Biharmonic maps between doubly warped product manifolds

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.