On 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 S2 of type (1,1), their properties will follow from the general properties of a symmetric divergence-free tensor field of type (1,1). We find the link between biconservativity, the property of the shape operator AH to be a Codazzi tensor field, the holomorphicity of a generalized Hopf function and the quality of the surface to have constant mean curvature. Finally we determine a Simons type formula for biconservative surfaces and then use it to study their geometry.