Biharmonic isometric immersions into and biharmonic Riemannian submersions from M^2 × R

In this paper,  we study  biharmonic isometric immersions of a surface  into  and biharmonic Riemannian submersions from the product space $M^2 times r$. We give a  classification of proper biharmonic isometric immersions of a surface with constant mean curvature into  $M^2 times r$. More precisely, we prove that a  surface with constant mean curvature $H$ in  $M^2 times r$ ;is proper biharmonic if and only if it is a part of $S^1 left( frac{1}{2 sqrt{2} H } right) times r$ in $S^2 left( frac{1}{2 H } right) times r$. We also obtain a complete classification of proper biharmonic Hopf cylinders in  $M^2 times r$. On the other hand, we give a classification of  biharmonic (including harmonic) Riemannian submersions $ pi:M^2 times r to (N^2,h)$ from  the product space, and  construct many family of proper biharmonic  Riemannian submersions $M^2 times r to r^2$.