  We consider vectorial problems in the calculus of variations with an
additional pointwise constraint. Our admissible mappings ${\bf
n}:\mathbb{R}^k\rightarrow \mathbb{R}^d$ satisfy ${\bf n}(x)\in M$, where $M$
is a manifold embedded in Euclidean space. The main results of the paper all
formulate necessary or sufficient conditions for a given mapping ${\bf n}$ to
be a weak or strong local minimizer. Our methods involve using projection
mappings in order to build on existing, unconstrained, local minimizer results.
We apply our results to a liquid crystal variational problem to quantify the
stability of the unwound cholesteric state under frustrated boundary
conditions.
