Instability for harmonic foliations on compact homogeneous spaces
Recall that one of the characterizations of a harmonic foliation \(\mathcal F\) of a compact Riemannian manifold is that it is an extremal, or critical point, of the energy functional. In other words, the first variation of energy vanishes for all variations of \(\mathcal F\). A harmonic foliation is said to be unstable if the second variation of energy is negative for some variation of the foliation. In this paper, the authors investigate the instability of harmonic foliations of compact subspaces of Euclidean space and of compact homogeneous spaces. They find sufficient conditions on the geometry of the compact space so that every harmonic foliation is unstable. In the case of submanifolds of Euclidean space, the condition is an inequality between the 2nd fundamental form and the Ricci curvature operator. In the case of homogeneous spaces, it is an inequality between the first eigenvalue of the Laplace operator on functions and the scalar curvature. As a corollary they classify the compact symmetric spaces which admit no non-trivial harmonic foliations.
scientific article; zbMATH DE number 5509729
scientific article