We present a vorticity method for simulating incompressible viscous flows on curved surfaces governed by the Navier-Stokes equations. Unlike previous approaches, our formulation incorporates the often-overlooked Gaussian-curvature-dependent term in the viscous force, which influences both the vorticity equation and the evolution of harmonic components. We show that these curvature-related terms are crucial for reproducing physically correct fluid behavior. We introduce an implicit-explicit (IMEX) scheme for solving the resulting system on triangle meshes and demonstrate its effectiveness on surfaces with arbitrary topology, including non-orientable surfaces, and under a variety of boundary conditions. Our theoretical contributions include several explicit formulas: a vorticity jump condition across curvature sheets, a geometric correspondence between friction coefficients and boundary curvature adjustments, and the influence of boundary curvature on harmonic modes. These results not only simplify the algorithmic design but also offer geometric insight into curvature-driven fluid phenomena, such as the emergence of the Kutta condition under free-slip boundaries.