  We develop a stabilized cut finite element method for the stationary
convection diffusion problem on a surface embedded in ${\mathbb{R}}^d$. The cut
finite element method is based on using an embedding of the surface into a
three dimensional mesh consisting of tetrahedra and then using the restriction
of the standard piecewise linear continuous elements to a piecewise linear
approximation of the surface. The stabilization consists of a standard
streamline diffusion stabilization term on the discrete surface and a so called
normal gradient stabilization term on the full tetrahedral elements in the
active mesh. We prove optimal order a priori error estimates in the standard
norm associated with the streamline diffusion method and bounds for the
condition number of the resulting stiffness matrix. The condition number is of
optimal order $O(h^{-1})$ for a specific choice of method parameters. Numerical
example supporting our theoretical results are also included.
