Biconservative Weingarten Surfaces With Flat Normal Bundle in N4(ε)

In this paper, we investigate the class of biconservative surfaces with non‐constant mean curvature in four‐dimensional space forms . Specifically, we focus on biconservative surfaces with non‐parallel normalized mean curvature vector field (non‐PNMC) that have flat normal bundle and are Weingarten. In our initial result, we obtain the compatibility conditions for this class of biconservative surfaces in terms of a first‐order ordinary differential equation system. Subsequently, by prescribing the flat connection in the normal bundle, we prove an existence result for the considered class of biconservative surfaces. Furthermore, we determine all non‐PNMC biconservative Weingarten surfaces with flat normal bundle that either exhibit a particular form of the shape operator in the direction of the mean curvature vector field or have constant Gaussian curvature . Finally, we prove that such surfaces cannot be biharmonic.