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Horizontal curvatures of surfaces in 3D contact sub-Riemannian Lie groups

Elia Bubani, Andrea Pinamonti, Ioannis D. Platis, Dimitrios Tsolis

Abstract

In this paper we study horizontal curvatures for surfaces embedded in three-dimensional contact sub-Riemannian Lie groups. Using a Riemannian approximation scheme, we derive explicit formulas for horizontal Gauss curvature, horizontal mean curvature, and symplectic distortion for surfaces embedded in three dimensional Lie groups with a sub-Riemannian structure obtained by a contact form. We focus on two primary examples: the Heisenberg group and the affine-additive group. We classify surfaces of revolution within these groups that exhibit constant horizontal curvatures, often expressing their profiles through elementary or elliptic integrals.

Horizontal curvatures of surfaces in 3D contact sub-Riemannian Lie groups

Abstract

In this paper we study horizontal curvatures for surfaces embedded in three-dimensional contact sub-Riemannian Lie groups. Using a Riemannian approximation scheme, we derive explicit formulas for horizontal Gauss curvature, horizontal mean curvature, and symplectic distortion for surfaces embedded in three dimensional Lie groups with a sub-Riemannian structure obtained by a contact form. We focus on two primary examples: the Heisenberg group and the affine-additive group. We classify surfaces of revolution within these groups that exhibit constant horizontal curvatures, often expressing their profiles through elementary or elliptic integrals.
Paper Structure (26 sections, 25 theorems, 217 equations, 2 figures)

This paper contains 26 sections, 25 theorems, 217 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Three dimensional contact sub-Riemannian Lie groups
  4. Sub-Riemannian isometries
  5. Notions of curvature
  6. The Riemannian approximation method
  7. Notions of curature
  8. Invariance
  9. Horizontal geodesic curvature and horizontal normal curvature
  10. $G$-cylindrical surfaces
  11. Applications: two characteristic examples
  12. Heisenberg group
  13. Surfaces embedded in the Heisenberg group
  14. Graphs
  15. Surfaces of revolution
  16. ...and 11 more sections

Key Result

Proposition 2.1

Let $X,Y,T$ be as above. Then there exist constants $a_i,b_i$, $i\in\{1,2,3\}$ and $c\neq 0$ such that Moreover, $a_i,b_i$, $i\in\{1,2,3\}$ and $c$ satisfy

Figures (2)

  • Figure 4.1: Plot of $\Sigma=\{(a,\rho):a=\frac{1}{2}\arctan(\rho)\}$ satifying $H^h_\Sigma=0$ and $K^h_\Sigma=-4$.
  • Figure 4.2: Plot of $\mathcal{F}(O,1)$.

Theorems & Definitions (62)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Theorem 3.1
  • Definition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Definition 3.5
  • Proposition 3.6
  • Proposition 3.7
  • ...and 52 more