  This article investigates the dynamical behaviours of the $n$-vortex problem
with vorticity $\mathbf{\Gamma}$ on a Riemann sphere $\mathbb{S}^2$ equipped
with an arbitrary metric $g$. From perspectives of Riemannian geometry and
symplectic geometry, we study the invariant orbits and prove that with some
constraints on vorticity $\mathbf{\Gamma}$, the $n$-vortex problem possesses
finitely many fixed points and infinitely many periodic orbits for generic $g$.
Moreover, we verify the contact structure on hyper-surfaces of the vortex
dipole, and exclude the existence of perverse symmetric orbits.
