Distributions on Riemannian manifolds, which are harmonic maps

Abstract

We find new examples of harmaonic maps between compact Riemannian manifolds. A section of a Riemannian fibration is called harmonic if it is harmonic as a map from the base manifold into the total space. When the fibres are totally geodesic, the Euler-Lagrange equation for such sections is formulated. In the case of distributions, which are sections of a Grassmannian bundle, this formula is described in terms of the geometry of base manifolds. Examples of harmonic distributions are constructed when the base manifolds are homogeneous spaces and the integral submanifolds are totally geodesic. In particular, we show all the generalized Hopf-fibrations define harmonic maps into the Grassmannian bundles with the standard metric.