Biconservative Weingarten surfaces with flat normal bundle in $N^4 (\epsilon)$

TL;DR

This paper studies a specific class of biconservative surfaces in 4D space forms, focusing on those with flat normal bundles and Weingarten properties, deriving conditions, existence results, and classifying particular cases.

Contribution

It extends the understanding of biconservative surfaces with non-constant mean curvature by analyzing flat normal bundles and Weingarten conditions, providing compatibility conditions, existence proofs, and classifications.

Findings

Derived compatibility conditions via an ODE system.

Proved existence of surfaces with prescribed flat normal connection.

Classified non-PNMC biconservative Weingarten surfaces with flat normal bundles.

Abstract

In this paper, we extend our investigation of the class of biconservative surfaces with non-constant mean curvature in 4-dimensional space forms $N^4(\epsilon)$. Specifically, we focus on biconservative surfaces with non-parallel normalized mean curvature vector fields (non-PNMC) that have flat normal bundles and are Weingarten. In our initial result we obtain the compatibility conditions for this class of biconservative surfaces in terms of an ODE 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 bundles that either exhibit a particular form of the shape operator in the direction of the mean curvature vector field or have constant Gaussian curvature $K = \epsilon$. Finally, we prove that such surfaces cannot be biharmonic.