Stability of Unit Hopf Vector Fields on Quotients of Spheres
Résumé
The volume of a unit vector field V of a Riemannian manifold (M, g) is the volume of its image V(M) in the unit tangent bundle endowed with the Sasaki metric. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fiber of a Hopf fibration S(n) -> CP(n-1/2) (n odd) are well known to be critical for the volume functional on the round n-dimensional sphere S(n)(r) for every radius r > 1. Regarding the Hessian, it turns out that its positivity actually depends on the radius. Indeed, in Borrelli and Gil-Medrano (2006) [2], it is proven that for n >= 5 there is a critical radius r(c) = 1/root n-4 such that Hopf vector fields are stable if and only if r <= r(c). In this paper we consider the question of the existence of a critical radius for space forms M(n)(c) (n odd) of positive curvature c. These space forms are isometric quotients S(n)(r)/Gamma of round spheres and naturally carry a unit Hopf vector field which is critical for the volume functional. We prove that r(c) = +infinity, unless Gamma is trivial. So, in contrast with the situation for the sphere, the Hopf field is stable on S(n)(r)/Gamma, Gamma not equal {Id}, whatever the radius