Instability for harmonic foliations on compact homogeneous spaces

In this paper we discuss the instability of harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces. We obtain a sufficient condition for a harmonic foliation to be unstable on compact submanifolds in a Euclidean space and on compact isotropy irreducible homogeneous spaces. We also classify compact symmetric spaces which have no non-trivial stable harmonic foliation.