# 数学代写|拓扑学代写Topology代考|Examples of contact manifolds

#### Doug I. Jones

Lorem ipsum dolor sit amet, cons the all tetur adiscing elit

couryes™为您提供可以保分的包课服务

couryes-lab™ 为您的留学生涯保驾护航 在代写拓扑学Topology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。我们的专家在代写拓扑学Topology代写方面经验极为丰富，各种代写拓扑学Topology相关的作业也就用不着说。

## 数学代写|拓扑学代写Topology代考|Examples of contact manifolds

Recall the definition of the standard contact structure on $\mathbb{R}^{2 n+1}$ given in Example 1.1.5. Here is a further example of a contact form on $\mathbb{R}^{2 n+1}$.

Example 2.1.1 On $\mathbb{R}^{2 n+1}$, with $\left(r_j, \varphi_j\right)$ denoting polar coordinates in the $\left(x_j, y_j\right)$-plane, $j=1, \ldots, n$, the following 1 -form is a contact form:
$$\alpha_2:=d z+\sum_{j=1}^n r_j^2 d \varphi_j=d z+\sum_{j=1}^n\left(x_j d y_j-y_j d x_j\right)$$

In fact, this contact form is not really ‘different’ from the standard contact form $\alpha_1$. The following definition gives a precise notion for the equivalence of contact structures or forms, generalising the concept of a contact transformation (Defn. 1.2.5).

Definition 2.1.2 Two contact manifolds $\left(M_1, \xi_1\right)$ and $\left(M_2, \xi_2\right)$ are said to be contactomorphic if there is a diffeomorphism $f: M_1 \rightarrow M_2$ with $\operatorname{Tf}\left(\xi_1\right)=$ $\xi_2$, where $T f: T M_1 \rightarrow T M_2$ denotes the differential of $f$. If $\xi_i=\operatorname{ker} \alpha_i$, $i=1,2$, this is equivalent to saying that $\alpha_1$ and $f^* \alpha_2$ determine the same hyperplane field, and hence equivalent to the existence of a nowhere zero function $\lambda: M_1 \rightarrow \mathbb{R} \backslash{0}$ such that $f^* \alpha_2=\lambda \alpha_1$. Occasionally one speaks of a strict contactomorphism between the strict contact manifolds $\left(M_1, \alpha_1\right)$ and $\left(M_2, \alpha_2\right)$ if $f^* \alpha_2=\alpha_1$.

Example 2.1.3 The contact manifolds $\left(\mathbb{R}^{2 n+1}, \xi_i=\operatorname{ker} \alpha_i\right), i=1,2$, from Example 1.1.5 and the preceding example are contactomorphic. An explicit contactomorphism $f$ with $f^* \alpha_2=\alpha_1$ is given by
$$f(\mathbf{x}, \mathbf{y}, z)=((\mathbf{x}+\mathbf{y}) / 2,(\mathbf{y}-\mathbf{x}) / 2, z+\mathbf{x y} / 2),$$
where $\mathbf{x}$ and $\mathbf{y}$ stand for $\left(x_1, \ldots, x_n\right)$ and $\left(y_1, \ldots, y_n\right)$, respectively, and $\mathbf{x y}$ stands for $\sum_j x_j y_j$. Similarly, both these contact structures are diffeomorphic to $\operatorname{ker}\left(d z-\sum_j y_j d x_j\right)$.

## 数学代写|拓扑学代写Topology代考|Gray stability and the Moser trick

The Gray stability theorem that we are going to prove in this section says that there are no non-trivial deformations of contact structures on closed manifolds. In fancy language, this means that contact structures on closed manifolds have discrete moduli. Here is a preparatory lemma.

Lemma 2.2.1 Let $\omega_t, t \in[0,1]$, be a smooth family of differential $k$-forms on a manifold $M$ and $\left(\psi_t\right){t \in[0,1]}$ an isotopy of $M$. Define a time-dependent vector field $X_t$ on $M$ by $X_t \circ \psi_t=\dot{\psi}_t$, where the dot denotes derivative with respect to $t$ (so that $\psi_t$ is the flow of $X_t$ ). Then $$\left.\frac{d}{d t}\left(\psi_t^* \omega_t\right)\right|{t=t_0}=\psi_{t_0}^\left(\left.\dot{\omega}t\right|{t=t_0}+\mathcal{L}{X{t_0}} \omega_{t_0}\right) .$$
Proof For a time-independent $k$-form $\omega$ we have
$$\left.\frac{d}{d t}\left(\psi_t^ \omega\right)\right|{t=t_0}=\psi{t_0}^*\left(\mathcal{L}{X{t_0}} \omega\right)$$
This follows directly from the definitions, see Appendix B.

We then compute
\begin{aligned} \frac{d}{d t}\left(\psi_t^* \omega_t\right) & =\lim {h \rightarrow 0} \frac{\psi{t+h}^* \omega_{t+h}-\psi_t^* \omega_t}{h} \ & =\lim {h \rightarrow 0} \frac{\psi{t+h}^* \omega_{t+h}-\psi_{t+h}^* \omega_t+\psi_{t+h}^* \omega_t-\psi_t^* \omega_t}{h} \ & =\lim {h \rightarrow 0} \psi{t+h}^\left(\frac{\omega_{t+h}-\omega_t}{h}\right)+\lim {h \rightarrow 0} \frac{\psi{t+h}^ \omega_t-\psi_t^* \omega_t}{h} \ & =\psi_t^*\left(\dot{\omega}t+\mathcal{L}{X_t} \omega_t\right) . \end{aligned}
This is the claimed identity.

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Examples of contact manifolds

$$\alpha_2:=d z+\sum_{j=1}^n r_j^2 d \varphi_j=d z+\sum_{j=1}^n\left(x_j d y_j-y_j d x_j\right)$$

## 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。

Days
Hours
Minutes
Seconds

# 15% OFF

## On All Tickets

Don’t hesitate and buy tickets today – All tickets are at a special price until 15.08.2021. Hope to see you there :)