Construction of normally biharmonic submanifolds

We examine biharmonic submanifolds within warped product structures. For a submanifold $(M,g)\subset (N,h)$ and a positive smooth function $f:I\to {\mathbb{R}}^{+}$, we study the inclusion $\phi :(I\times M,\tilde{g})\to (I\times N,\bar{h})$, where $\tilde{g}=d{t}^2+f^{2}g$ and $\bar{h}=d{t}^2+f^{2}h$. We relate the tension and bitension fields of $\phi$ to the warping function and the geometry of $M$. We further characterize tangentially and normally biharmonic cases via differential conditions on $f$, and interpret these conditions in terms of the Ricci curvature of $(M,g)$ and $(I\times M,\tilde{g})$.