# 数学代写|拓扑学代写Topology代考|Contact structures and Reeb vector fields

#### 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|Contact structures and Reeb vector fields

Let $M$ be a differential manifold, $T M$ its tangent bundle, and $\xi \subset T M$ a field of hyperplanes on $M$, that is, a smooth $\dagger$ sub-bundle of codimension 1. The term codimension 1 distribution is quite common for such a tangent hyperplane field (and not to be confused with distributions in the analysts’ sense, of course). In order to describe special types of hyperplane fields, it is useful to present them as the kernel of a differential 1-form.

Lemma 1.1.1 Locally, $\xi$ can be written as the kernel of a differential 1form $\alpha$. It is possible to write $\xi=\operatorname{ker} \alpha$ with a 1-form $\alpha$ defined globally on all of $M$ if and only if $\xi$ is coorientable, which by definition means that the quotient line bundle $T M / \xi$ is trivial.

Proof Choose an auxiliary Riemannian metric $g$ on $M$ and define the line bundle $\xi^{\perp}$ as the orthogonal complement of $\xi$ in $T M$ with respect to that metric. Then $T M \cong \xi \oplus \xi^{\perp}$ and $T M / \xi \cong \xi^{\perp}$. Around any given point $p$ of $M$, there is a neighbourhood $U=U_p$ over which the line bundle $\xi^{\perp}$ is trivial. Let $X$ be a non-zero section of $\left.\xi^{\perp}\right|_U$ and define a 1 -form $\alpha_U$ on $U$ by $\alpha_U=g(X,-)$. Then clearly $\left.\xi\right|_U=\operatorname{ker} \alpha_U$.

Saying that $\xi$ is coorientable is the same as saying that $\xi^{\perp}$ is orientable and hence (being a line bundle) trivial. In that case, $X$ and thus also $\alpha$ exist globally. Conversely, if $\xi=\operatorname{ker} \alpha$ with a globally defined 1 -form $\alpha$, one can define a global section of $\xi^{\perp}$ by the conditions $g(X, X) \equiv 1$ and $\alpha(X)>0$, hence $\xi$ is coorientable.

## 数学代写|拓扑学代写Topology代考|The space of contact elements

In 1872, Lie [159] (see also [160], [161]) introduced the notion of contact transformation (Berührungstransformation) as a geometric tool for studying systems of differential equations. This may be regarded as the earliest precursor of modern contact geometry.

Contact transformations constitute a particular case of a local transformation group defined by the integrals of a system of differential equations. These transformations were studied extensively during the later part of the nineteenth century and the beginning of the twentieth century by, amongst others, Engel, Poincaré, Goursat, and Cartan.

In the present section we phrase in modern language some of the contact geometric notions that can be traced back to the work of Lie.

Definition 1.2.1 Let $B$ be a smooth $n$-dimensional manifold. A contact element is a hyperplane in a tangent space to $B$. The space of contact elements of $B$ is the collection of pairs $(b, V)$ consisting of a point $b \in B$ and a contact element $V \subset T_b B$.

Lemma 1.2.2 The space of contact elements of $B$ can be naturally identified with the projectivised cotangent bundle $\mathbb{P} T^* B$, which is a manifold of dimension $2 n-1$.

Proof A hyperplane $V$ in the tangent space $T_b B$ is defined as the kernel of a non-trivial linear map $u_V: T_b B \rightarrow \mathbb{R}$, and $u_V$ is determined by $V$ up to multiplication by a non-zero scalar. So the space of contact elements at $b \in B$ may be thought of as the projectivisation of the dual space $T_b^* B$. It is standard bundle theory that this fibrewise projectivisation yields a smooth bundle, see [38].

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|The space of contact elements

1872年，Lie159引入了接触变换(ber hrungtransform)的概念，作为研究微分方程组的几何工具。这可以看作是现代接触几何的最早的先驱。

## 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 :)