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# 数学代写|黎曼曲面代写Riemann surface代考|MAT565 $S \ell(2)$ connections

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## 数学代写|黎曼曲面代写Riemann surface代考|$S \ell(2)$ connections

In the case of connections with structure group $S \ell(2)$, we will give a topological interpretation of the real part of the exponent $\xi=\Re \lambda$ in the asymptotic expansion (essentially, $\xi=\xi_{\text {path }}$ ). This interpretation shows that the exponent has some interesting geometric significance, and it suggests that it would be interesting to explore the significance in the case of connections of higher rank.
The condition that the structure group is $S \ell(2)$ means that the bundle in question has rank 2. Thus the matrices $A$ and $B$ are $2 \times 2$. The condition that the connection has determinant one translates infinitesimally to the condition that the matrix of one-forms has trace zero, in other words $\operatorname{Tr}(A)=0$. Note that $\operatorname{Tr}(B)=0$ automatically since $b_{i i}=0$. Thus our matrices of one-forms contain three one-forms, $a, b_{12}$, and $b_{21}$ :
\begin{aligned} & A=\left(\begin{array}{cc} a & 0 \ 0 & -a \end{array}\right) \ & B=\left(\begin{array}{cc} 0 & b_{12} \ b_{21} & 0 \end{array}\right) . \end{aligned}

## 数学代写|黎曼曲面代写Riemann surface代考|The geodesics in this metric are not unique

The geodesics in this metric are not unique, but they do exist. If $P$ and $Q$ are two points in $Z$, and if $a$ is generic with respect to $P$ and $Q$, then there is a path $\gamma$ from $P$ to $Q$ which minimizes length in the metric $|d \Re g|$, and which has the property that any local maxima or minima of $\Re g$ along the path are attained at critical points of $\Re g$ on $Z$. Furthermore we may assume that near such points, $\gamma$ is a path of steepest descent or ascent.

Proposition 11.6 Let $\xi=\xi_{\text {path }}(\gamma)$ be the length of the geodesic $\gamma$ from $P$ to $Q$. Suppose $a$ and $B$ are generic. Then in the asymptotic expansion for the monodromy $m(t)$ for positive real $t \rightarrow \infty$, the real parts of the exponents $\lambda_i$ are equal to $\xi$.

## 数学代写|黎曼曲面代写Riemann surface代考 $S \ell(2)$ 连接

$$A=\left(\begin{array}{llll} a & 0 & 0 & -a \end{array}\right) \quad B=\left(\begin{array}{llll} 0 & b_{12} & b_{21} & 0 \end{array}\right) .$$

## 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。