Stability of Unit Hopf Vector Fields on Quotients of Spheres

Abstract. – The volume of a unit vector field V of a Riemannian manifold (M, g) (n odd) is the volume of its image V (Mn) 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 Sn → CP n−12 are well known to be critical for the volume functional on the round n-dimensional sphere Sn(r) for every radius r> 1. Regarding the Hessian, it turns out that its positivity actually depends on the radius. Indeed, in [2], it is proven that for n ≥ 5 there is a critical radius rc = 1√n−4 such that Hopf vector fields are stable if and only if r ≤ rc. In this paper we consider the question of the existence of a critical radius for space forms Mn(c) (n odd) of positive curvature c. These space forms are isometric quotients Sn(r)/Γ of round spheres and naturally carry a unit Hopf vector field which is critical for the volume functional. We prove that rc = +∞, unless Γ is trivial. So, in contrast with the situation for the sphere, the Hopf field is stable on Sn(r)/Γ, Γ 6 = {Id}, whatever the radius