# On Harmonic Unit Vector Fields

When we restrict the Dirichlet energy functional to the set of all unit vector fields on a compact Riemannian manifold $(M,g)$, then the critical points of this functional are satisfied in $\Delta_g X=||\nabla X||^2 X$ where $X$ is a unit vector field on $M$ and a critical point. ($\Delta _g$ is the Laplace-Beltrami operator).

Is it true that such a vector field is killing or conformal or affine-killing vector field? I mean how can i investigate the 1-parameter group of the vector field $X$ by the equation $\Delta_g X=||\nabla X||^2 X$?

Update: Thanks to Peter's introduced reference. I found out a killing vector field $X$ is a critical point if and only if $Ric(X,V)=0$ for all vector fields $V \in X^{\perp}$.

• When you have a new question, please make a new post, possibly referencing the earlier via a link. – user9072 Nov 18 '15 at 22:11
• @quid Thanks for mentioning the point. Sorry I'm a new user here and I'm not quite familiar with the way things work here. I try to have your point in mind for my future posts. – user82800 Nov 19 '15 at 5:01
• See also this related question of mine. – user82800 Nov 19 '15 at 13:48