  An almost contact metric structure is parametrized by a section of an
associated homogeneous fibre bundle, and conditions for this to be a harmonic
section, and a harmonic map, are studied. These involve the characteristic
vector field, and the almost complex structure in the contact subbundle.
Several examples are given where the harmonic section equations reduce to those
for the characteristic field to be a harmonic section of the unit tangent
bundle. These include trans-Sasakian structures, and certain nearly
cosymplectic structures. On the other hand, we obtain examples where the
characteristic field is harmonic but the almost contact structure is not. Many
of our examples are obtained by considering hypersurfaces of almost Hermitian
manifolds, with the induced almost contact structure, and comparing the
harmonic section equations for both structures.
