## $$f$$-biharmonic maps between Riemannian manifolds.(English)Zbl 1267.58012

Summary: We show that if $$\psi$$ is an $$f$$-biharmonic map from a compact Riemannian manifold into a Riemannian manifold with non-positive curvature satisfying a condition, then $$\psi$$ is an $$f$$-harmonic map. We prove that if the $$f$$-tension field $$\tau_f (\psi)$$ of a map $$\psi$$ of Riemannian manifolds is a Jacobi field and $$\phi$$ is a totally geodesic map of Riemannian manifolds, then $$\tau_f (\phi \circ \psi)$$ is a Jacobi field. We finally investigate the stress $$f$$-bienergy tensor, and relate the divergence of the stress $$f$$-bienergy of a map $$\psi$$ of Riemannian manifolds with the Jacobi field of the $$\tau_f (\psi)$$ of the map.

### MSC:

 5.8e+21 Harmonic maps, etc.