Constant angle surfaces in the Heisenberg group (original) (raw)
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Straight ruled surfaces in the Heisenberg group
Journal of Geometry, 2014
We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface inside the Heisenberg group is locally a (straight) ruled surface, i.e. it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally contactomorphic to the complex plane.
Geodesics in Heisenberg groups
The Heisenberg group H^(2n+1) naturally arises as a boundary sphere S"(2n+1) of a unit ball B^(2n+2) in a complex hyperbolic space H^(2n+2)_C with one point deleted. This sphere carries a CR-structure coming from a complex structure in H^(2n+2)_C which in turns defines Carnot-Caratheodory metric on S^(2n+1), and also contact structure and Carnot-Caratheodory metric on H^(2n+1) = S^(2n+1)-(pt). As a Lie-group, H^(2n+1) admits also a left-invariant Riemannian metric. In this note we consider geodesic lines in the Heisenberg group with a left-invariant Riemannian metric. We will obtain equations of them and prove that the ideal boundary of the Heisenberg group H^(2n+1) is a sphere S^(2n-1) with a natural CR-structure and corresponding Carnot-Caratheodory metric, i.e., it is a one-point compactification of the Heisenberg group H^(2n-1) of the next dimension in a row. For n = 1 the ideal boundary of H^3 is a circle with Tit's metric.
Notion of ℍ-orientability for surfaces in the Heisenberg group ℍ^n
2020
This paper aims to define and study a notion of orientability in the Heisenberg sense (ℍ-orientability) for the Heisenberg group ℍ^n. In particular, we define such notion for ℍ-regular 1-codimensional surfaces. Analysing the behaviour of a Möbius Strip in ℍ^1, we find a 1-codimensional ℍ-regular, but not Euclidean-orientable, subsurface. Lastly we show that, for regular enough surfaces, ℍ-orientability implies Euclidean-orientability. As a consequence, we conclude that non-ℍ-orientable ℍ-regular surfaces exist in ℍ^1.
SUB-RIEMANNIAN CURVATURE AND A GAUSS–BONNET THEOREM IN THE HEISENBERG GROUP
We use a Riemannnian approximation scheme to define a notion of sub-Riemannian Gaussian curvature for a Euclidean C 2-smooth surface Σ ⊂ H away from characteristic points, and a notion of sub-Riemannian signed geodesic curvature for Euclidean C 2-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss–Bonnet theorem. An application to Steiner's formula for the Carnot-Carathéodory distance in H is provided.
Steiner Formula and Gaussian Curvature in the Heisenberg Group
Bruno Pini Mathematical Analysis Seminar, 2017
The classical Steiner formula expresses the volume of the ∈-neighborhood Ω ∈ of a bounded and regular domain Ω⊂R n as a polynomial of degree n in ∈. In particular, the coefficients of this polynomial are the integrals of functions of the curvatures of the boundary ∂Ω. The aim of this note is to present the Heisenberg counterpart of this result. The original motivation for studying this kind of extension is to try to identify a suitable candidate for the notion of horizontal Gaussian curvature. The results presented in this note are contained in the paper [4] written in collaboration with Zoltan Balogh, Fausto Ferrari, Bruno Franchi and Kevin Wildrick
The Hessian of the distance from a surface in the Heisenberg group
2006
We compute the horizontal Hessian of the signed Carnot-Charatheodory distance from a surface S in the Heisenberg group H. The expression for the Hessian is in terms of the surface's intrinsic curvatures. As an application, we compute the horizontal Hessian of the Carnot-Charatheodory distance from a point in H.
Annali di Matematica Pura ed Applicata (1923 -)
In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space {\mathbb {R}}^3$$ R 3 to the case of helicoidal surfaces in the Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.