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数学代写|黎曼几何代写Riemannian geometry代考|MATH5061 Bimodule Quantum Levi-Civita Connections

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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$$\partial_r g{m n}=g_{i n} \Gamma_{r m}^i+g_{i j} C_q^i\left(\Gamma_{p n}^j\right) \sigma_{r m}^{q p}$$

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