https://haifei-home.pages.dev/en/tags/topology/Recent content in Topology on Haifei's HomeHugoenhfwang132@gmail.com (hfwang132)hfwang132@gmail.com (hfwang132)This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.Sun, 30 Mar 2025 01:13:18 +0800https://haifei-home.pages.dev/en/post_20250330_%E7%8B%84%E6%8B%89%E5%85%8B%E7%A3%81%E5%8D%95%E6%9E%81%E5%AD%90-%E9%99%88%E7%B1%BB%E4%B8%8E%E9%99%88%E6%95%B0-%E6%9B%B4%E9%AB%98%E6%9B%B4%E5%A6%99%E7%9A%84%E7%94%B5%E5%8A%A8%E5%8A%9B%E5%AD%A6-4/Sun, 30 Mar 2025 01:13:18 +0800hfwang132@gmail.com (hfwang132)https://haifei-home.pages.dev/en/post_20250330_%E7%8B%84%E6%8B%89%E5%85%8B%E7%A3%81%E5%8D%95%E6%9E%81%E5%AD%90-%E9%99%88%E7%B1%BB%E4%B8%8E%E9%99%88%E6%95%B0-%E6%9B%B4%E9%AB%98%E6%9B%B4%E5%A6%99%E7%9A%84%E7%94%B5%E5%8A%A8%E5%8A%9B%E5%AD%A6-4/<p>Previous article in this series:</p> <p><a href="https://zhuanlan.zhihu.com/p/32173040413" target="_blank" rel="noopener noreffer ">Godfly: The electromagnetic field is more than just electric and magnetic fields—The AB effect and Berry connection [Higher and More Wonderful Electrodynamics · 3]</a></p> <hr> <p>In the <a href="https://zhuanlan.zhihu.com/p/32173040413" target="_blank" rel="noopener noreffer ">previous article</a>, we mentioned that if a manifold is topologically trivial, then integrating the Berry curvature over a closed surface \(\Sigma\) should yield zero. This is because, according to Stokes&rsquo; theorem:</p> \[\int_\Sigma F = \int_{\partial \Sigma } A = 0\]<p>However, on a topologically nontrivial manifold, the integral \(\int_\Sigma F\) need not be zero. This is because a globally single-valued connection \(A\) cannot be defined in this case, so Stokes&rsquo; theorem no longer applies. What does this mean?</p>