# Helix surfaces in the special linear group

@article{Montaldo2013HelixSI,
title={Helix surfaces in the special linear group},
author={S. Montaldo and I. Onnis and A. Passos Passamani},
journal={Annali di Matematica Pura ed Applicata (1923 -)},
year={2013},
volume={195},
pages={59-77}
}
• Published 2013
• Mathematics
• Annali di Matematica Pura ed Applicata (1923 -)
We characterize helix surfaces (constant angle surfaces) in the special linear group $$\mathrm {SL}(2,{\mathbb {R}})$$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 $$\mathrm {SL}(2,{\mathbb {R}})$$SL(2,R).
8 Citations
New developments on constant angle property in $${\mathbb {S}}^2\times {\mathbb {R}}$$S2×R
In the present paper, we classify curves and surfaces in $${\mathbb {S}}^2\times \mathbb {R}$$S2×R, which make constant angle with a rotational Killing vector field. We obtain the explicitExpand
A new approach on constant angle surfaces in the special linear group
AbstractWe give a new approach to a global classification of surfaces for which the unit normal makes a constant angle with a fixed direction in the special linear group. In particular, we give anExpand
Constant Angle Surfaces in Lorentzian Berger Spheres
• Mathematics
• 2017
In this work, we study helix spacelike and timelike surfaces in the Lorentzian Berger sphere $${\mathbb S}_{\varepsilon }^3$$Sε3, that is the three-dimensional sphere endowed with a 1-parameterExpand
Constant angle surfaces in the Lorentzian Heisenberg group
• Mathematics
• 2017
In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics.Expand
Rotationally invariant constant Gauss curvature surfaces in Berger spheres
• Mathematics
• 2019
We give a full classification of complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres: they are either Clifford tori, which are flat, or spheres of Gauss curvatureExpand
Generalization of the Cayley transform in 3D homogeneous geometries
The Cayley transform maps the unit disk onto the upper half-plane, conformally and isometrically. In this paper, we generalize the Cayley transform in three-dimensional homogeneous geometries whichExpand
Constant Angle Surfaces in the Lorentzian Warped Product Manifold $$-I \times _{f} \mathbb {E}^2$$
• Mathematics
• 2021
In this work, we study constant angle space-like and time-like surfaces in the 3-dimensional Lorentzian warped product manifold $$-I \times _{f} \mathbb {E}^2$$ with the metric $${\tilde{g}} = -Expand On Killing Magnetic Curves in Sl(2, ℝ) Geometry Killing magnetic curves are trajectories of charged particles on a Riemannian manifold under action of a Killing magnetic field. In this paper we study Killing magnetic curves in SL(2, ℝ) geometry. #### References SHOWING 1-10 OF 13 REFERENCES Constant angle surfaces in the Heisenberg group • Mathematics • 2009 AbstractIn this article we extend the notion of constant angle surfaces in$$ \mathbb{S}^2 $$× ℝ and ℍ2 × ℝ to general Bianchi-Cartan-Vranceanu spaces. We show that these surfaces have constantExpand Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds Abstract We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl ( 2 , R ) . In particular, all constantExpand Helix surfaces in the Berger sphere • Mathematics • 2012 We characterize helix surfaces in the Berger sphere, that is surfaces which form a constant angle with the Hopf vector field. In particular, we show that, locally, a helix surface is determined by aExpand Helix submanifolds of euclidean spaces • Mathematics • 2009 A submanifold of$${\mathbb {R}^n} whose tangent space makes constant angle with a fixed direction d is called a helix. In the first part of the paper we study helix hypersurfaces. We give a localExpand
Constant angle surfaces in ${\Bbb S}^2\times {\Bbb R}$
• Mathematics
• 2007
Abstract.In this article we study surfaces in ${\Bbb S}^2\times {\Bbb R}$ for which the unit normal makes a constant angle with the ${\Bbb R}$-direction. We give a complete classification forExpand
Minimal helix surfaces in Nn×ℝ
An immersed surface M in Nn×ℝ is a helix if its tangent planes make constant angle with ∂t. We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it isExpand
On the Geometry of Constant Angle Surfaces in $Sol_3$
• Mathematics
• 2010
In this paper we classify all surfaces in the 3-dimensional Lie group $Sol_3$ whose normals make constant angle with a left invariant vector field.
Isometric immersions into 3-dimensional homogeneous manifolds
We give a necessary and sufficient condition for a 2-dimensional Riemannian manifold to be locally isometrically immersed into a 3-dimensional homogeneous Riemannian manifold with a 4-dimensionalExpand
Constant-angle surfaces in liquid crystals
• Chemistry
• 2007
We discuss some properties of surfaces in whose unit normal vector has constant angle with an assigned direction field. The constant-angle condition can be rewritten as a Hamilton–Jacobi equationExpand
Higher codimensional Euclidean helix submanifolds
• Mathematics
• 2010
A submanifold of Rn whose tangent space makes constant angle with a fixed direction d is called a helix. Helix submanifolds are related with the eikonal PDE equation. We give a method to find everyExpand