now gives
\begin{aligned} -V^{\prime}(t) &=\int_{\partial D_{(R-t)}}\left|\nabla^{P} r\right|^{-1} d A \ & \geq \operatorname{Vol}\left(\partial D_{(R-t)}\right) \ &=\operatorname{Vol}\left(\partial D_{u}\right) \end{aligned}
and this proves the statement.
Exercise 9.2. Explain why the non-smoothness of the function $f$ at $p$ does not create problems for the application of equation (6.4) in this proof although smoothness is one of the assumptions in Theorem 6.1.

## 数学代写黎曼几何代写Riemannian geometry代 考|lsoperimetric Relations

$$L q^{b, m}(t)=q\left(\frac{\operatorname{Vol}\left(S_{t}^{b, m-1}\right)}{m h_{b}(t)}-\operatorname{Vol}\left(B_{t}^{b, m}\right)\right)$$

$$L_{q}^{b, m}(0)=0 \text { for all } b, q, \text { and } m$$

$$\operatorname{sign}\left(\frac{d}{d t} L_{q}^{b, m}(t)\right)=\operatorname{sign}(b q) \text { for all } b, q, m, \text { and } t>0$$

$$\operatorname{Vol}\left(B_{t}^{b, m}\right)=\operatorname{Vol}\left(S_{1}^{0, m-1}\right) \cdot \int_{0}^{t}\left(Q_{b}(u)\right)^{m-1} d u \operatorname{Vol}\left(S_{t}^{b, m-1}\right)=\operatorname{Vol}\left(S_{1}^{0, m-1}\right) \cdot\left(Q_{b}(t)\right)^{m-1}$$

## 数学代写黎曼几何代写Riemannian geometry代考|A Consequence of the Co-area Formula

$$\frac{d}{d u} \operatorname{Vol}\left(D_{u}\right) \geq \operatorname{Vol}\left(\partial D_{u}\right) \text { for all } u \leq R$$

$$\Omega(t)=D_{(R-t)} V(t)=\operatorname{Vol}\left(D_{(R-t)}\right) \text { and } \Sigma(t)=\partial D_{(R-t)}$$

$$\operatorname{Vol}\left(D_{u}\right)=V(R-u) \text { so that } \frac{d}{d u} \operatorname{Vol}\left(D_{u}\right) \quad=-V^{\prime}(t) \mid i=n-u$$

$$-V^{\prime}(t)=\int_{\partial D_{(R-t)}}\left|\nabla^{P} r\right|^{-1} d A \geq \operatorname{Vol}\left(\partial D_{(R-t)}\right)=\operatorname{Vol}\left(\partial D_{u}\right)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|黎曼几何代写Riemannian geometry代考|MATH3342

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Analysis of Lorentzian Distance Functions

For comparison, and before going further into the Riemannian setting, we briefly present the corresponding Hessian analysis of the distance function from a point in a Lorentzian manifold and its restriction to a spacelike hypersurface. The results can be found in [AHP], where the corresponding Hessian analysis was also carried out, i.e., the analysis of the Lorentzian distance from an achronal spacelike hypersurface in the style of Proposition 3.9. Recall that in Section 3 we also considered

the analysis of the distance from a totally geodesic hypersurface $P$ in the ambient Riemannian manifold $N$.

Let $\left(N^{n+1}, g\right)$ denote an $(n+1)$-dimensional spacetime, that is, a timeoriented Lorentzian manifold of dimension $n+1 \geq 2$. The metric tensor $g$ has index 1 in this case, and, as we did in the Riemannian context, we shall denote it alternatively as $g=\langle,$,$rangle (see, e.g., [O’N] as a standard reference for this section).$
Given $p, q$ two points in $N$, one says that $q$ is in the chronological future of $p$, written $p \ll q$, if there exists a future-directed timelike curve from $p$ to $q$. Similarly, $q$ is in the causal future of $p$, written $p<q$, if there exists a future-directed causal (i.e., nonspacelike) curve from $p$ to $q$.
Then the chronological future $I^{+}(p)$ of a point $p \in N$ is defined as
$$I^{+}(p)={q \in N: p \ll q} .$$
The Lorentzian distance function $d: N \times N \rightarrow[0,+\infty]$ for an arbitrary spacetime may fail to be continuous in general, and may also fail to be finite-valued. But there are geometric restrictions that guarantee a good behavior of $d$. For example, globally hyperbolic spacetimes turn out to be the natural class of spacetimes for which the Lorentzian distance function is finite-valued and continuous.

Given a point $p \in N$, one can define the Lorentzian distance function $d_{p}$ :
$M \rightarrow[0,+\infty]$ with respect to $p$ by
$$d_{p}(q)=d(p, q) .$$
In order to guarantee the smoothness of $d_{p}$, we need to restrict this function on certain special subsets of $N$. Let $\left.T_{-1} N\right|{p}$ be the following set $$\left.T{-1} N\right|{p}=\left{v \in T{p} N: v \text { is a future-directed timelike unit vector }\right} .$$
Define the function $s_{p}:\left.T_{-1} N\right|{p} \rightarrow[0,+\infty]$ by $$s{p}(v)=\sup \left{t \geq 0: d_{p}\left(\gamma_{v}(t)\right)=t\right},$$
where $\gamma_{v}:[0, a) \rightarrow N$ is the future inextendible geodesic starting at $p$ with initial velocity $v$. Then we define
$$\tilde{\mathcal{I}}^{+}(p)=\left{t v: \text { for all }\left.v \in T_{-1} N\right|{p} \text { and } 0{p}(v)\right}$$
and consider the subset $\mathcal{I}^{+}(p) \subset N$ given by
$$\mathcal{I}^{+}(p)=\exp {p}\left(\operatorname{int}\left(\tilde{\mathcal{I}}^{+}(p)\right)\right) \subset I^{+}(p) .$$ Observe that the exponential map $$\exp {p}: \operatorname{int}\left(\tilde{\mathcal{I}}^{+}(p)\right) \rightarrow \mathcal{I}^{+}(p)$$
is a diffeomorphism and $\mathcal{I}^{+}(p)$ is an open subset (possible empty).
Remark 4.1. When $b \geq 0$, the Lorentzian space form of constant sectional curvature $b$, which we denote as $N_{b}^{n+1}$, is globally hyperbolic and geodesically complete, and every future directed timelike unit geodesic $\gamma_{b}$ in $N_{b}^{n+1}$ realizes the Lorentzian distance between its points. In particular, if $b \geq 0$ then $\mathcal{I}^{+}(p)=I^{+}(p)$ for every point $p \in N_{b}^{n+1}$ (see [EGK, Remark 3.2]).

## 数学代写|黎曼几何代写Riemannian geometry代考|Concerning the Riemannian Setting and Notation

Returning now to the Riemannian case: Although we indeed do have the possibility of considering 4 basically different settings determined by the choice of $p$ or $V$ as the ‘base’ of our normal domain and the choice of $K_{N} \leq b$ or $K_{N} \geq b$ as the curvature assumption for the ambient space $N$, we will, however, mainly consider the ‘first’ of these. Specifically we will (unless otherwise explicitly stated) apply the following assumptions and denotations:
Definition 5.1. A standard situation encompasses the following:
(1) $P^{m}$ denotes an $m$-dimensional complete minimally immersed submanifold of the Riemannian manifold $N^{n}$. We always assume that $P$ has dimension $m \geq 2 .$
(2) The sectional curvatures of $N$ are assumed to satisfy $K_{N} \leq b, b \in \mathbb{R}$, cf. Proposition $3.10$, equation (3.13).
(3) The intersection of $P$ with a regular ball $B_{R}(p)$ centered at $p \in P$ (cf. Definition 3.4) is denoted by
$$D_{R}=D_{R}(p)=P^{m} \cap B_{R}(p)$$
and this is called a minimal extrinsic $R$-ball of $P$ in $N$, see the Figures 3-7 of extrinsic balls, which are cut out from some of the well-known minimal surfaces in $\mathbb{R}^{3}$.
(4) The totally geodesic $m$-dimensional regular $R$-ball centered at $\tilde{p}$ in $\mathbb{K}^{n}(b)$ is denoted by
$$B_{R}^{b, m}=B_{R}^{b, m}(\tilde{p})$$
whose boundary is the $(m-1)$-dimensional sphere
$$\partial B_{R}^{b, m}=S_{R}^{b, m-1}$$
(5) For any given smooth function $F$ of one real variable we denote
$$W_{F}(r)=F^{\prime \prime}(r)-F^{\prime}(r) h_{b}(r) \text { for } 0 \leq r \leq R$$
We may now collect the basic inequalities from our previous analysis as follows.

## 数学代写|黎曼几何代写Riemannian geometry代考|Green’s Formulae and the Co-area Formula

Now we recall the coarea formula. We follow the lines of [Sa] Chapter II, Section 5. Let $(M, g)$ denote a Riemannian manifold and $\Omega$ a precompact domain in $M$. Let $\psi: \Omega \rightarrow \mathbb{R}$ be a smooth function such that $\psi(\Omega)=[a, b]$ with $a<b$. Denote by $\Omega_{0}$ the set of critical points of $\psi$. By Sard’s theorem, the set of critical values $S_{\psi}=\psi\left(\Omega_{0}\right)$ has null measure, and the set of regular values $R_{\psi}=[a, b]-S_{\psi}$ is open. In particular, for any $t \in R_{\psi}=[a, b]-S_{\psi}$, the set $\Gamma(t):=\psi^{-1}(t)$ is a smooth embedded hypersurface in $\Omega$ with $\partial \Gamma(t)=\emptyset$. Since $\Gamma(t) \subseteq \Omega-\Omega_{0}$ then $\nabla \psi$ does not vanish along $\Gamma(t)$; indeed, a unit normal along $\Gamma(t)$ is given by $\nabla \psi /|\nabla \psi|$.
Now we let
\begin{aligned} &A(t)=\operatorname{Vol}(\Gamma(t)) \ &\Omega(t)={x \in \bar{\Omega} \mid \psi(x)<t} \ &V(t)=\operatorname{Vol}(\Omega(t)) \end{aligned}
Theorem 6.1.
i) For every integrable function $u$ on $\bar{\Omega}$ :
$$\int_{\Omega} u \cdot|\nabla \psi| d V=\int_{a}^{b}\left(\int_{\Gamma(t)} u d A_{t}\right) d t,$$
where $d A_{t}$ is the Riemannian volume element defined from the induced metric $g_{t}$ on $\Gamma(t)$ from $g$.
ii) The function $V(t)$ is a smooth function on the regular values of $\psi$ given by:
$$V(t)=\operatorname{Vol}\left(\Omega_{0} \cap \Omega(t)\right)+\int_{a}^{t}\left(\int_{\Gamma(t)}|\nabla \psi|^{-1} d A_{t}\right)$$
and its derivative is
$$\frac{d}{d t} V(t)=\int_{\Gamma(t)}|\nabla \psi|^{-1} d A_{t}$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Analysis of Lorentzian Distance Functions

dp(q)=d(p,q).

\left.T{-1} N\right|{p}=\left{v \in T{p} N: v \text { 是一个面向未来的类时单位向量 }\right} 。\left.T{-1} N\right|{p}=\left{v \in T{p} N: v \text { 是一个面向未来的类时单位向量 }\right} 。

s{p}(v)=\sup \left{t \geq 0: d_{p}\left(\gamma_{v}(t)\right)=t\right},s{p}(v)=\sup \left{t \geq 0: d_{p}\left(\gamma_{v}(t)\right)=t\right},

\tilde{\mathcal{I}}^{+}(p)=\left{t v: \text { for all }\left.v \in T_{-1} N\right|{p} \text { 和} 0{p}(v)\right}\tilde{\mathcal{I}}^{+}(p)=\left{t v: \text { for all }\left.v \in T_{-1} N\right|{p} \text { 和} 0{p}(v)\right}

## 数学代写|黎曼几何代写Riemannian geometry代考|Concerning the Riemannian Setting and Notation

(1)磷米表示一个米黎曼流形的一维完全最小浸没子流形ñn. 我们总是假设磷有维度米≥2.
(2) 截面曲率ñ假设满足ķñ≤b,b∈R，参见。主张3.10，等式（3.13）。
(3) 交集磷用普通球乙R(p)以p∈磷（参见定义 3.4）表示为

DR=DR(p)=磷米∩乙R(p)

(4) 完全测地线米维规则R- 球为中心p~在ķn(b)表示为

∂乙Rb,米=小号Rb,米−1
(5) 对于任何给定的平滑函数F我们表示的一个实变量

## 数学代写|黎曼几何代写Riemannian geometry代考|Green’s Formulae and the Co-area Formula

i) 对于每个可积函数在上Ω¯ :

∫Ω在⋅|∇ψ|d在=∫一个b(∫Γ(吨)在d一个吨)d吨,

ii) 功能在(吨)是一个关于正则值的平滑函数ψ给出：

dd吨在(吨)=∫Γ(吨)|∇ψ|−1d一个吨

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|黎曼几何代写Riemannian geometry代考|MATH3405

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Appetizer and Introduction

It is a natural and indeed a classical question to ask: “What is the effective resistance of, say, a hyperboloid or a helicoid if the surface is made of a homogeneous conducting material?”.

In these notes we will study the precise meaning of this and several other related questions and analyze how the answers depend on the curvature and topology of the given surfaces and manifolds. We will focus mainly on minimal submanifolds in ambient spaces which are assumed to have a well-defined upper (or lower) bound on their sectional curvatures.

One key ingredient is the comparison theory for distance functions in such spaces. In particular we establish and use a comparison result for the Laplacian of geometrically restricted distance functions. It is in this setting that we obtain information about such diverse phenomena as diffusion processes, isoperimetric inequalities, Dirichlet eigenvalues, transience, recurrence, and effective resistance of the spaces in question. In this second edition of the present notes we extend those previous findings in four ways: Firstly, we include comparison results for the exit time moment spectrum for compact domains in Riemannian manifolds; Secondly, and most substantially, we report on very recent results obtained by the first and third author together with C. Rosales concerning comparison results for the capacities and the type problem (transient versus recurrent) in weighted Riemannian manifolds; Thirdly we survey how some of the purely Riemannian results on transience and recurrence can be lifted to the setting of spacelike submanifolds in Lorentzian manifolds; Fourthly, the comparison spaces that we employ for some of the new results are typically so-called model spaces, i.e., warped products (gen= eralized surfaces of revolution) where ‘all the geometry’ in each case is determined by a given radial warping function and a given weight function.In a sense, all the different phenomena that we consider are ‘driven’ by the Laplace operator which in turn depends on the background curvatures and the weight function. One key message of this report is that the Laplacian is a particularly ‘swift’ operator – for example on minimal submanifolds in ambient spaces with small sectional curvatures – but depending on the weight functions. Specifically, we observe and report new findings about this behaviour in the contexts of both Riemannian, Lorentzian, and weighted geometries, see Sections 12 and $20-27$. Similar results generally hold true within the intrinsic geometry of the manifolds themselves – often even with Ricci curvature lower bounds (see, e.g., the survey [Zhu]) as a substitute for the specific assumption of a lower bound on sectional curvatures.

## 数学代写|黎曼几何代写Riemannian geometry代考|The Comparison Setting and Preliminaries

We consider a complete immersed submanifold $P^{m}$ in a Riemannian manifold $N^{n}$, and denote by $\mathrm{D}^{P}$ and $\mathrm{D}^{N}$ the Riemannian connections of $P$ and $N$, respectively. We refer to the excellent general monographs on Riemannian geometry – e.g., [Sa], [CheeE], and [Cha2] – for the basic notions, that will be applied in these notes. In particular we shall be concerned with the second-order behavior of certain functions on $P$ which are obtained by restriction from the ambient space $N$ as displayed in Proposition $3.1$ below. The second-order derivatives are defined in terms of the Hessian operators Hess ${ }^{N}$, Hess ${ }^{P}$ and their traces $\Delta^{N}$ and $\Delta^{P}$, respectively (see, e.g., [Sa] p. 31). The difference between these operators quite naturally involves geometric second-order information about how $P^{m}$ actually sits inside $N^{n}$. This information is provided by the second fundamental form $\alpha$ (resp. the mean curvature $H$ ) of $P$ in $N$ (see [Sa] p. 47). If the functions under consideration are essentially distance functions in $N$ – or suitably modified distance functions then their second-order behavior is strongly influenced by the curvatures of $N$, as is directly expressed by the second variation formula for geodesics ([Sa] p. 90).

As is well known, the ensuing and by now classical comparison theorems for Jacobi fields give rise to the celebrated Toponogov theorems for geodesic triangles and to powerful results concerning the global structure of Riemannian spaces ([Sa], Chapters IV-V). In these notes, however, we shall mainly apply the Jacobi field comparison theory only off the cut loci of the ambient space $N$, or more precisely, within the regular balls of $N$ as defined in Definition $3.4$ below. On the other hand, from the point of view of a given (minimal) submanifold $P$ in $N$, our results for $P$ are semi-global in the sense that they apply to domains which are not necessarily distance-regular within $P$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Analysis of Riemannian Distance Functions

Let $\mu: N \mapsto \mathbb{R}$ denote a smooth function on $N$. Then the restriction $\tilde{\mu}=\mu_{\left.\right|{P}}$ is a smooth function on $P$ and the respective Hessians $\operatorname{Hess}^{N}(\mu)$ and $\operatorname{Hess}^{P}(\tilde{\mu})$ are related as follows: Proposition $3.1([\mathrm{JK}]$ p. 713$)$. \begin{aligned} \operatorname{Hess}^{P}(\tilde{\mu})(X, Y)=& \operatorname{Hess}^{N}(\mu)(X, Y) \ &+\left\langle\nabla^{N}(\mu), \alpha(X, Y)\right\rangle \end{aligned} for all tangent vectors $X, Y \in T P \subseteq T N$, where $\alpha$ is the second fundamental form of $P$ in $N$. Proof. \begin{aligned} \operatorname{Hess}^{P}(\tilde{\mu})(X, Y) &=\left\langle\mathrm{D}{X}^{P} \nabla^{P} \tilde{\mu}, Y\right\rangle \ &=\left\langle\mathrm{D}{X}^{N} \nabla^{P} \tilde{\mu}-\alpha\left(X, \nabla^{P} \tilde{\mu}\right), Y\right\rangle \ &=\left\langle\mathrm{D}{X}^{N} \nabla^{P} \tilde{\mu}, Y\right\rangle \ &=X\left(\left\langle\nabla^{P} \tilde{\mu}, Y\right\rangle\right)-\left\langle\nabla^{P} \tilde{\mu}, \mathrm{D}{X}^{N} Y\right\rangle \ &=\left\langle\mathrm{D}{X}^{N} \nabla^{N} \mu, Y\right\rangle+\left\langle\left(\nabla^{N} \mu\right)^{\perp}, \mathrm{D}_{X}^{N} Y\right\rangle \ &=\operatorname{Hess}^{N}(\mu)(X, Y)+\left\langle\left(\nabla^{N} \mu\right)^{\perp}, \alpha(X, Y)\right\rangle \ &=\operatorname{Hess}^{N}(\mu)(X, Y)+\left\langle\nabla^{N} \mu, \alpha(X, Y)\right\rangle \end{aligned}
If we modify $\mu$ to $F \circ \mu$ by a smooth function $F: \mathbb{R} \mapsto \mathbb{R}$, then we get
Lemma 3.2.
\begin{aligned} \operatorname{Hess}^{N}(F \circ \mu)(X, X)=& F^{\prime \prime}(\mu) \cdot\left\langle\nabla^{N}(\mu), X\right\rangle^{2} \ &+F^{\prime}(\mu) \cdot \operatorname{Hess}^{N}(\mu)(X, X) \end{aligned}
for all $X \in T N^{n}$

In the following we write $\mu=\tilde{\mu}$. Combining (3.1) and (3.3) then gives
Corollary 3.3.
\begin{aligned} \operatorname{Hess}^{P}(F \circ \mu)(X, X)=& F^{\prime \prime}(\mu) \cdot\left\langle\nabla^{N}(\mu), X\right\rangle^{2} \ &+F^{\prime}(\mu) \cdot \operatorname{Hess}^{N}(\mu)(X, X) \ &+\left\langle\nabla^{N}(\mu), \alpha(X, X)\right\rangle \end{aligned}
for all $X \in T P^{m}$.
In what follows the function $\mu$ will always be a distance function in $N$-either from a point $p$ in which case we set $\mu(x)=\operatorname{dist}{N}(p, x)=r(x)$, or from a totally geodesic hypersurface $V^{n-1}$ in $N$ in which case we let $\mu(x)=$ dist ${N}(V, x)=$ $\eta(x)$. The function $F$ will always be chosen, so that $F \circ \mu$ is smooth inside the respective regular balls around $p$ and inside the regular tubes around $V$, which we now define. The sectional curvatures of the two-planes $\Omega$ in the tangent bundle of the ambient space $N$ are denoted by $K_{N}(\Omega)$, see, e.g., [Sa], Section II.3. Concerning the notation: In the following both Hess $^{N}$ and Hess will be used invariantly for both the Hessian in the ambient manifold $N$, as well as in a purely intrinsic context where only $N$ and not any of its submanifolds is under consideration.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|黎曼几何代写Riemannian geometry代考|MATH3342

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|The Minimal Control and the Length of an Admissible Curve

We start by defining the sub-Riemannian norm for vectors that belong to the distribution of a sub-Riemannian manifold.

Definition 3.8 Let $v \in \mathcal{D}{q}$. We define the sub-Riemannian norm of $v$ as follows: $$|v|:=\min \left{|u|, u \in U{q} \text { s.t. } v=f(q, u)\right} .$$
Notice that since $f$ is linear with respect to $u$, the minimum in $(3.9)$ is always attained at a unique point. Indeed, the condition $f(q, \cdot)=v$ defines an affine subspace of $U_{q}$ (which is nonempty since $v \in \mathcal{D}_{q}$ ) and the minimum in (3.9) is uniquely attained at the orthogonal projection of the origin onto this subspace (see Figure 3.2).

Exercise 3.9 Show that $|\cdot|$ is a norm in $\mathcal{D}{q}$. Moreover prove that it satisfies the parallelogram law, i.e., it is induced by a scalar product $\langle\cdot \mid \cdot\rangle{q}$ on $\mathcal{D}{q}$ that can be recovered by the polarization identity $$\langle v \mid w\rangle{q}=\frac{1}{4}|v+w|^{2}-\frac{1}{4}|v-w|^{2}, \quad v, w \in \mathcal{D}{q} .$$ Exercise $3.10$ Let $u{1}, \ldots, u_{m} \in U_{q}$ be an orthonormal basis for $U_{q}$. Define $v_{i}=f\left(q, u_{i}\right)$. Show that if $f(q, \cdot)$ is injective then $v_{1}, \ldots, v_{m}$ is an orthonormal basis for $\mathcal{D}_{q}$.

An admissible curve $\gamma:[0, T] \rightarrow M$ is Lipschitz, hence differentiable at almost every point. Hence the unique control $t \mapsto u^{*}(t)$ associated with $\gamma$ and realizing the minimum in $(3.9)$ is well defined a.e. on $[0, T]$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Equivalence of Sub-Riemannian Structures

In this section we introduce the notion of the equivalence of sub-Riemannian structures on the same base manifold $M$ and the notion of isometry between sub-Riemannian manifolds.

Definition $3.18$ Let $(\mathbf{U}, f),\left(\mathbf{U}^{\prime}, f^{\prime}\right)$ be two sub-Riemannian structures on a smooth manifold $M$. They are said to be equivalent as distributions if the following conditions hold:

(i) there exist a Euclidean bundle $\mathbf{V}$ and two surjective vector bundle morphisms $p: \mathbf{V} \rightarrow \mathbf{U}$ and $p^{\prime}: \mathbf{V} \rightarrow \mathbf{U}^{\prime}$ such that the following diagram is commutative:

The structures $(\mathbf{U}, f)$ and $\left(\mathbf{U}^{\prime}, f^{\prime}\right)$ are said to be equivalent as sub-Riemannian structures (or simply equivalent) if (i) is satisfied and moreover
(ii) the projections $p, p^{\prime}$ are compatible with the scalar product, i.e., it holds that
\begin{aligned} |u| &=\min {|v|, p(v)=u}, & \forall u \in \mathbf{U}, \ \left|u^{\prime}\right| &=\min \left{|v|, p^{\prime}(v)=u^{\prime}\right}, & \forall u^{\prime} \in \mathbf{U}^{\prime} . \end{aligned}
Remark $3.19$ If $(\mathbf{U}, f)$ and $\left(\mathbf{U}^{\prime}, f^{\prime}\right)$ arc cquivalcnt as sub-Ricmannian structures on $M$ then:
(a) the distributions $\mathcal{D}{q}$ and $\mathcal{D}{q}^{\prime}$ defined by $f$ and $f^{\prime}$ coincide, since $f\left(U_{q}\right)=f^{\prime}\left(U_{q}^{\prime}\right)$ for all $q \in M$;
(b) for each $w \in \mathcal{D}_{q}$ we have $|w|=|w|^{\prime}$, where $|\cdot|$ and $|\cdot|^{\prime}$ are the norms induced by $(\mathbf{U}, f)$ and $\left(\mathbf{U}^{\prime}, f^{\prime}\right)$ respectively.
In particular the lengths of admissible curves for two equivalent subRiemannian structures are the same.

Exercise 3.20 Prove that $(M, \mathbf{U}, f)$ and $\left(M, \mathbf{U}^{\prime}, f^{\prime}\right)$ are equivalent as distributions if and only if the moduli of the horizontal vector fields $\mathcal{D}$ and $\mathcal{D}^{\prime}$ coincide.

## 数学代写|黎曼几何代写Riemannian geometry代考|Sub-Riemannian Distance

In this section we introduce the sub-Riemannian distance and prove the Rashevskii-Chow theorem.

Recall that, thanks to the results of Section 3.1.4, in what follows we can assume that the sub-Riemannian structure on $M$ is free, with generating family $\mathcal{F}=\left{f_{1}, \ldots, f_{m}\right}$. Notice that, by the definition of a sub-Riemannian manifold, $M$ is assumed to be connected and $\mathcal{F}$ is assumed to be bracketgenerating.

Definition 3.30 Let $M$ be a sub-Riemannian manifold and $q_{0}, q_{1} \in M$. The sub-Riemannian distance (or Carnot-Carathéodory distance) between $q_{0}$ and $q_{1}$ is
$d\left(q_{0}, q_{1}\right)=\inf \left{\ell(\gamma) \mid \gamma:[0, T] \rightarrow M\right.$ admissible, $\left.\gamma(0)=q_{0}, \gamma(T)=q_{1}\right} .$
We now state the main result of this section.
Theorem $3.31$ (Rashevskii-Chow) Let $M$ be a sub-Riemannian manifold. Then
(i) $(M, d)$ is a metric space,
(ii) the topology induced by $(M, d)$ is equivalent to the manifold topology.
In particular, $d: M \times M \rightarrow \mathbb{R}$ is continuous.
One of the main consequences of this result is that, thanks to the bracketgenerating condition, for every $q_{0}, q_{1} \in M$ there exists an admissible curve that joins them. Hence $d\left(q_{0}, q_{1}\right)<+\infty$.

In what follows $B(q, r)$ (sometimes denoted also $B_{r}(q)$ ) is the (open) subRiemannian ball of radius $r$ and center $q$ :
$$B(q, r):=\left{q^{\prime} \in M \mid d\left(q, q^{\prime}\right)<r\right} .$$

## 数学代写|黎曼几何代写Riemannian geometry代考|The Minimal Control and the Length of an Admissible Curve

|v|:=\min \left{|u|, u \in U{q} \text { st } v=f(q, u)\right} 。|v|:=\min \left{|u|, u \in U{q} \text { st } v=f(q, u)\right} 。

⟨在∣在⟩q=14|在+在|2−14|在−在|2,在,在∈Dq.锻炼3.10让在1,…,在米∈在q是一个正交基在q. 定义在一世=F(q,在一世). 证明如果F(q,⋅)是内射的在1,…,在米是一个正交基Dq.

## 数学代写|黎曼几何代写Riemannian geometry代考|Equivalence of Sub-Riemannian Structures

(i) 存在欧几里得丛在和两个满射向量丛态射p:在→在和p′:在→在′使得下图是可交换的：

(ii)p,p′与标量积兼容，即它认为

\begin{对齐} |u| &=\min {|v|, p(v)=u}, & \forall u \in \mathbf{U}, \ \left|u^{\prime}\right| &=\min \left{|v|, p^{\prime}(v)=u^{\prime}\right}, & \forall u^{\prime} \in \mathbf{U}^{\主要} 。\end{对齐}\begin{对齐} |u| &=\min {|v|, p(v)=u}, & \forall u \in \mathbf{U}, \ \left|u^{\prime}\right| &=\min \left{|v|, p^{\prime}(v)=u^{\prime}\right}, & \forall u^{\prime} \in \mathbf{U}^{\主要} 。\end{对齐}

(a) 分布Dq和Dq′被定义为F和F′巧合，因为F(在q)=F′(在q′)对所有人q∈米;
(b) 对于每个在∈Dq我们有|在|=|在|′， 在哪里|⋅|和|⋅|′规范是由(在,F)和(在′,F′)分别。

## 数学代写|黎曼几何代写Riemannian geometry代考|Sub-Riemannian Distance

d\left(q_{0}, q_{1}\right)=\inf \left{\ell(\gamma) \mid \gamma:[0, T] \rightarrow M\right.$可接受，$\left .\gamma(0)=q_{0}, \gamma(T)=q_{1}\right} 。d\left(q_{0}, q_{1}\right)=\inf \left{\ell(\gamma) \mid \gamma:[0, T] \rightarrow M\right.$可接受，$\left .\gamma(0)=q_{0}, \gamma(T)=q_{1}\right} 。

（一）(米,d)是一个度量空间，
（ii）由(米,d)等价于流形拓扑。

B(q, r):=\left{q^{\prime} \in M \mid d\left(q, q^{\prime}\right)<r\right} 。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|黎曼几何代写Riemannian geometry代考|MATH3405

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Vector Bundles

Heuristically, a smooth vector bundle on a smooth manifold $M$ is a smooth family of vector spaces parametrized by points in $M$.

Definition 2.47 Let $M$ be an $n$-dimensional manifold. A smooth vector bundle of rank $k$ over $M$ is a smooth manifold $E$ with a surjective smooth map $\pi: E \rightarrow M$ such that:
(i) the set $E_{q}:=\pi^{-1}(q)$, the $f$ iber of $E$ at $q$, is a $k$-dimensional vector space;
(ii) for every $q \in M$ there exist a neighborhood $O_{q}$ of $q$ and a linear-on-fibers diffeomorphism (called a local trivialization) $\psi: \pi^{-1}\left(O_{q}\right) \rightarrow O_{q} \times \mathbb{R}^{k}$ such that the following diagram commutes:
The space $E$ is called total space and $M$ is the base of the vector bundle. We will refer to $\pi$ as the canonical projection, and rank $E$ will denote the rank of the bundle.
Remark $2.48$ A vector bundle $E$, as a smooth manifold, has dimension
$$\operatorname{dim} E=\operatorname{dim} M+\operatorname{rank} E=n+k .$$
In the case when there exists a global trivialization map (i.e., when one can choose a local trivialization with $O_{q}=M$ for all $q \in M$ ), then $E$ is diffeomorphic to $M \times \mathbb{R}^{k}$ and we say that $E$ is trivializable.

Example 2.49 For any smooth $n$-dimensional manifold $M$, the tangent bundle $T M$, defined as the disjoint union of the tangent spaces at all points of $M$,
$$T M=\bigcup_{q \in M} T_{q} M,$$
has the natural structure of a $2 n$-dimensional smooth manifold, equipped with the vector bundle structure (of rank $n$ ) induced by the canonical projection map
$$\pi: T M \rightarrow M, \quad \pi(v)=q \quad \text { if } \quad v \in T_{q} M .$$
In the same way one can consider the cotangent bundle $T^{} M$, defined as $$T^{} M=\bigcup_{q \in M} T_{q}^{*} M .$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Submersions and Level Sets of Smooth Maps

If $\varphi: M \rightarrow N$ is a smooth map, we define the rank of $\varphi$ at $q \in M$ to be the rank of the linear map $\varphi_{*, q}: T_{q} M \rightarrow T_{\varphi(q)} N$. It is, of course, just the rank of the matrix of partial derivatives of $\varphi$ in any coordinate chart, or the dimension

of $\operatorname{im}\left(\varphi_{*, q}\right) \subset T_{\varphi(q)} N$. If $\varphi$ has the same rank $k$ at every point, we say $\varphi$ has constant rank and write rank $\varphi=k$.

An immersion is a smooth map $\varphi: M \rightarrow N$ with the property that $\varphi_{}$ is injective at each point (or equivalently $\operatorname{rank} \varphi=\operatorname{dim} M$ ). Similarly, a submersion is a smooth map $\varphi: M \rightarrow N$ such that $\varphi_{}$ is surjective at each point (equivalently, $\operatorname{rank} \varphi=\operatorname{dim} N$ ).

Theorem $2.56$ (Rank theorem) Suppose that $M$ and $N$ are smooth manifolds of dimensions $m$ and $n$ respectively and that $\varphi: M \rightarrow N$ is a smooth map with constant rank $k$ in a neighborhood of $q \in M$. Then there exist coordinates $\left(x_{1}, \ldots, x_{m}\right)$ centered at $q$ and $\left(y_{1}, \ldots, y_{n}\right)$ centered at $\varphi(q)$ in which $\varphi$ has the following coordinate representation:
$$\varphi\left(x_{1}, \ldots, x_{m}\right)=\left(x_{1}, \ldots, x_{k}, 0, \ldots, 0\right) .$$
Remark $2.57$ The previous theorem can be rephrased in the following way.
Let $\varphi: M \rightarrow N$ be a smooth map between two smooth manifolds. Then the following are equivalent:
(i) $\varphi$ has constant rank in a neighborhood of $q \in M$;
(ii) there exist coordinates near $q \in M$ and $\varphi(q) \in N$ in which the coordinate representation of $\varphi$ is linear.

In the case of a submersion, from Theorem $2.56$ one can deduce the following result.

## 数学代写|黎曼几何代写Riemannian geometry代考|Basic Definitions

We start by introducing a bracket-generating family of vector fields.
Definition $3.1$ Let $M$ be a smouth manifold and let $\mathcal{F} \subset \operatorname{Vec}(M)$ be a family of smooth vector fields. The Lie algebra generated by $\mathcal{F}$ is the smallest subalgebra of $\operatorname{Vec}(M)$ containing $\mathcal{F}$, namely
$$\operatorname{Lie} \mathcal{F}:=\operatorname{span}\left{\left[X_{1}, \ldots,\left[X_{j-1}, X_{j}\right]\right], X_{i} \in \mathcal{F}, j \in \mathbb{N}\right}$$
We will say that $\mathcal{F}$ is bracket-generating (or that it satisfies the Hörmander condition) if
$$\operatorname{Lie}{q} \mathcal{F}:={X(q), X \in \text { Lie } \mathcal{F}}=T{q} M, \quad \forall q \in M$$

Moreover, for $s \in \mathbb{N}$, we define
$$\operatorname{Lie}^{s} \mathcal{F}:=\operatorname{span}\left{\left[X_{1}, \ldots,\left[X_{j-1}, X_{j}\right]\right], X_{i} \in \mathcal{F}, j \leq s\right}$$
We say that the family $\mathcal{F}$ has step s at $q$ if $s \in \mathbb{N}$ is the minimal integer satisfying
$$\operatorname{Lie}{q}^{s} \mathcal{F}:=\left{X(q) X \in \operatorname{Lie}^{s} \mathcal{F}\right}=T{q} M$$
Notice that, in general, the step $s$ may depend on the point on $M$ and that $s=s(q)$ can be unbounded on $M$ even for bracket-generating families.

## 数学代写|黎曼几何代写Riemannian geometry代考|Vector Bundles

(i) 集合和q:=圆周率−1(q)， 这F伊伯尔和在q, 是一个ķ维向量空间；
(ii) 对于每个q∈米有一个社区○q的q和纤维上的线性微分同胚（称为局部平凡化）ψ:圆周率−1(○q)→○q×Rķ使得下图通勤：

## 数学代写|黎曼几何代写Riemannian geometry代考|Submersions and Level Sets of Smooth Maps

(i)披在附近有恒定的排名q∈米;
(ii) 附近有坐标q∈米和披(q)∈ñ其中的坐标表示披是线性的。

## 数学代写|黎曼几何代写Riemannian geometry代考|Basic Definitions

\operatorname{Lie} \mathcal{F}:=\operatorname{span}\left{\left[X_{1}, \ldots,\left[X_{j-1}, X_{j}\right]\right ], X_{i} \in \mathcal{F}, j \in \mathbb{N}\right}\operatorname{Lie} \mathcal{F}:=\operatorname{span}\left{\left[X_{1}, \ldots,\left[X_{j-1}, X_{j}\right]\right ], X_{i} \in \mathcal{F}, j \in \mathbb{N}\right}

\operatorname{Lie}^{s} \mathcal{F}:=\operatorname{span}\left{\left[X_{1}, \ldots,\left[X_{j-1}, X_{j}\对]\right], X_{i} \in \mathcal{F}, j \leq s\right}\operatorname{Lie}^{s} \mathcal{F}:=\operatorname{span}\left{\left[X_{1}, \ldots,\left[X_{j-1}, X_{j}\对]\right], X_{i} \in \mathcal{F}, j \leq s\right}

\operatorname{Lie}{q}^{s} \mathcal{F}:=\left{X(q) X \in \operatorname{Lie}^{s} \mathcal{F}\right}=T{q } 米\operatorname{Lie}{q}^{s} \mathcal{F}:=\left{X(q) X \in \operatorname{Lie}^{s} \mathcal{F}\right}=T{q } 米

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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,

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|黎曼几何代写Riemannian geometry代考|MAST90029

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Nonautonomous Vector Fields

Definition $2.13$ A nonautonomous vector field is family of vector fields $\left{X_{t}\right}_{t \in \mathbb{R}}$ such that the $\operatorname{map} X(t, q)=X_{t}(q)$ satisfies the following properties:
(C1) the map $t \mapsto X(t, q)$ is measurable for every fixed $q \in M$;
(C2) the map $q \mapsto X(t, q)$ is smooth for every fixed $t \in \mathbb{R}$;
(C3) for every system of coordinates defined in an open set $\Omega \subset M$ and every compact $K \subset \Omega$ and compact interval $I \subset \mathbb{R}$ there exist two functions $c(t), k(t)$ in $L^{\infty}(I)$ such that, for all $(t, x),(t, y) \in I \times K$,
$$|X(t, x)| \leq c(t), \quad|X(t, x)-X(t, y)| \leq k(t)|x-y|$$
Conditions (C1) and (C2) are equivalent to requiring that for every smooth function $a \in C^{\infty}(M)$ the scalar function $\left.(t, q) \mapsto X_{t} a\right|_{q}$ defined on $\mathbb{R} \times M$ is measurable in $t$ and smooth in $q$.

Remark $2.14$ In what follows we are mainly interested in nonautonomous vector fields of the following form:
$$X_{t}(q)=\sum_{i=1}^{m} u_{i}(t) f_{i}(q)$$
where the $u_{i}$ are $L^{\infty}$ functions and the $f_{i}$ are smooth vector fields on $M$. For this class of nonautonomous vector fields, assumptions (C1)-(C2) are trivially satisfied. Regarding $(\mathrm{C} 3)$, thanks to the smoothness of $f_{i}$, for every compact set $K \subset \Omega$ we can find two positive constants $C_{K}, L_{K}$ such that, for all $i=1, \ldots, m$, and $j=1, \ldots, n$, we have

$$\left|f_{i}(x)\right| \leq C_{K}, \quad\left|\frac{\partial f_{i}}{\partial x_{j}}(x)\right| \leq L_{K}, \quad \forall x \in K,$$
and we obtain, for all $(t, x),(t, y) \in I \times K$,
$$|X(t, x)| \leq C_{K} \sum_{i=1}^{m}\left|u_{i}(t)\right|, \quad|X(t, x)-X(t, y)| \leq L_{K} \sum_{i=1}^{m}\left|u_{i}(t)\right| \cdot|x-y| .$$
The existence and uniqueness of integral curves of a nonautonomous vector field are guaranteed by the following theorem (see [BP07]).

## 数学代写|黎曼几何代写Riemannian geometry代考|Differential of a Smooth Map

A smooth map between manifolds induces a map between the corresponding tangent spaces.

Definition $2.17$ Let $\varphi: M \rightarrow N$ be a smooth map between smooth manifolds and let $q \in M$. The differential of $\varphi$ at the point $q$ is the linear map
$$\varphi_{, q}: T_{q} M \rightarrow T_{\varphi(q)} N$$ defined as follows: $$\varphi_{, q}(v)=\left.\frac{d}{d t}\right|{t=0} \varphi(\gamma(t)) \quad \text { if } \quad v=\left.\frac{d}{d t}\right|{t=0} \gamma(t), \quad q=\gamma(0) .$$
It is easily checked that this definition depends only on the equivalence class of $\gamma$.

The differential $\varphi_{: q}$ of a smooth map $\varphi: M \rightarrow N$ (see Figure 2.1), also called its pushforward, is sometimes denoted by the symbols $D_{q} \varphi$ or $d_{q} \varphi$. Exercise 2.18 Let $\varphi: M \rightarrow N, \psi: N \rightarrow Q$ be smooth maps between manifolds. Prove that the differential of the composition $\psi \circ \varphi: M \rightarrow Q$ satisfies $(\psi \circ \varphi){}=\psi{} \circ \varphi_{}$.

As we said, a smooth map induces a transformation of tangent vectors. If we deal with diffeomorphisms, we can also obtain a pushforward for a vector field.

## 数学代写|黎曼几何代写Riemannian geometry代考|Lie Brackets

In this section we introduce a fundamental notion for sub-Riemannian geometry, the Lie bracket of two vector fields $X$ and $Y$. Geometrically it is defined as an infinitesimal version of the pushforward of the second vector field along the flow of the first. As explained below, it measures how much $Y$ is modified by the flow of $X$.

Definition 2.22 Let $X, Y \in \operatorname{Vec}(M)$. We define their Lie bracket as the vector field
$$[X, Y]:=\left.\frac{\partial}{\partial t}\right|{t=0} e{}^{-t X} Y .$$ Remark $2.23$ The geometric meaning of the Lie bracket can be understood by writing explicitly \begin{aligned} {\left.[X, Y]\right|{q} } &=\left.\left.\frac{\partial}{\partial t}\right|{t=0} e_{}^{-t X} Y\right|{q}=\left.\frac{\partial}{\partial t}\right|{t=0} e_{}^{-t X}\left(\left.Y\right|{e^{t} X}(q)\right.\ &=\left.\frac{\partial}{\partial s \partial t}\right|{t=s=0} e^{-t X} \circ e^{s Y} \circ e^{t X}(q) \end{aligned}
Proposition 2.24 As derivations on functions, one has the identity
$$\lfloor X, Y \mid=X Y-Y X$$
Proof By definition of the Lie bracket we have $[X, Y] a=\left.(\partial / \partial t)\right|{t=0}$ $\left(e{}^{-t X} Y\right) a$. Hence we need to compute the first-order term in the expansion, with respect to $t$, of the map

$t \mapsto\left(e_{}^{-t X} Y\right) a .$ Using formula (2.28), we have $$\left(e_{}^{-t X} Y\right) a=Y\left(a \circ e^{-t X}\right) \circ e^{t X} .$$
By Remark 2.9, we have $a \circ e^{-t X}=a-t X a+O\left(t^{2}\right)$, hence
\begin{aligned} \left(e_{}^{-t X} Y\right) a &=Y\left(a-t X a+O\left(t^{2}\right)\right) \circ e^{t X} \ &=\left(Y a-t Y X a+\bar{O}\left(t^{2}\right)\right) \circ e^{t X} . \end{aligned} Denoting $b=Y a-t Y X a+O\left(t^{2}\right), b_{t}=b \circ e^{t X}$, and using again the above expansion, we get \begin{aligned} \left(e_{}^{-t X} Y\right) a &=\left(Y a-t Y X a+O\left(t^{2}\right)\right)+t X\left(Y a-t Y X a+O\left(t^{2}\right)\right)+O\left(t^{2}\right) \ &=Y a+t(X Y-Y X) a+O\left(t^{2}\right) \end{aligned}
which proves that the first-order term with respect to $t$ in the expansion is $(X Y-Y X) a$.
Proposition $2.24$ shows that $(\operatorname{Vec}(M),[\cdot, \cdot])$ is a Lie algebra.

## 数学代写|黎曼几何代写Riemannian geometry代考|Nonautonomous Vector Fields

（C1）地图吨↦X(吨,q)是可测量的每个固定的q∈米;
(C2) 地图q↦X(吨,q)对于每个固定的都是平滑的吨∈R;
(C3) 对于在开放集中定义的每个坐标系统Ω⊂米和每一个契约ķ⊂Ω和紧区间我⊂R存在两个功能C(吨),ķ(吨)在大号∞(我)这样，对于所有人(吨,X),(吨,是)∈我×ķ,

|X(吨,X)|≤C(吨),|X(吨,X)−X(吨,是)|≤ķ(吨)|X−是|

X吨(q)=∑一世=1米在一世(吨)F一世(q)

|F一世(X)|≤Cķ,|∂F一世∂Xj(X)|≤大号ķ,∀X∈ķ,

|X(吨,X)|≤Cķ∑一世=1米|在一世(吨)|,|X(吨,X)−X(吨,是)|≤大号ķ∑一世=1米|在一世(吨)|⋅|X−是|.

## 数学代写|黎曼几何代写Riemannian geometry代考|Lie Brackets

[X,是]:=∂∂吨|吨=0和−吨X是.评论2.23李括号的几何意义可以通过显式书写来理解

[X,是]|q=∂∂吨|吨=0和−吨X是|q=∂∂吨|吨=0和−吨X(是|和吨X(q) =∂∂s∂吨|吨=s=0和−吨X∘和s是∘和吨X(q)

⌊X,是∣=X是−是X

(和−吨X是)一个=是(一个∘和−吨X)∘和吨X.

(和−吨X是)一个=是(一个−吨X一个+○(吨2))∘和吨X =(是一个−吨是X一个+○¯(吨2))∘和吨X.表示b=是一个−吨是X一个+○(吨2),b吨=b∘和吨X，并再次使用上述展开式，我们得到

(和−吨X是)一个=(是一个−吨是X一个+○(吨2))+吨X(是一个−吨是X一个+○(吨2))+○(吨2) =是一个+吨(X是−是X)一个+○(吨2)

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|黎曼几何代写Riemannian geometry代考|MAST90143

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Negative Curvature: The Hyperbolic Plane

The negative constant curvature model is the hyperbolic plane $H_{r}^{2}$ obtained as the surface of $\mathbb{R}^{3}$, endowed with the hyperbolic metric, defined as the zero level set of the function
$$a(x, y, z)=x^{2}+y^{2}-z^{2}+r^{2} .$$
Indeed, this surface is a two-fold hyperboloid, so we can restrict our attention to the set of points $H_{r}^{2}=a^{-1}(0) \cap{z>0}$.

In analogy with the positive constant curvature model (which is the set of points in $\mathbb{R}^{3}$ whose Euclidean norm is constant) the negative constant curvature model can be seen as the set of points whose hyperbolic norm is constant in $\mathbb{R}^{3}$. In other words,
$$H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{2}\right} \cap{z>0}$$
The hyperbolic Gauss map associated with this surface can be easily computed, since it is explicitly given by
$$\mathcal{N}: H_{r}^{2} \rightarrow H^{2}, \quad \mathcal{N}(q)=\frac{1}{r} \nabla_{q} a$$
Exercise 1.63 Prove that the Gaussian curvature of $H_{r}^{2}$ is $\kappa=-1 / r^{2}$ at every point $q \in H_{r}^{2}$.

We can now discuss the structure of geodesics and curves with constant geodesic curvature on the hyperbolic space. We start with a result that can be proved in an analogous way to Proposition $1.60$. The proof is left to the reader.
Proposition 1.64 Let $\gamma:[0, T] \rightarrow H_{r}^{2}$ be a curve with unit speed and constant geodesic curvature equal to $c \in \mathbb{R}$. For every vector $w \in \mathbb{R}^{3}$, the function $\alpha(t)=\langle\dot{\gamma}(t) \mid w\rangle_{h}$ is a solution of the differential equation
$$\ddot{\alpha}(t)+\left(c^{2}-\frac{1}{r^{2}}\right) \alpha(t)=0 .$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Tangent Vectors and Vector Fields

Let $M$ be a smooth $n$-dimensional manifold and let $\gamma_{1}, \gamma_{2}: I \rightarrow M$ be two smooth curves based at $q=\gamma_{1}(0)=\gamma_{2}(0) \in M$. We say that $\gamma_{1}$ and $\gamma_{2}$ are equivalent if they have the same first-order Taylor polynomial in some (or, equivalently, in every) coordinate chart. This defines an equivalence relation on the space of smooth curves based at $q$.

Definition 2.1 Let $M$ be a smooth $n$-dimensional manifold and let $\gamma: I \rightarrow$ $M$ be a smooth curve such that $\gamma(0)=q \in M$. Its tangent vector at $q=\gamma(0)$, denoted by
$$\left.\frac{d}{d t}\right|_{t=0} \gamma(t) \quad \text { or } \quad \dot{\gamma}(0),$$
is the equivalence class in the space of all smooth curves in $M$ such that $\gamma(0)=$ $q$ (with respect to the equivalence relation defined above).

It is easy to check, using the chain rule, that this definition is well posed (i.e., it does not depend on the representative curve).

Definition $2.2$ Let $M$ be a smooth $n$-dimensional manifold. The tangent space to $M$ at a point $q \in M$ is the set
$$T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \gamma(0)=q\right} .$$ It is a standard fact that $T{q} M$ has a natural structure of an $n$-dimensional vector space, where $n=\operatorname{dim} M$.

Definition 2.3 A smooth vector field on a smooth manifold $M$ is a smooth map
$$X: q \mapsto X(q) \in T_{q} M$$
that associates with every point $q$ in $M$ a tangent vector at $q$. We denote by $\operatorname{Vec}(M)$ the set of smooth vector fields on $M$.

In coordinates we can write $X=\sum_{i=1}^{n} X^{i}(x) \partial / \partial x_{i}$, and the vector field is smooth if its components $X^{i}(x)$ are smooth functions. The value of a vector field $X$ at a point $q$ is denoted, in what follows, by both $X(q)$ and $\left.X\right|_{q}$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Flow of a Vector Field

Given a complete vector field $X \in \operatorname{Vec}(M)$ we can consider the family of maps
$$\phi_{t}: M \rightarrow M, \quad \phi_{t}(q)=\gamma(t ; q), \quad t \in \mathbb{R}{2}$$ where $\gamma(t ; q)$ is the integral curve of $X$ starting at $q$ when $t=0$. By Theorem $2.5$ it follows that the map $$\phi: \mathbb{R} \times M \rightarrow M{,} \quad \phi(t, q)=\phi_{t}(q)$$
is smooth in both variables and the family $\left{\phi_{t}, t \in \mathbb{R}\right}$ is a one-parametric subgroup of Diff $(M)$; namely, it satisfies the following identities:
\begin{aligned} \phi_{0} &=\mathrm{Id}{+} \ \phi{t} \circ \phi_{s} &=\phi_{s} \circ \phi_{t}=\phi_{t+s}, \quad \forall t, s \subset \mathbb{R}, \ \left(\phi_{t}\right)^{-1} &=\phi_{-t}, \quad \forall t \in \mathbb{R} . \end{aligned}

Moreover, by construction, we have
$$\frac{\partial \phi_{t}(q)}{\partial t}=X\left(\phi_{t}(q)\right), \quad \phi_{0}(q)=q, \quad \forall q \in M$$
The family of maps $\phi_{t}$ defined by $(2.5)$ is called the flow generated by $X$. For the flow $\phi_{t}$ of a vector field $X$ it is convenient to use the exponential notation $\phi_{t}:=e^{t X}$, for every $t \in \mathbb{R}$. Using this notation, the group properties (2.6) take the form
$$\begin{gathered} e^{0 X}=\mathrm{Id}, \quad e^{t X} \circ e^{s X}=e^{s X} \circ e^{t X}=e^{(t+s) X}, \quad\left(e^{t X}\right)^{-1}=e^{-t X} \ \frac{d}{d t} e^{t X}(q)=X\left(e^{t X}(q)\right), \quad \forall q \in M \end{gathered}$$
Remark $2.8$ When $X(x)=A x$ is a linear vector field on $\mathbb{R}^{n}$, where $A$ is an $n \times n$ matrix, the corresponding flow $\phi_{t}$ is the matrix exponential $\phi_{t}(x)=e^{t A} x$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Negative Curvature: The Hyperbolic Plane

H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{ 2}\right} \cap{z>0}H_{r}^{2}=\left{q=(x, y, z) \in \mathbb{R}^{3} \mid|q|_{h}^{2}=-r^{ 2}\right} \cap{z>0}

ñ:Hr2→H2,ñ(q)=1r∇q一个

## 数学代写|黎曼几何代写Riemannian geometry代考|Tangent Vectors and Vector Fields

dd吨|吨=0C(吨) 或者 C˙(0),

T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \伽马(0)=q\right} 。T_{q} M:=\left{\left.\frac{d}{d t}\right|{t=0} \gamma(t) \mid \gamma: I \rightarrow M \text { smooth, } \伽马(0)=q\right} 。一个标准的事实是吨q米有一个自然的结构n维向量空间，其中n=暗淡⁡米.

X:q↦X(q)∈吨q米

## 数学代写|黎曼几何代写Riemannian geometry代考|Flow of a Vector Field

φ吨:米→米,φ吨(q)=C(吨;q),吨∈R2在哪里C(吨;q)是积分曲线X开始于q什么时候吨=0. 按定理2.5随之而来的是地图

φ:R×米→米,φ(吨,q)=φ吨(q)

φ0=我d+ φ吨∘φs=φs∘φ吨=φ吨+s,∀吨,s⊂R, (φ吨)−1=φ−吨,∀吨∈R.

∂φ吨(q)∂吨=X(φ吨(q)),φ0(q)=q,∀q∈米

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|黎曼几何代写Riemannian geometry代考|MATH3903

statistics-lab™ 为您的留学生涯保驾护航 在代写黎曼几何Riemannian geometry方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写黎曼几何Riemannian geometry代写方面经验极为丰富，各种代写黎曼几何Riemannian geometry相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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|Model Spaces of Constant Curvature

In this section we briefly discuss surfaces embedded in $\mathbb{R}^{3}$ (with Euclidean or Minkowski inner product) that have constant Gaussian curvature and play the role of model spaces. For each model space we are interested in describing the geodesics and, more generally, the curves of constant geodesic curvature. These results will be useful in the study of sub-Riemannian model spaces in dimension 3 (see Chapter 7 ).

Assume that the surface $M$ has constant Gaussian curvature $\kappa \in \mathbb{R}$. We already know that $\kappa$ is a metric invariant of the surface, i.e., it does not depend on the embedding of the surface in $\mathbb{R}^{3}$. We will distinguish the following three cases:
(i) $\kappa=0$ : this is the flat model, corresponding to the Euclidean plane,
(ii) $\kappa>0$ : this corresponds to the sphere,
(iii) $\kappa<0$ : this corresponds to the hyperbolic plane.
We will briefly discuss case (i), since it is trivial, and study in more detail cases
(ii) and (iii), of spherical and hyperbolic geometry respectively.

## 数学代写|黎曼几何代写Riemannian geometry代考|Zero Curvature: The Euclidean Plane

The Euclidean plane can be realizéd as the surface of $\mathbb{R}^{3}$ defined by the zero level set of the function
$$a: \mathbb{R}^{3} \rightarrow \mathbb{R}, \quad a(x, y, z)=z$$
It is an easy exercise, applying the results of the previous sections, to show that the Gaussian curvature of this surface is zero (the Gauss map is constant) and to characterize geodesics and curves with constant geodesic curvature.

Exercise 1.59 Prove that geodesics on the Euclidean plane are lines. Moreover, show that curves with constant geodesic curvature $c \neq 0$ are circles of radius $1 / c$.

## 数学代写|黎曼几何代写Riemannian geometry代考|Positive Curvature: The Sphere

Let us consider the sphere $S_{r}^{2}$ of radius $r$ as the surface of $\mathbb{R}^{3}$ defined as the zero level set of the function
$$S_{r}^{2}=a^{-1}(0), \quad a(x, y, z)=x^{2}+y^{2}+z^{2}-r^{2} .$$
If we denote, as usual, by $\langle\cdot \mid \cdot\rangle$ the Euclidean inner product in $\mathbb{R}^{3}, S_{r}^{2}$ can be viewed also as the set of points $q=(x, y, z)$ whose Euclidean norm is constant:
$$S_{r}^{2}=\left{q \in \mathbb{R}^{3} \mid\langle q \mid q\rangle=r^{2}\right} .$$
The Gauss map associated with this surface can be easily computed, and it is explicitly given by
$$\mathcal{N}: S_{r}^{2} \rightarrow S^{2}, \quad \mathcal{N}(q)=\frac{1}{r} q$$
It follows immediately from (1.75) that the Gaussian curvature of the sphere is $\kappa=1 / r^{2}$ at every point $q \in S_{r}^{2}$. Let us now recover the structure of geodesics and curves with constant geodesic curvature on the sphere.

Proposition $1.60$ Let $\gamma:[0, T] \rightarrow S_{r}^{2}$ be a curve with unit speed and constant geodesic curvature equal to $c \in \mathbb{R}$. Then, for every $w \in \mathbb{R}^{3}$, the function $\alpha(t)=\langle\dot{\gamma}(t) \mid w\rangle$ is a solution of the differential equation
$$\ddot{\alpha}(t)+\left(c^{2}+\frac{1}{r^{2}}\right) \alpha(t)=0 .$$
Proof Differentiating twice the equality $a(\gamma(t))=0$, where $a$ is the function defined in (1.74), we get (in matrix notation):
$$\dot{\gamma}(t)^{T}\left(\nabla_{\gamma(t)}^{2} a\right) \dot{\gamma}(t)+\ddot{\gamma}(t)^{T} \nabla_{\gamma(t)} a=0 .$$
Moreover, since $|\dot{\gamma}(t)|$ is constant and $\gamma$ has constant geodesic curvature equal to $c$, there exists a function $b(t)$ such that
$$\ddot{\gamma}(t)=b(t) \nabla_{\gamma(t)} a+c \eta(t),$$
where $c$ is the gcodesic curvature of the curve and $\eta(t)=\dot{\gamma}(t)^{\perp}$ is the vector orthogonal to $\dot{\gamma}(t)$ in $T_{\gamma(t)} S_{r}^{2}$ (defined in such a way that $\dot{\gamma}(t)$ and $\eta(t)$ form a positively oriented frame). Reasoning as in the proof of Proposition $1.8$ and noticing that $\nabla_{\gamma(t)} a$ is proportional to the vector $\gamma(t)$, one can compute $b(t)$ and obtain that $\gamma$ satisfies the differential equation
$$\ddot{\gamma}(t)=-\frac{1}{r^{2}} \gamma(t)+c \eta(t) .$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Model Spaces of Constant Curvature

（i）ķ=0：这是平面模型，对应于欧几里得平面，
(ii)ķ>0：这对应于球体，
（iii）ķ<0：这对应于双曲平面。

（ii）和（iii）。

## 数学代写|黎曼几何代写Riemannian geometry代考|Positive Curvature: The Sphere

S_{r}^{2}=\left{q \in \mathbb{R}^{3} \mid\langle q \mid q\rangle=r^{2}\right} 。S_{r}^{2}=\left{q \in \mathbb{R}^{3} \mid\langle q \mid q\rangle=r^{2}\right} 。

ñ:小号r2→小号2,ñ(q)=1rq

C˙(吨)吨(∇C(吨)2一个)C˙(吨)+C¨(吨)吨∇C(吨)一个=0.

C¨(吨)=b(吨)∇C(吨)一个+C这(吨),

C¨(吨)=−1r2C(吨)+C这(吨).

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|黎曼几何代写Riemannian geometry代考|MATH3968

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• Longitudinal Data Analysis 纵向数据分析
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## 数学代写|黎曼几何代写Riemannian geometry代考|Gauss–Bonnet Theorem: Global Version

Now we state the global version of the Gauss-Bonnet theorem. In other words we want to generalize $(1.33)$ to the case when $\Gamma$ is a region of $M$ that is not

necessarily homeomorphic to a disk; see for instance Figure 1.4. As we will find, the result depends on the Euler characteristic $\chi(\Gamma)$ of this region.

In what follows, by a triangulation of $M$ we mean a decomposition of $M$ into curvilinear polygons (see Definition $1.31$ ). Notice that every compact surface admits a triangulation. 3

Definition 1.34 Let $M \subset \mathbb{R}^{3}$ be a compact oriented surface with piecewise smooth boundary $\partial M$. Consider a triangulation of $M$. We define the Euler characteristic of $M$ as
$$\chi(M):=n_{2}-n_{1}+n_{0},$$
where $n_{i}$ is the number of $i$-dimensional faces in the triangulation.
The Euler characteristic can be defined for every region $\Gamma$ of $M$ in the same way. Here, by a region $\Gamma$ on a surface $M$ we mean a closed domain of the manifold with piecewise smooth boundary.

## 数学代写|黎曼几何代写Riemannian geometry代考|Consequences of the Gauss–Bonnet Theorems

Definition $1.39$ Let $M, M^{\prime}$ be two surfaces in $\mathbb{R}^{3}$. A smooth map $\phi: \mathbb{R}^{3} \rightarrow$ $\mathbb{R}^{3}$ is called a local isometry between $M$ and $M^{\prime}$ if $\phi(M)=M^{\prime}$ and for every $q \in M$ it satisfies
$$\langle v \mid w\rangle=\left\langle D_{q} \phi(v) \mid D_{q} \phi(w)\right\rangle, \quad \forall v, w \in T_{q} M$$
If, moreover, the map $\phi$ is a bijection then $\phi$ is called a global isometry. Two surfaces $M$ and $M^{\prime}$ are said to be locally isometric (resp. globally isometric) if there exists a local isometry (resp. global isometry) between $M$ and $M^{\prime}$. Notice that the restriction $\phi$ of an isometry of $\mathbb{R}^{3}$ to a surface $M \subset \mathbb{R}^{3}$ always defines a global isometry between $M$ and $M^{\prime}=\phi(M)$.

Formula (1.52) says that a local isometry between two surfaces $M$ and $M^{\prime}$ preserves the angles between tangent vectors and, a fortiori, the lengths of curves and the distances between points.

By Corollary $1.33$, thanks to the fact that the angles and the volumes are preserved by isometries, one obtains that the Gaussian curvature is invariant under local isometries, in the following sense.

Theorem 1.40 (Gauss’ theorema egregium) Let $\phi$ be a local isometry between $M$ and $M^{\prime}$. Then for every $q \in M$ one has $\kappa(q)=\kappa^{\prime}(\phi(q))$, where $\kappa$ (resp. $\kappa^{\prime}$ ) is the Gaussian curvature of $M$ (resp. $\left.M^{\prime}\right)$.

This result says that the Gaussian curvature $\kappa$ depends only on the metric structure on $M$ and not on the specific fact that the surface is embedded in $\mathbb{R}^{3}$ with the induced inner product.

## 数学代写|黎曼几何代写Riemannian geometry代考|The Gauss Map

We end this section with a geometric characterization of the Gaussian curvature of a manifold $M$, using the Gauss map. The Gauss map is a map from the surface $M$ to the unit sphere $S^{2}$ of $\mathbb{R}^{3}$.

Definition 1.44 Let $M$ be an oriented surface. We define the Gauss map associated with $M$ as
$$\mathcal{N}: M \rightarrow S^{2}, \quad q \mapsto v_{q}$$
where $v_{q} \in S^{2} \subset \mathbb{R}^{3}$ denotes the external unit normal vestor to $M$ at $q$.
Let us consider the differential of the Gauss map at the point $q$,
$$D_{q} \mathcal{N}: T_{q} M \rightarrow T_{\mathcal{N}(q)} S^{2}$$

Notice that a tangent vector to the sphere $S^{2}$ at $\mathcal{N}(q)$ is by construction orthogonal to $\mathcal{N}(q)$. Hence it is possible to identify $T_{\mathcal{N}(q)} S^{2}$ with $T_{q} M$ and to think of the differential of the Gauss map $D_{q} \mathcal{N}$ as an endomorphism of $T_{q} M$

Theorem 1.45 Let $M$ be a surface of $\mathbb{R}^{3}$ with Gauss map $\mathcal{N}$ and Gaussian curvature к. Then
$$\kappa(q)=\operatorname{det}\left(D_{q} \mathcal{N}\right),$$
where $D_{q} \mathcal{N}$ is interpreted as an endomorphism of $T_{q} M$.
We start by proving an important property of the Gauss map.
Lemma $1.46$ For every $q \in M$, the differential $D_{q} \mathcal{N}$ of the Gauss map is a symmetric operator, i.e., it satisfies
$$\left\langle D_{q} \mathcal{N}(\xi) \mid \eta\right\rangle=\left\langle\xi \mid D_{q} \mathcal{N}(\eta)\right\rangle, \quad \forall \xi, \eta \in T_{q} M .$$
Proof The statement is local, hence it is not restrictive to assume that $M$ is parametrized by a function $\phi: \mathbb{R}^{2} \rightarrow M$. In this case $T_{q} M=\operatorname{Im} D_{u} \phi$, where $\phi(u)=q$. Let $v, w \in \mathbb{R}^{2}$ such that $\xi=D_{u} \phi(v)$ and $\eta=D_{u} \phi(w)$. Since $\mathcal{N}(q) \in T_{q} M^{\perp}$ we have
$$\langle\mathcal{N}(q) \mid \eta\rangle=\left\langle\mathcal{N}(q) \mid D_{u} \phi(w)\right\rangle=0$$

χ(米):=n2−n1+n0,

## 数学代写|黎曼几何代写Riemannian geometry代考|Consequences of the Gauss–Bonnet Theorems

⟨在∣在⟩=⟨Dqφ(在)∣Dqφ(在)⟩,∀在,在∈吨q米

## 数学代写|黎曼几何代写Riemannian geometry代考|The Gauss Map

ñ:米→小号2,q↦在q

Dqñ:吨q米→吨ñ(q)小号2

ķ(q)=这⁡(Dqñ),

⟨Dqñ(X)∣这⟩=⟨X∣Dqñ(这)⟩,∀X,这∈吨q米.

⟨ñ(q)∣这⟩=⟨ñ(q)∣D在φ(在)⟩=0

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。