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 ) 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 → C P 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 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 ) / Γ of round spheres and naturally carry a unit Hopf vector field which is critical for the volume functional. We prove that r c = + ∞ , unless Γ is trivial. So, in contrast with the situation for the sphere, the Hopf field is stable on S n ( r ) / Γ , Γ ≠ { Id } , whatever the radius.