  When a material surface is functionalized so as to acquire some type of
order, functionalization of which soft condensed matter systems have recently
provided many interesting examples, the modeller faces an alternative. Either
the order is described on the curved, physical surface where it belongs, or it
is described on a flat surface that is unrolled as pre-image of the physical
surface under a suitable height function. This paper proposes a general method
that pursues the latter avenue by lifting whatever order tensor is deemed
appropriate from a flat to a curved surface. To produce a specific application,
we specialize this method to nematic shells, for which it also provides a
simple, but convincing interpretation of the outcomes of some
molecular-dynamics experiments on ellipsoidal shells.
