We investigate the dynamical behaviours of the n-vortex problem with circulation vector \(\varvec{\Gamma }\) on a Riemann sphere \({\mathbb {S}}^2\), equipped with an arbitrary metric g. By mixing perspectives from Riemannian geometry and symplectic geometry, we prove that for any given \(\varvec{\Gamma }\), the Hamiltonian is a Morse function for \({\mathcal {C}}^2\) generic g. If some constraints are put on \(\varvec{\Gamma }\), then for such g the n-vortex problem possesses finitely many fixed points and infinitely many periodic orbits. Moreover, we exclude the existence of perverse symmetric orbits.

我们研究了配备有任意度量g的黎曼球面\（{\ mathbb {S}} ^ 2 \）上带有循环矢量\（\ varvec {\ Gamma} \）的n涡问题的动力学行为。通过混合黎曼几何和辛几何的观点，我们证明对于任何给定\（\ varvec {\ Gamma} \），哈密顿量是\（{\ mathcal {C}} ^ 2 \）泛型g的莫尔斯函数。如果某些约束放在\（\ varvec {\伽玛} \） ，那么对于这样克的Ñ-涡旋问题具有无限多个固定点和无限多个周期轨道。此外，我们排除了不对称对称轨道的存在。