Harmonicity and minimality of distributions on Riemannian manifolds via the intrinsic torsion

Abstract

We consider a $q$-dimensional distribution as a section of the Grassmannian bundle $G_q(M^n)$ of $q$-planes and we derive, in terms of the intrinsic torsion of the corresponding $S(O(q)\times O(n-q))$-structure, the conditions that this map must satisfy in order to be critical for the functionals energy and volume. Using this it is shown that invariant Riemannian foliations of homogeneous Riemannian manifolds which are transversally symmetric determine harmonic maps and minimal immersions. In particular, canonical homogeneous fibrations on rank one normal homogeneous spaces or on compact irreducible 3-symmetric spaces provide many examples of harmonic maps and minimal immersions of compact Riemannian manifolds.