On transversally harmonic maps of foliated Riemannian manifolds (Q2914421)

Tranversally harmonic maps of foliated Riemann manifolds give harmonic maps between the leaf spaces, thus being considered as generalizations of harmonic maps. The authors first review some well-known facts on foliated Riemann manifolds and prove some basic results concerning the transversally harmonic maps. Next, they give a new proof of the first variational formula for the transversal energy. Finally, they study the generalized Weitzenböck formula. As a consequence, they show that if \((M,g, \mathcal F)\) is a compact foliated Riemannian manifold of non negative transversal Ricci curvature and if \((M',g', \mathcal F')\) is a foliated Riemannian manifold of non positive transversal sectional curvature, then a transversally harmonic map between these two manifolds is transversally totally geodesic. Other results ensuring that such a map is transversally constant are also proven. An interesting example concerning these facts on the flat 2-torus is also presented.