We characterize helix surfaces (constant angle surfaces) in the special linear group SL(2,R). In particular, we give an explicit local description of these surfaces by means of a suitable curve and a 1-parameter family of isometries of SL(2,R)

Abstract. Biconservative hypersurfaces are hypersurfaces with conservative stress-energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3-dimensional Bianchi-Cartan-Vranceanu spaces, obtaining their characterization in the following cases: when they form a constant angle with the Hopf vector field; when they are SO(2)-invariant.

We characterize the biharmonic curves in the special linear group SL(2, R). In particular, we show that all proper biharmonic curves in SL(2, R) are helices and we give their explicit parametrizations as curves in the pseudo-Euclidean space R 4 2 .Therefore, we obtain that J 1 e 1 = e 2 , J 1 e 3 = e 4 .Consequently, if we consider the orthonormal basis {E i } 4 i=1 of R 4 2 given by E 1 = (1, 0, 0, 0) , E 2 = (0, 1, 0, 0) , E 3 = (0, 0, 1, 0) , E 4 = (0, 0, 0, 1) , there must exists A ∈ O 2 (4), with J 1 A = AJ 1 , such that e i = A E i , i ∈ {1, 2, 3, 4}.

In this work, we study helix spacelike and timelike surfaces in the Lorentzian Berger sphere S 3 ε , that is the three-dimensional sphere endowed with a 1-parameter family of Lorentzian metrics, obtained by deforming the round metric on S 3 along the fibers of the Hopf fibration S 3 → S 2 (1/2) by −ε 2 . Our main result provides a characterization of the helix surfaces in S 3 ε using the symmetries of the ambient space and a general helix in S 3 ε , with axis the infinitesimal generator of the Hopf fibers. Also, we construct some explicit examples of helix surfaces in S 3 ε .1991 Mathematics Subject Classification. 53B25, 53C50.

A 3-dimensional Riemannian manifold is called Killing submersion if it admits a Riemannian submersion over a surface such that its fibers are the trajectories of a complete unit Killing vector field. In this paper, we give a characterization of proper biharmonic CMC surfaces in a Killing submersion. In the last part, we also classify the proper biharmonic Hopf cylinders in a Killing submersion.2000 Mathematics Subject Classification. 53C42, 58E20.1 is the curvature operator on (N, h). Equation (4) is a fourth order semi-linear elliptic system of differential equations. We also note that any harmonic map is an absolute minimum of the bienergy and, so, it is trivially biharmonic. Therefore, a general working plan is to study the existence of proper biharmonic maps, i.e. biharmonic maps which are not harmonic. We refer to [4,12] for existence results and general properties of biharmonic maps.In this paper, we study biharmonic surfaces in the total space of a Killing submersion. A Riemannian submersion π : E → M of a 3-dimensional Riemannian manifold E over a surface M will be called a Killing submersion if its fibers are the trajectories of a complete unit Killing vector field ξ. Killing submersions are determined by two functions on M, its Gaussian curvature G and the so called bundle curvature r (for more details, see [9,11] and Section 2) and, for this reason, we denote them by E(G, r). A remarkable family of Killing submersions are the homogeneous 3-spaces. In fact, with the exception of the hyperbolic 3-space H 3 (−1), simply-connected homogeneous Riemannian 3-manifolds with isometry group of dimension 4 or 6 can be represented by a 2-parameter family E(c, b), where c, b ∈ R. These E(c, b)-spaces are 3-manifolds admitting a global unit Killing vector field whose integral curves are the fibers of a certain Riemannian submersion over the simply-connected constant Gaussian curvature surface M(c) and, therefore, they determine Killing submersions π : E(c, b) → M(c) (for more details, see [6]). A local description of these examples can be given by using the so called Bianchi-Cartan-Vranceanu spaces (see Section 2). As it turns out, the E(c, b)-spaces are the only simply-connected homogeneous 3-manifolds admitting the structure of a Killing submersion (see [11]).