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 \(2m+2\).
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