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 $V \in X^{\perp}$.

  • $\begingroup$ When you have a new question, please make a new post, possibly referencing the earlier via a link. $\endgroup$ – user9072 Nov 18 '15 at 22:11
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    $\begingroup$ @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. $\endgroup$ – user82800 Nov 19 '15 at 5:01
  • $\begingroup$ See also this related question of mine. $\endgroup$ – user82800 Nov 19 '15 at 13:48


  • MR2105148 Reviewed Gil-Medrano, Olga Unit vector fields that are critical points of the volume and of the energy: characterization and examples. Complex, contact and symmetric manifolds, 165–186, Progr. Math., 234, Birkhäuser Boston, Boston, MA, 2005. (Reviewer: Eric Boeckx) 53C43 (53C42)

and other papers by Olga Gil-Medrano.

  • $\begingroup$ Thank you very much for your nice answer and references. $\endgroup$ – user82800 Nov 17 '15 at 9:50

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