On the biharmonicity of product maps

Leonard Todjihounde

doi:10.1155/ijmms/2006/73094

Abstract

We introduce the warped product of maps defined between Riemannian warped product spaces and we give necessary and sufficient conditions for warped product maps to be (bi)harmonic. We obtain then some characterizations of nontrivial harmonic metrics and nonharmonic biharmonic metrics on warped product spaces.

Highlights

ON THE BIHARMONICITY OF PRODUCT MAPSLEONARD TODJIHOUNDE Received 29 November 2005; Revised 9 May 2006; Accepted 11 May 2006
Let f : (Mm, g) → (Nn, h) be a map between the m-dimensional Riemannian manifold (M,g) and the n-dimensional Riemannian manifold (N,h).The energy of the map f is given byE( f ) = e( f )vg, M (1.1)where vg is the volume form on (M, g )and e( f )(x) := (1/2) df (x)2 T∗M⊗ f −1TN energy density of f at the point x ∈ M
We introduce the warped product of maps defined between Riemannian warped product spaces and we give necessary and sufficient conditions for warped product maps to beharmonic

Summary

ON THE BIHARMONICITY OF PRODUCT MAPS

LEONARD TODJIHOUNDE Received 29 November 2005; Revised 9 May 2006; Accepted 11 May 2006. We obtain some characterizations of nontrivial harmonic metrics and nonharmonic biharmonic metrics on warped product spaces

The energy of the map f is given by

Γki j

It follows