Second variation of volume and energy of vector fields. Stability of Hopf vector fields

Abstract.

In this paper we compute the Hessian of the volume of unit vector fields at a minimal one. We also find the Hessians of a family of functionals thus generalizing the known results concerning second variation of the energy or total bending. We use them to study the stability of Hopf vector fields on $S^{2m+1}(r)$ and to show that they are stable for $m=1$, but that for $m> 1$ there is $r_0(m)\le 1$ such that for $r>r_0$ the index is at least $2m+2$.