Posted on Categories:Riemannian geometry, 数学代写, 黎曼几何

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## 数学代写|黎曼几何代写Riemannian geometry代考|Bimodule Quantum Levi-Civita Connections

Let $(A, \Omega$, d) be an exterior algebra in the sense of Chap. 1 , specified at least to degree 2 . We have already met the notion of a connection on a general $A$-module in Chap. 3 but now we focus exclusively on connections $\nabla: \Omega^1 \rightarrow \Omega^1 \otimes_A \Omega^1$ on $\Omega^1$. As usual, a left connection obeys a left Leibniz rule
$$\nabla(a \eta)=\mathrm{d} a \otimes \eta+a \nabla \eta, \quad a \in A, \eta \in \Omega^1$$
and has curvature $R_{\nabla}$ and torsion $T_{\nabla}$ given by left $A$-module maps
$$\begin{gathered} R_{\nabla}: \Omega^1 \rightarrow \Omega^2 \otimes_A \Omega^1, \quad R_{\nabla}=(\mathrm{d} \otimes \mathrm{id}-\mathrm{id} \wedge \nabla) \nabla \ T_{\nabla}: \Omega^1 \rightarrow \Omega^2, \quad T_{\nabla}=\wedge \nabla-\mathrm{d} \end{gathered}$$
where $\wedge: \Omega^1 \otimes_A \Omega^1 \rightarrow \Omega^2$ is the exterior product. We have met both formulae before in Chap. 3 and then again in Chap. 5. The concept of a connection itself requires only $\Omega^1$, while curvature and torsion require $\Omega^2$ in the role, classically, of defining curvature and torsion on antisymmetric combinations of vector fields. In Chap. 5 , coming from quantum frame bundles, we were also led to introduce a new tensor built from a metric and a connection, the cotorsion, defined as
$$\operatorname{co} T_{\nabla} \in \Omega^2 \otimes_A \Omega^1, \quad \operatorname{co} T_{\nabla}=(\mathrm{d} \otimes \mathrm{id}-\mathrm{id} \wedge \nabla) g$$

## 数学代写|黎曼几何代写Riemannian geometry代考|More Examples of Bimodule Riemannian Geometries

Here we use bases of 1-forms to write explicit formulae for ideas which we previously discussed in a basis-free fashion. The existence of the basis $\left{e^i\right}$ corresponds to the assumption that $\Omega^1$ is finitely generated projective as a left module as in $\S 3.1$, and the uniqueness of the coefficients of the basis elements in the following formulae corresponds to $\Omega^1$ being left-parallelisable as in Definition 1.2. Without the latter we would have to insert a projection matrix in various places (this generality is discussed in Chap. 3) so to keep things simple here we proceed under the assumption that $\Omega^1$ is left-parallelisable. To fix conventions, we write basis 1-forms $e^i$ with indices $u p$, which has not been our preference in most of the book where we have tended to use lower indices where possible as upper ones clash with powers. This is needed to fit conventions in physics and we combine this with Einstein’s summation convention where repeated up-down pairs of indices are to be summed. Thus the defining formulae for ‘partial derivatives’ from Chap. 1 and left connections in terms of Christoffel symbols from $\S 3.2$ now appear as,
$$e^i a=C^i{ }j(a) e^j, \quad \mathrm{~d} a=\left(\partial_i a\right) e^i, \quad \nabla\left(e^i\right)=-\Gamma^i{ }{j k} e^j \otimes e^k$$
for all $a \in A$ in our coordinate algebra. If $e^i$ and $a$ commute (e.g. if $a$ is an element of the field $\mathbb{k}$, which we refer to loosely as a constant) then $C^i{ }j(a)=a \delta^i{ }_j$. For a bimodule connection we write $\sigma$ as $$\sigma\left(e^i \otimes e^j\right)=\sigma^{i j}{ }{m n} e^m \otimes e^n$$
with coefficients determined from $\Gamma^i{ }{j k}$ and $C^i{ }_j$ and such that $\sigma$ extends as a bimodule map, which will depend on $\Gamma^i{ }{j k}$ as not every left connection is necessarily a bimodule connection. We next suppose that there is a central metric $g=g_{i j} e^i \otimes e^j$ and define the inverse-metric tensor as $g^{i j}=\left(e^i, e^j\right)$. This is inverse in the sense that
$$g_{i j} C^i{ }n\left(g^{j k}\right)=\delta^k{ }_n, \quad C^k{ }_p\left(g{i j}\right) g^{p i}=\delta^k{ }j$$ while centrality of $g$ comes down to $$a g{i j}=g_{q s} C^q\left(C^s{ }_j(a)\right)$$

for all $a \in A$. We give one detailed calculation of converting tensor product notation to index notation and leave the rest to the reader. Namely, the equation for metric compatibility $\nabla g=0$ is
$$\begin{gathered} \mathrm{d} g_{i j} \otimes e^i \otimes e^j=g_{i j} \Gamma_{p k}^i e^p \otimes e^k \otimes e^j+g_{i j} \sigma\left(e^i \otimes \Gamma_{p k}^j e^p\right) \otimes e^k \ \left(\partial_r g_{i j}\right) e^r \otimes e^i \otimes e^j=g_{i j} \Gamma_{p k}^i e^p \otimes e^k \otimes e^j+g_{i j} C_q^i\left(\Gamma_{p k}^j\right) \sigma^{q p}{ }{r m} e^r \otimes e^m \otimes e^k \end{gathered}$$ so on re-indexing and taking coefficients of the basis elements we get the equation $$\partial_r g{m n}=g_{i n} \Gamma_{r m}^i+g_{i j} C_q^i\left(\Gamma_{p n}^j\right) \sigma_{r m}^{q p}$$

## 数学代写|黎曼几何代写riemanannian geometry代考|双模量子Levi-Civita连接

$$\nabla(a \eta)=\mathrm{d} a \otimes \eta+a \nabla \eta, \quad a \in A, \eta \in \Omega^1$$
，并具有由左$A$ -模块映射
$$\begin{gathered} R_{\nabla}: \Omega^1 \rightarrow \Omega^2 \otimes_A \Omega^1, \quad R_{\nabla}=(\mathrm{d} \otimes \mathrm{id}-\mathrm{id} \wedge \nabla) \nabla \ T_{\nabla}: \Omega^1 \rightarrow \Omega^2, \quad T_{\nabla}=\wedge \nabla-\mathrm{d} \end{gathered}$$

$$\operatorname{co} T_{\nabla} \in \Omega^2 \otimes_A \Omega^1, \quad \operatorname{co} T_{\nabla}=(\mathrm{d} \otimes \mathrm{id}-\mathrm{id} \wedge \nabla) g$$

## 数学代写|黎曼几何代写黎曼几何代考|更多双模黎曼几何的例子

$$e^i a=C^i{ }j(a) e^j, \quad \mathrm{~d} a=\left(\partial_i a\right) e^i, \quad \nabla\left(e^i\right)=-\Gamma^i{ }{j k} e^j \otimes e^k$$

，其中的系数由$\Gamma^i{ }{j k}$和$C^i{ }_j$确定，并使$\sigma$扩展为一个双模块映射，这将依赖于$\Gamma^i{ }{j k}$，因为并非每个左连接都一定是一个双模块连接。我们接下来假设有一个中心度规$g=g{i j} e^i \otimes e^j$，并定义逆度规张量$g^{i j}=\left(e^i, e^j\right)$。这与
$$g_{i j} C^i{ }n\left(g^{j k}\right)=\delta^k{ }n, \quad C^k{ }_p\left(g{i j}\right) g^{p i}=\delta^k{ }j$$相反，而$g$的中心性可归结为$$a g{i j}=g{q s} C^q\left(C^s{ }_j(a)\right)$$

for all $a \in A$。我们给出了一个将张量积表示法转换为索引表示法的详细计算，其余的留给读者。也就是说，度规兼容性的方程$\nabla g=0$是
$$\begin{gathered} \mathrm{d} g_{i j} \otimes e^i \otimes e^j=g_{i j} \Gamma_{p k}^i e^p \otimes e^k \otimes e^j+g_{i j} \sigma\left(e^i \otimes \Gamma_{p k}^j e^p\right) \otimes e^k \ \left(\partial_r g_{i j}\right) e^r \otimes e^i \otimes e^j=g_{i j} \Gamma_{p k}^i e^p \otimes e^k \otimes e^j+g_{i j} C_q^i\left(\Gamma_{p k}^j\right) \sigma^{q p}{ }{r m} e^r \otimes e^m \otimes e^k \end{gathered}$$，所以重新索引并取基本元素的系数，我们得到方程$$\partial_r g{m n}=g_{i n} \Gamma_{r m}^i+g_{i j} C_q^i\left(\Gamma_{p n}^j\right) \sigma_{r m}^{q p}$$

## 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。

Posted on Categories:Riemannian geometry, 数学代写, 黎曼几何

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## 数学代写|黎曼几何代写Riemannian Geometry代考|Relative Cohomology and Cofibrations

There are two short topics left to discuss in this chapter, one relatively straightforward and one for discussion. We begin with relative cohomology. Given cochain complexes $(F, \mathrm{~d})$ and $(G, \mathrm{~d})$ with a cochain $\operatorname{map} \phi: F^{n} \rightarrow G^{n}$ for all $n$ (i.e., d $\phi=$ $\phi \mathrm{d})$, we form a new complex $E^{n}=F^{n} \oplus G^{n-1}$ with $\mathrm{d}(f, g)=(\mathrm{d} f, \phi(f)-\mathrm{d} g)$. Then
$$\mathrm{d}^{2}(f, g)=\mathrm{d}(\mathrm{d} f, \phi(f)-\mathrm{d} g)=\left(\mathrm{d}^{2} f, \phi(\mathrm{d} f)-\mathrm{d} \phi(f)+\mathrm{d}^{2} g\right)=(0,0) .$$
We call the cohomology of $(E$, d) the relative cohomology $\mathrm{H}(F, G, \phi)$.
Proposition 4.90 Given a cochain map $\phi: F^{n} \rightarrow G^{n}$ for all $n$ we have a long exact relative cohomology sequence
$$\ldots \mathrm{H}^{n-1}(G) \stackrel{i_{2}^{}}{\longrightarrow} \mathrm{H}^{n}(F, G, \phi) \stackrel{\pi_{1}^{}}{\longrightarrow} \mathrm{H}^{n}(F) \stackrel{\phi^{}}{\longrightarrow} \mathrm{H}^{n}(G) \stackrel{i_{2}^{}}{\longrightarrow} \mathrm{H}^{n+1}(F, G, \phi) \ldots,$$
where $\pi_{1}: E^{n} \rightarrow F^{n}$ is $\pi(f, g)=f$ and $i_{2}: G^{n} \rightarrow E^{n+1}$ is $i_{2}(g)=(-1)^{n}(0, g)$.
Proof Standard algebraic manipulation. Looking at the kernel of $\phi^{*}: \mathrm{H}^{n}(F) \rightarrow$ $\mathrm{H}^{n}(G)$ shows that $\mathrm{H}^{n}(F, G, \phi)$ is defined precisely to make this work.

## 数学代写|黎曼几何代写Riemannian Geometry代考|Quantum Principal Bundles and Framings

Vector bundles in classical geometry typically arise as associated to something deeper, a principal bundle. A connection on this then induces covariant derivatives on all associated bundles in a coherent way. This is the situation in Riemannian geometry where a ‘spin connection’ on the frame bundle induces the Levi-Civita connection on tensor fields but also a covariant derivative on the spinor bundle in the case of a spin manifold, leading to the Dirac operator. Similarly in gauge theory, a principal connection induces covariant derivatives on all associated matter fields.
Briefly, a principal $G$-bundle $P$ over a manifold $X$ is defined exactly like a vector bundle with a surjection $\pi: P \rightarrow X$ but each fibre $P_{x}=\pi^{-1}(x)$ now has the structure of a fixed group $G$. This is achieved by starting with a free right action of $G$ on the manifold $P$ such that $X=P / G$. Free here means any non-identity element of $G$ acts without fixed points, which is equivalent to saying that the map
$$P \times G \rightarrow P \times P, \quad(p, g) \mapsto\left(p, p^{g}\right)$$
is an inclusion, where $p^{g}$ denotes the right action of $g \in G$ on $p \in P$. A connection on $P$ is defined concretely as an equivariant complement in $\Omega^{1}(P)$ to the ‘horizontal forms’ (those pulled back from $\Omega^{1}(X)$ ). This is, however, equivalent to $\omega \in \Omega^{1}(P) \otimes \mathfrak{g}$ with certain properties, where $\mathfrak{g}$ is the Lie algebra of $G$. We will see details in the noncommutative case. Given this data, there is an associated vector bundle $E=P \times_{G} V$ and a connection $\nabla$ on it, for any representation $V$ of $G$. We will give the algebraic and potentially ‘quantum’ version of this notion where the structure group is now a Hopf algebra or ‘quantum group’ as in Chap. 2. We will then use this theory to understand the geometry of quantum homogeneous spaces and framed quantum manifolds more generally.

# 黎曼几何代写

## 数学代写|黎曼几何代写Riemannian Geometry代考|Relative Cohomology and Cofibrations

$$\mathrm{d}^{2}(f, g)=\mathrm{d}(\mathrm{d} f, \phi(f)-\mathrm{d} g)=\left(\mathrm{d}^{2} f, \phi(\mathrm{d} f)-\mathrm{d} \phi(f)+\mathrm{d}^{2} g\right)=(0,0) .$$

$$\ldots \mathrm{H}^{n-1}(G) \stackrel{i_{2}}{\longrightarrow} \mathrm{H}^{n}(F, G, \phi) \stackrel{\pi_{1}}{\longrightarrow} \mathrm{H}^{n}(F) \stackrel{\phi}{\longrightarrow} \mathrm{H}^{n}(G) \stackrel{i_{2}}{\longrightarrow} \mathrm{H}^{n+1}(F, G, \phi) \ldots$$

## 数学代写|黎曼几何代写Riemannian Geometry代考|Quantum Principal Bundles and Framing

$$P \times G \rightarrow P \times P, \quad(p, g) \mapsto\left(p, p^{g}\right)$$

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## 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。