Hausdorff dimensions of self-similar and self-affine fractals in the Heisenberg group (original) (raw)

GEOMETRIC MECHANICS ON THE HEISENBERG GROUP

2000

We give detailed discussion of subRiemannian geometry which arised from the sub-LaplacianH on the Heisen- berg group. In particular, we calculate the subRiemannian dis- tances along the geodesics. We also find the complex action func- tion and the volume element on the group. Using this action func- tion and the volume element, we obtain the fundamental solution and the heat

Geometry of left-invariant Randers metric on the Heisenberg groups

Arab Journal of Mathematical Sciences, 2021

Purpose In this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics. Design/methodology/approach In the first part, we retrieve the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature. Findings In the second part, we present our main results. We show that the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained which shows that there exist flags of strictly negative and strictly positive curvatures. Originality/value In this work, we present complete Reimannian geometry of left invarint-metrics on Heisenberg groups. Also, some geometric properties of...

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.

Lifts of Lipschitz maps and horizontal fractals in the Heisenberg group

2004

We consider horizontal iterated function systems in the Heisenberg group H¹, i.e., collections of Lipschitz contractions of H¹ with respect to the Heisenberg metric. The invariant sets for such systems are so-called horizontal fractals. We study questions related to connectivity of horizontal fractals, and regularity of functions whose graph lies within a horizontal fractal. Our construction yields examples of horizontal BV surfaces in H¹ that is in contrast with the nonexistence of horizontal Lipschitz surfaces which was recently proved by Ambrosio and Kirchheim.

Horizontal submanifolds of groups of Heisenberg type

2005

We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution ("lagrangians") and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a consequence.