Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function
$$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^2dvol_g.$$Definition 2: A 1-form $\omega$ side to be harmonic if it is in kernel of Laplace operator. i.e. $\Delta\omega=(d\delta+\delta d)\omega=0$.
Question: Is there relation between two above definitions? Please give a simple example.
Update: I find some theorem in this topic:
Theorem 1. If $\omega$ is harmonic and $X$ is the dual vector field, we have that $\mathrm{div}X = 0$.
Theorem 2. If $X$ is a vector field on $(M,g)$ and $\omega(v) = g(X,v)$ is the dual 1-form, then
$$\mathrm{div}X = −\delta\omega.$$
Thanks.
Best Answer
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.