The index of harmonic foliations on spheres

Abstract:

For foliations on a compact oriented manifold there is a natural energy functional, defined with respect to a Riemannian metric. Harmonic Riemannian foliations are then the critical foliations for this functional under an appropriate class of special variations. The index of the title is the index of the Hessian of the energy functional at a critical, i.e., harmonic foliation. It is a finite number. In this note it is shown that for a harmonic Riemannian foliation $\mathcal {F}$ of codimension $q$ on the $n$-sphere ($n > 2$) this index is greater or equal to $q + 1$. Thus $\mathcal {F}$ is unstable. Moreover the given bound is best possible.