Transversally f-harmonic maps between foliated Riemannian manifolds generalize transversally harmonic maps [22, 23]. Transversally f-harmonic maps and f-harmonic maps are different. However, if  y is an f-harmonic map under certain circumstance, then it is a transversally f-harmonic map. We also investigate transversally f-biharmonic maps, which generalize transversally biharmonic maps [10]. We show that if y is a transversally f-biharmonic map from a compact foliated manifold into a foliated manifold with non-positive transverse curvature satisfying a condition, then it is a transversally f‑harmonic map.

foliation, transversally f-harmonic map, transversally f-biharmonic map.