On Naturally Reductive Homogeneous Spaces Harmonically Embedded into Spheres

This note continues earlier studies [9, 10] concerning rigidity properties of harmonic maps into spheres. Given a harmonic map / : M -> S", n ^ 2 [5] of a compact Riemannian manifold M into the Euclidean n-sphere S" the (finite dimensional) vector space K(f) of all divergence free Jacobi fields along / [7] contains the vector space of infinitesimal isometric deformations so{n +1) o / , where so(n +1) is identified with the Lie algebra of Killing vector fields on S" [8, 9]. Recall t h a t t h e h a r m o n i c m a p / i s s a i d t o b e infinitesimally rigid if so(n + \ ) o f = PK(f), where PK(j') <= K(f) is the projectable part, that is