  We consider a numerical approach for the incompressible surface Navier-Stokes
equation on surfaces with arbitrary genus $g(\mathcal{S})$. The approach is
based on a reformulation of the equation in Cartesian coordinates of the
embedding $\mathbb{R}^3$, penalization of the normal component, a Chorin
projection method and discretization in space by surface finite elements for
each component. The approach thus requires only standard ingredients which most
finite element implementations can offer. We compare computational results with
discrete exterior calculus (DEC) simulations on a torus and demonstrate the
interplay of the flow field with the topology by showing realizations of the
Poincar\'e-Hopf theorem on $n$-tori.
