Biharmonic maps between doubly warped product manifolds

TL;DR

This paper investigates biharmonic maps between doubly warped product manifolds, establishing conditions for their harmonicity and non-harmonicity, and extending previous results in the special case of warped products.

Contribution

It provides new characterizations of biharmonic maps in doubly warped products, including conditions for inclusion and projection maps, and generalizes known results for warped products.

Findings

Inclusion maps of B and F are not proper biharmonic.

Conditions for biharmonicity of projection maps are derived.

Results extend to the case where the warping function equals one.

Abstract

In this paper biharmonic maps between doubly warped product manifolds are studied. We show that the inclusion maps of Riemannian manifolds $B$ and $F$ into the doubly warped product $_{f}B\times_{b}F$ can not be proper biharmonic maps. Also we analyze the conditions for the biharmonicity of projections $_{f}B\times_{b}F\to B$ and $_{f}B\times_{b}F\to F$, respectively. Some characterizations for non-harmonic biharmonic maps are given by using product of harmonic maps and warping metric. Specially, in the case of $f=1$, the results for warped product in \cite{Balmus-mont} are obtained.