Harmonic maps of foliated Riemannian manifolds

We study (F, G)-harmonic maps between foliated Riemannian manifolds (M,F, g) and (N, G, h) i.e. smooth critical points φ : M → N of the transverse energy functional with respect to variations through foliated maps. In particular we study (F, G)-harmonic morphisms i.e. smooth foliated maps preserving the basic Laplace equation. We show that CR maps of compact Sasakian manifolds preserving the Reeb flows are weakly stable (F, G)-harmonic maps. We study (F, G0)-harmonic maps into spheres and give foliated analogs to Solomon’s (cf., J Differ Geom 21:151–162, 1985) results.