Construction of normally biharmonic submanifolds

TL;DR

This paper investigates biharmonic submanifolds in warped product spaces, deriving conditions for tangential and normal biharmonicity based on the warping function and curvature properties.

Contribution

It provides a characterization of normally biharmonic submanifolds in warped products through differential conditions involving the warping function and Ricci curvature.

Findings

Conditions for tangential and normal biharmonicity are explicitly derived.

Relations between tension, bitension fields, and the warping function are established.

Curvature conditions influence the biharmonicity of submanifolds.

Abstract

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 $\varphi:(I\times M,\widetilde{g})\to (I\times N,\overline{h})$, where $\widetilde{g}=dt^2+f^2g$ and $\overline{h}=dt^2+f^2h$. We relate the tension and bitension fields of $\varphi$ 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 $(I\times M,\widetilde{g})$.