Chern-Weil Theorem

Recall that given a vector bundle on $M$, there exists (many) connections $\nabla^E\mathpunct{:}\Gamma(E)\to\Gamma(T^\ast M\otimes E)$, which can be extended to $\nabla^E\mathpunct{:}\Omega^\cdot(M;E)\to\Omega^{\cdot+1}(M;E)$, and we defined the curvature (operator) $R^E$ of $\nabla^E$ as
\[
R^E=(\nabla^E)^2\in\Omega^{\cdot+2}(M;\End(E)),
\]
it can be viewed as a matrix of 2-forms. What is more, it satisfy the Bianchi identity $[\nabla^E,R^E]=0$.
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Connections and Curvatures on Vector Bundles

Let $E\to M$ be a smooth complex vector bundle over a smooth compact manifold $M$. Denote $\Omega^\cdot(M;E):=\Gamma(\Lambda^\cdot(T^\ast M)\otimes E)$ be the space of smooth sections of the tensor product vector bundle $\Lambda^\cdot(T^\ast M)\otimes E$.

A connection on $E$ is an extension of exterior differential operator $\rd$ to include the coefficient $E$.

定义 1. A connection $\nabla^E$ on $E$ is a $C^\infty(M)$-linear operator from $\Gamma(E)$ to $\Omega^1(M;E)$ such that for any $f\in C^\infty(M)$ and $s\in\Gamma(E)$, satisfy the Leibniz rule
\[
\nabla^E(fs)=(\rd f)\otimes s+f\nabla^Es.
\]
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Review of the Rham Cohomology Theory

Let $M$ be a smooth closed (i.e., compact and without boundary) manifold. Let $TM$ and $T^\ast M$ denote the tangent and cotangent (vector) bundle of $M$, respectively. Denote $\Lambda^\cdot(T^\ast M)$ the complex exterior algebra bundle of $T^\ast M$ and $\Omega^\cdot(M,\C)=\Gamma(\Lambda^\cdot(T^\ast M))$ be the space of smooth sections of $\Lambda^\cdot(T^\ast M)$. Particularly, for any integer $p$ with $0\leq p\leq n=\dim M$, we denote by
\[
\Omega^p(M)=\Gamma(\Lambda^p(T^\ast M))
\]
be the space of smooth $p$-forms over $M$.
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关于Latex 转成MS Word 2007的探讨

我们时不时的会收到编辑要求转换我们自以为很漂亮的Latex文档为Word文档. 关于这一要求的提出, 我不想多说, 只是还是忍不住发句牢骚: 妈的, 你就不能把打麻将, 聊qq 或者玩农场的时间学下TeX排版. 其次, 我想稍微的探讨下已经存在的方法. 我认为比较权威的参考可以见这里(Converters from LaTeX to PC Texprocessors—overview) 这是tug.org上的一篇文章, 我觉得比较中肯. Continue Reading