  Nematic elastomers and glasses deform spontaneously when subjected to
temperature changes. This property can be exploited in the design of
heterogeneously patterned thin sheets that deform into a non-trivial shape when
heated or cooled. In this paper, we start from a variational formulation for
the entropic elastic energy of liquid crystal elastomers and we derive an
effective two-dimensional metric constraint, which links the deformation and
the heterogeneous director field. Our main results show that satisfying the
metric constraint is both necessary and sufficient for the deformation to be an
approximate minimizer of the energy. We include several examples which show
that the class of deformations satisfying the metric constraint is quite rich.
