
> 
> **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.



Relation between harmonic vector field and harmonic 1-form
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.


