On biconservative surfaces in $3$-dimensional space forms

Abstract

We consider biconservative surfaces M 2 , g in a space form N 3 (c), with mean curvature function f satisfying f > 0 and ∇f = 0 at any point, and determine a certain Riemannian metric gr on M such that M 2 , gr is a Ricci surface in N 3 (c). We also obtain an intrinsic characterization of these biconservative surfaces.2010 Mathematics Subject Classification. 53A10, 53C42.

Biharmonic hypersurfaces with constant scalar curvature in space forms

“…In view of Lemma 3.3 and the Codazzi equation (2.14), we see at once that ω 3 22 = ω 2 33 = 0. In addition, it follows from (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) in [12] that ω 1 23 = ω 1 32 = 0. Therefore, we obtain that R(e 2 , e 3 )e 2 , e 3 = ω 1 22 ω 1 33 .…”

“…The local parametric equations of biconservative hypersurfaces in 4-dimensional space forms were obtained in [15,29]. Furthermore, the global and uniqueness properties of biconservative surfaces (or hypersurfaces) have been investigated in a series of papers [8,9,[18][19][20].…”

Global Properties of Biconservative Surfaces in $\mathbb{R}^3$ and $\mathbb{S}^3$

“…The proof follows by direct computations and by using Remark 4.3 in [9] and Proposition 3.4 in [21]. where α ∈ R is a positive constant and β ∈ R, and define φ = ϕ (αũ + β) + log α.…”

Biharmonic and biconservative hypersurfaces in space forms

“…Now, we present a uniqueness result for biconservative surfaces with nowhere vanishing grad f . Theorem 3.8 ( [28]). Let M 2 , g be an abstract surface and c ∈ R an arbitrarily fixed constant.…”

“…Even if the notion of a biconservative submanifold belongs, obviously, to extrinsic geometry, in the particular case of biconservative surfaces in N 3 (c) one can give an intrinsic characterization of such surfaces. Theorem 3.9 ( [28]). Let M 2 , g be an abstract surface.…”