# On the critical points of the energy functional on vector fields of a Riemannian manifold

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## Abstract

Given a compact Lie subgroup on the space of \(\ G\)-invariant vector fields are critical points of

*G*of the isometry group of a compact Riemannian manifold*M*with a Riemannian connection \(\nabla ,\) a*G*-symmetrization process of a vector field of*M*is introduced and it is proved that the critical points of the energy functional$$\begin{aligned} F(X):=\frac{\int _{M}\left\| \nabla X\right\| ^{2}\mathrm{d}M}{\int _{M}\left\| X\right\| ^{2}\mathrm{d}M} \end{aligned}$$

*F*on the space of all vector fields of*M*and that this inclusion may be strict in general. One proves that the infimum of*F*on \({\mathbb {S}}^{3}\) is not assumed by a \({\mathbb {S}}^{3}\)-invariant vector field. It is proved that the infimum of*F*on a sphere \({\mathbb {S}}^{n},\)\(n\ge 2,\) of radius 1 /*k*, is \(k^{2},\) and is assumed by a vector field invariant by the isotropy subgroup of the isometry group of \({\mathbb {S}}^{n}\) at any given point of \({\mathbb {S}} ^{n}\). It is proved that if*G*is a compact Lie subgroup of the isometry group of a compact rank 1 symmetric space*M*which leaves pointwise fixed a totally geodesic submanifold of dimension bigger than or equal to 1, then the infimum of*F*is assumed by a*G*-invariant vector field.## Keywords

Energy Rough laplacian Infimum Lie group## References

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