Let \(\Sigma\) be a compact manifold without boundary whose first homology is nontrivial. The Hodge decomposition of the incompressible Euler equation in terms of 1-forms yields a coupled PDE-ODE system. The \(L^{2}\)-orthogonal components are a “pure” vorticity flow and a potential flow (harmonic, with the dimension of the homology). In this paper we focus on \(N\) point vortices on a compact Riemann surface without boundary of genus \(g\), with a metric chosen in the conformal class. The phase space has finite dimension \(2N+2g\). We compute a surface of section for the motion of a single vortex (\(N=1\)) on a torus (\(g=1\)) with a nonflat metric that shows typical features of nonintegrable 2 degrees of freedom Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces (\(g\geqslant 2\)) having constant curvature \(-1\), with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincaré disk. Finally, we consider multiply connected planar domains. The image method due to Green and Thomson is viewed in the Schottky double. The Kirchhoff – Routh Hamiltonian given in C. C. Lin’s celebrated theorem is recovered by Marsden – Weinstein reduction from \(2N+2g\) to \(2N\). The relation between the electrostatic Green function and the hydrodynamic Green function is clarified. A number of questions are suggested.

设\(\Sigma\)是一个无边界的紧流形，其第一同调性是非平凡的。不可压缩欧拉方程以 1-形式进行 Hodge 分解产生耦合的 PDE-ODE 系统。 \ (L^{2}\) -正交分量是“纯”涡量流和势流（谐波，具有同调性的维度）。在本文中，我们重点研究没有约束边界的紧致黎曼曲面上的\(N\)点涡旋\(g\)，并在共形类中选择度量。相空间具有有限维度\(2N+2g\)。我们使用非平坦度量计算环面 ( \(g=1\) ) 上单个涡旋 ( \(N=1\) )运动的截面表面，该度量显示不可积 2 自由度哈密顿量的典型特征。相比之下，对于平环面，谐波部分是恒定的。接下来，我们转向具有恒定曲率\(-1\)且具有离散对称性的双曲曲面 ( \(g\geqslant 2\) )。卷合不动点在庞加莱盘中产生涡旋晶体。最后，我们考虑多重连通的平面域。格林和汤姆森提出的图像方法可以在肖特基双像中查看。林CC林著名定理中给出的基尔霍夫-劳斯哈密顿量通过马斯登-韦恩斯坦从\(2N+2g\)还原到\(2N\)恢复。阐明了静电格林函数和流体动力学格林函数之间的关系。建议提出一些问题。