Posted on Categories:数学代写, 示性类

# 数学代写|示性类代考Characteristic Classes代考|Math8230 Flat bundles

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|示性类代考Characteristic Classes代考|Flat bundles

2.1.1. Chern-Weil theory. Let $G$ be a Lie group and let $\pi$ : $P \rightarrow M$ be a principal $G$-bundle over a $C^{\infty}$ manifold $M$. Namely there is given a right action
$$P \times G \longrightarrow P$$
of the structure group $G$ on the total space $P$ satisfying the following condition.

Local triviality: For any point $p \in M$, there exist an open neighborhood $U \ni p$ and a diffeomorphism $\varphi: \pi^{-1}(U) \cong U \times G$ such that
$$\pi(u g)=\pi(u), \varphi(u g)=\varphi(u) g \quad\left(u \in \pi^{-1}(U), g \in G\right) .$$
For example, the tangent frame bundle $\pi: P(M) \rightarrow M$ of $M$ becomes a principal bundle with structure group $G L(n, \mathbb{R})$, where $\operatorname{dim} M=n$, and it is a very important principal bundle for the investigation of the structure of $M$. In fact, for the study of manifolds, it is one of the main tools to consider various bundles over $M$, not merely the tangent frame bundle, and then to examine the structure of them.

## 数学代写|示性类代考Characteristic Classes代考|Definition of flat bundles

Definition 2.1. A connection $\omega$ on a principal $G$-bundle is called a flat connection if its curvature $\Omega$ is identically 0 . A principal $G$ bundle equipped with a flat connection is called a flat $G$-bundle.
EXAMPLE 2.2. If we put the trivial connection on a product bundle $M \times G$, it is clearly a flat bundle. This is called a trivial flat bundle. The connection form $\omega_0$ of this bundle is given by $\omega_0=q^* \theta$ where $q: M \times G \rightarrow G$ is the natural projection and $\theta \in A^1(G ; \mathfrak{g})$ denotes the Maurer-Cartan form of $G$.

Example 2.3. Let $\pi: P \rightarrow M$ be a flat $G$-bundle and let $f:$ $N \rightarrow M$ be a $C^{\infty}$ map. Then the pullback bundle $f^* P \rightarrow N$ by $f$ becomes a flat $G$-bundle.

By virtue of the Chern-Weil theory, which we recalled in the previous subsection, any real characteristic class of a flat bundle vanishes. However, such bundle is not necessarily a trivial bundle as a principal bundle and furthermore, even if it were so, the flat connection on it is not necessarily a trivial one. Depending on the base space $M$, it may happen that there are many flat $G$-bundles on it. In such a situation, it often becomes an important problem to consider all flat bundles on $M$ and then classify them. Accordingly we first give a criterion of classification of flat bundles.

## 数学代写|示性类代考Characteristic Classes代考|Flat bundles

2.1.1. 陈-魏尔理论。让 $G$ 是一个李群并且让 $\pi: P \rightarrow M$ 当校长 $G$-㧽绑在 $C^{\infty}$ 歧管 $M$. 即珨出一个正确的动作
$$P \times G \longrightarrow P$$

Local triviality: 对于任何一点 $p \in M$ ，存在一个开邻域 $U \ni p$ 和微分同顺 $\varphi: \pi^{-1}(U) \cong U \times G$ 这样
$$\pi(u g)=\pi(u), \varphi(u g)=\varphi(u) g \quad\left(u \in \pi^{-1}(U), g \in G\right) .$$

## 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。