The two notions are related, but they are not the same.

The condition for a *unit* vector field $X$ on a Riemannian manifold $(M,g)$ to be harmonic is *not* the same as the condition that the dual $1$-form $X^\flat$ be harmonic. The point is that, for unit vector fields, one defines the energy as the energy of the map $X:M\to S(M)$ where $S(M)$ is the unit sphere bundle of $(M,g)$ endowed with the Sasaki metric and one says that a unit vector field is harmonic if it is a critical point of this energy. This is not the same as the energy of the $1$-form $X^\flat$ in general (though it can be sometimes, for example, if the metric is flat).

A simple example is to take $(M,g)$ to be $S^3=\mathrm{SU}(2)$ endowed with its natural bi-invariant metric. Then any unit left-invariant (or right-invariant) tangent vector field $X$ is harmonic in the above sense, but the dual $1$-form $\omega = X^\flat$ is not harmonic as a $1$-form because the only harmonic $1$-form on $S^3$ is the zero $1$-form. (Since $H^1(S^3) = 0$, this follows, for instance, from the Hodge Theorem.)

There are several good sources for study of this notion of harmonicity of unit vector fields. There is a whole book, *Harmonic Vector Fields: Variational Principles and Differential Geometry*, by S. Dragomir and Domenico Perrone (Elsevier, 2012), but there are also articles that you may find useful: For example, see the survey article *Volume, energy and generalized energy of unit vector fields on Berger spheres. Stability of Hopf vector fields* by Olga Gil-Medrano and Ana Hurtado (http://www.ugr.es/~ahurtado/PDF/correcciones.pdf) and the references therein.

very strangeto require that $X$ is unit in the definition. $\endgroup$ – Willie Wong May 3 '16 at 19:46unitvector fields, there is another notion of 'harmonic': Regard $S(M)$, the unit sphere bundle of $(M,g)$, as a Riemannian manifold in the natural way and then ask whether a unit vector field $X:M\to S(M)$ is harmonic as a mapping between two Riemannian manifolds. There is actually an additional subtlety, in that one can ask that $X$ be a critical point of the energy functionalwhen one only varies $X$ through sections of $S(M)\to M$(instead of through all maps). $\endgroup$ – Robert Bryant May 6 '16 at 9:11