  Polyharmonic maps of order k (briefly, k-harmonic maps) are a natural
generalization of harmonic and biharmonic maps. These maps are defined as the
critical points of suitable higher order functionals which extend the classical
energy functional for maps between Riemannian manifolds. The main aim of this
paper is to investigate the so-called unique continuation principle. More
precisely, assuming that the domain is connected, we shall prove the following
extensions of results known in the harmonic and biharmonic case: (i) if a
k-harmonic map is harmonic on an open subset, then it is harmonic everywhere;
(ii) if two k-harmonic maps agree on a open subset, then they agree everywhere;
(iii) if, for a k-harmonic map to the n-dimensional sphere, an open subset of
the domain is mapped into the equator, then all the domain is mapped into the
equator.
