# On biconservative surfaces

## TL;DR

This paper provides a comprehensive study of biconservative surfaces in Riemannian manifolds, linking their properties to tensor divergence, shape operator conditions, and curvature formulas, advancing understanding of their geometric structure.

## Contribution

It introduces a unified approach to analyze biconservative surfaces, establishing connections with shape operator properties, holomorphic functions, and deriving a Simons type formula.

## Key findings

- Biconservative surfaces characterized by divergence-free symmetric tensor fields.
- Link between biconservativity, Codazzi shape operator, and constant mean curvature.
- Derived a Simons type formula for biconservative surfaces.

## Abstract

We study in a uniform manner the properties of biconservative surfaces in arbitrary Riemannian manifolds. Biconservative surfaces being characterized by the vanishing of the divergence of a symmetric tensor field $S_2$ of type $(1,1)$, their properties will follow from general properties of a symmetric tensor field of type $(1,1)$ with free divergence. We find the link between the biconservativity, the property of the shape operator $A_H$ to be a Codazzi tensor field, the holomorphicity of a generalized Hopf function and the quality of the surface to have constant mean curvature. Then we determine the Simons type formula for biconservative surfaces and use it to study their geometry.