  In this paper we study the problem of the optimal distribution of two
materials on smooth submanifolds $M$ of dimension $d-1$ in $\mathbf R^d$
without boundary by means of the topological derivative. We consider a class of
shape optimisation problems which are constrained by a linear partial
differential equation on the surface. We examine the singular perturbation of
the differential operator and material coefficients and derive the topological
derivative. Finally, we show how the topological derivative in conjunction with
a level set method on the surface can be used to solve the topology
optimisation problem numerically.
