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

Chiang, Yuan-Jen

J. Geom. Symmetry Phys. 27, 45-58 (2012).

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:

58E20

Harmonic maps, etc.

Keywords:

\(f \)-biharmonic map; Riemannian manifold; Jacobi field; stress \(f \)-bienergy tensor