  In this work we consider the numerical solution of incompressible flows on
two-dimensional manifolds. Whereas the compatibility demands of the velocity
and the pressure spaces are known from the flat case one further has to deal
with the approximation of a velocity field that lies only in the tangential
space of the given geometry. Abandoning $H^1$-conformity allows us to construct
finite elements which are -- due to an application of the Piola transformation
-- exactly tangential. To reintroduce continuity (in a weak sense) we make use
of (hybrid) discontinuous Galerkin techniques. To further improve this
approach, $H(\operatorname{div}_{\Gamma})$-conforming finite elements can be
used to obtain exactly divergence-free velocity solutions. We present several
new finite element discretizations. On a number of numerical examples we
examine and compare their qualitative properties and accuracy.
