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

This paper employs a Riemannian approximation scheme to derive explicit formulas for horizontal curvatures and symplectic distortion of surfaces in 3D contact sub-Riemannian Lie groups, specifically classifying surfaces of revolution with constant curvatures in the Heisenberg and affine-additive groups.

The Gist

Imagine you are walking through a very strange, magical city. In this city, you are not allowed to walk in any direction you please. You can only walk forward, backward, left, or right along specific "highways" painted on the ground. If you try to walk diagonally or sideways across the street, you simply can't; the laws of physics (or in this case, the geometry) forbid it.

This is the world of Sub-Riemannian Geometry. It's a place where movement is constrained, like a car that can only drive forward or turn, but never slide sideways.

The paper you provided is a mathematical expedition into this constrained world, specifically looking at surfaces (like the skin of a balloon or the side of a hill) inside a 3D version of this city. The authors want to answer a fundamental question: How do we measure the "curvature" (how bent or twisted a surface is) when we are forced to move only along these specific highways?

Here is a breakdown of their journey using simple analogies:

1. The Problem: The "Slippery" Surface

In normal geometry (like on a beach ball), measuring curvature is easy. You can roll a marble in any direction, and it tells you how the surface bends.

But in this "magical city" (the Contact Sub-Riemannian Lie Groups), the surface has a special "slippery" property. There are points where the surface aligns perfectly with the allowed highways. At these points, the usual way of measuring curvature breaks down. It's like trying to measure the slope of a hill using a compass that only works if you are walking North; if you are standing still or facing East, the compass spins wildly and gives you no information.

The authors needed a new way to measure curvature that works even when the surface is "aligned" with the city's traffic rules.

2. The Solution: The "Slow-Motion" Camera

To solve this, the authors used a clever trick called Riemannian Approximation.

Imagine you are trying to measure the speed of a race car, but your camera is too slow to capture it clearly. So, you decide to film the race in slow motion.

• The Trick: They imagined that the "forbidden" sideways movement wasn't impossible, but just extremely expensive or very slow. They introduced a "speed dial" (a parameter called $\epsilon$).
• The Process:
  1. They turned the dial up, making sideways movement possible but very difficult (like driving through deep mud).
  2. They calculated the curvature using standard math (which works fine in the mud).
  3. Then, they slowly turned the dial down, making the mud thinner and thinner, until sideways movement became impossible again.
  4. By watching what happened to the numbers as the mud disappeared, they found the "true" curvature that exists in the constrained world.

This allowed them to derive new formulas for Horizontal Mean Curvature (how much the surface bends on average) and Horizontal Gaussian Curvature (how the surface twists and turns).

3. The Two Main Cities (Examples)

The authors tested their new formulas in two specific "cities" (mathematical groups):

• The Heisenberg Group: Think of this as a city where the "highways" twist around a central vertical pole (like a spiral staircase). If you walk in a circle around the pole, you actually end up at a different height than where you started. This is a classic model for quantum mechanics and robotics.
• The Affine-Additive Group: This is a different kind of city, more like a landscape that stretches and shrinks as you move. It's related to how we model hyperbolic space (like the inside of a saddle).

4. The Discoveries: Shapes with Constant Curvature

Once they had their new rulers, they asked: "What do surfaces look like if they have a constant curvature in this weird world?"

In normal geometry, a sphere has constant curvature everywhere. In these constrained cities, the shapes are much stranger.

• The "Bubbles": They found shapes that look like bubbles, but their profiles are described by complex mathematical curves (elliptic integrals).
• The "Flasks": In the second city, they found shapes that look like flasks or bottles, which are formed by the paths of shortest travel (geodesics).
• The "Rotating" Surfaces: They classified all the surfaces of revolution (shapes made by spinning a curve around an axis) that have constant curvature. They found that these shapes can be described using specific equations, some of which involve elementary math and others involving more complex "elliptic" math.

5. Why Does This Matter?

You might wonder, "Who cares about math in a city where you can't walk sideways?"

• Robotics: Robots often have constraints (like a car that can't slide). Understanding the geometry of their movement helps engineers design better paths.
• Physics: This geometry appears in quantum mechanics and the study of heat diffusion in strange environments.
• Economics & Control Theory: Many real-world problems involve optimizing a path under strict constraints.

The Takeaway

The authors of this paper built a new "ruler" for a world where movement is restricted. They showed that even in a world with strict traffic laws, surfaces still have a shape, a bend, and a twist. By using a "slow-motion" mathematical trick, they were able to map out these shapes, discovering new types of "bubbles" and "flasks" that only exist in this constrained universe.

In short: They taught us how to measure the curve of a surface when you are only allowed to walk in a straight line or turn, but never slide.

Technical Summary

Here is a detailed technical summary of the paper "Horizontal Curvatures of Surfaces in 3D Contact Sub-Riemannian Lie Groups" by Bubani, Pinamonti, Platis, and Tsolis.

1. Problem Statement

The paper addresses a fundamental challenge in sub-Riemannian geometry: defining intrinsic notions of curvature for surfaces embedded in 3-dimensional contact sub-Riemannian Lie groups.

• Context: In classical Riemannian geometry, mean and Gaussian curvatures are well-defined using the full ambient metric and the Levi-Civita connection. However, in sub-Riemannian settings, the metric is defined only on a horizontal distribution (a non-integrable sub-bundle of the tangent bundle).
• The Difficulty: Standard Riemannian definitions fail because the ambient metric is degenerate in the vertical direction. Furthermore, the geometry must account for characteristic points, where the tangent plane of the surface aligns with the horizontal distribution, causing the horizontal normal to degenerate.
• Goal: The authors aim to derive explicit formulas for horizontal mean curvature ($H^h_\Sigma$), horizontal Gaussian curvature ($K^h_\Sigma$), and symplectic distortion ($Q^h_\Sigma$) that are:
  1. Intrinsic to the sub-Riemannian structure.
  2. Invariant under the group of sub-Riemannian isometries.
  3. Consistent with known results in model spaces (like the Heisenberg group).
  4. Applicable to classification problems (e.g., surfaces of revolution with constant curvature).

2. Methodology: Riemannian Approximation Scheme

The core methodological innovation is the use of a Riemannian approximation scheme combined with Cartan's method of moving frames.

• Approximation: The authors introduce a one-parameter family of Riemannian metrics $(g_\epsilon)_{\epsilon > 0}$ on the Lie group $G$.
  • They define a frame $\{X, Y, T_\epsilon\}$ where $T_\epsilon = \epsilon T$ (with $T$ being the Reeb vector field).
  • As $\epsilon \to 0^+$, the Riemannian distance $d_\epsilon$ converges to the Carnot-Carathéodory (sub-Riemannian) distance $d_{CC}$ in the Gromov-Hausdorff sense.
• Moving Frames: For a regular surface $\Sigma$ defined by $u=0$, they construct an adapted orthonormal frame $\{E_1, E_2, n_\Sigma\}$ relative to the metric $g_\epsilon$.
  • $n_\Sigma$ is the Riemannian unit normal.
  • $E_1$ and $E_2$ are tangent vectors adapted to the CR structure.
• Asymptotic Analysis:
  1. They compute the Riemannian connection forms, second fundamental form ($II_\epsilon$), and sectional curvatures ($K_\epsilon$) using Cartan's structural equations.
  2. They perform a careful asymptotic analysis of these quantities as $\epsilon \to 0$.
  3. The horizontal curvatures are defined as the finite limits of these Riemannian quantities, effectively extracting the sub-Riemannian behavior from the singular limit.

3. Key Definitions and Contributions

The paper establishes rigorous definitions for three key geometric quantities away from the characteristic set $C(\Sigma)$ (where the horizontal gradient vanishes):

1. Horizontal Mean Curvature ($H^h_\Sigma$):
   Derived as the limit of the Riemannian mean curvature. It is expressed in terms of the horizontal derivatives of the defining function $u$:
   $$H^h_\Sigma = \lim_{\epsilon \to 0} H^\epsilon_\Sigma = X\left(\frac{Xu}{\|\nabla_H u\|}\right) + Y\left(\frac{Yu}{\|\nabla_H u\|}\right) + \frac{a_3 Yu - b_3 Xu}{\|\nabla_H u\|}$$
   (Where $a_i, b_i$ are structure constants of the Lie algebra).

2. Horizontal Gaussian Curvature ($K^h_\Sigma$):
   Derived via the Gauss equation and the limit of the Riemannian Gaussian curvature. The paper proves a Sub-Riemannian Theorema Egregium, showing that $K^h_\Sigma$ depends only on the horizontal derivatives of $u$ and is invariant under sub-Riemannian isometries.
   $$K^h_\Sigma = E_1(Q^h_\Sigma) - E_1 E_1(\log \|\nabla_H u\|) - \left(Q^h_\Sigma - E_1(\log \|\nabla_H u\|)\right)^2$$

3. Symplectic Distortion ($Q^h_\Sigma$):
   A new quantity introduced to capture how the surface geometry is "twisted" relative to the ambient contact structure. It plays a crucial role in the geometric identities derived from the approximation scheme.
   $$Q^h_\Sigma = X(q) - Y(p) - a_3 p - b_3 q$$
   (Where $p, q$ are normalized horizontal derivatives).

4. Main Results and Applications

The authors apply their general framework to two specific Lie groups, deriving explicit formulas and classification theorems.

A. The Heisenberg Group ($\mathbb{H}$)

• Structure: The standard contact structure with $[X, Y] = -4T$.
• Explicit Formulas: The paper provides closed-form expressions for $H^h_\Sigma$, $K^h_\Sigma$, and $Q^h_\Sigma$ for general surfaces, graphs, and cylindrical surfaces.
• Surfaces of Revolution: The authors classify surfaces of revolution ($t = f(r)$) with constant curvature:
  • Constant $K^h_\Sigma$: Solutions involve elementary functions (for $k=0$) and elliptic integrals (for $k=\pm 1$).
  • Constant $H^h_\Sigma$: Solutions involve hyperboloids (for $h=0$) and integrals involving arcsine functions.
  • Constant $Q^h_\Sigma$: Solutions are expressed via integrals involving arccosine functions.
• Inequalities: A sharp inequality $(H^h_\Sigma)^2 - K^h_\Sigma > 0$ is proven for surfaces of revolution in $\mathbb{H}$.

B. The Affine-Additive Group ($AA$)

• Structure: Defined as $\mathbb{R} \times \mathbb{H}^1_{\mathbb{C}}$ (right half-plane model of the hyperbolic plane).
• Surfaces of Revolution: The authors define "surfaces of revolution" relative to specific one-parameter subgroups (acting as rotational symmetry).
• Classification: Similar to the Heisenberg case, they classify surfaces with constant $K^h_\Sigma$, $H^h_\Sigma$, and $Q^h_\Sigma$.
  • Notably, they identify a surface with zero horizontal mean curvature ($H^h_\Sigma = 0$) that simultaneously has constant horizontal Gaussian curvature $K^h_\Sigma = -4$. This surface is given by $a = \frac{1}{2}\arctan(\rho)$.
• New Geometric Objects: The paper discusses the Carnot-Carathéodory sphere and the "Flask" (a surface foliated by horizontal curves), calculating their curvatures (which are generally non-constant).

5. Significance and Impact

• Unification: The paper provides a unified, systematic framework for defining and computing sub-Riemannian curvatures in 3D contact Lie groups, extending previous results that were often specific to the Heisenberg group.
• Intrinsic Nature: By proving the invariance of these curvatures under the isometry group and establishing the "Theorema Egregium" (dependence only on horizontal data), the authors confirm these are genuine intrinsic geometric invariants.
• Classification: The explicit classification of surfaces of revolution with constant curvature fills a gap in the literature, providing a "blueprint" for solving similar problems in other sub-Riemannian geometries.
• Methodological Rigor: The use of the Riemannian approximation scheme allows for the derivation of formulas that are consistent with classical limits while capturing the unique anisotropic behavior of sub-Riemannian spaces.
• Future Directions: The results support further investigations into isoperimetric problems and the study of constant curvature surfaces in more general sub-Riemannian settings.

In summary, this paper successfully bridges the gap between Riemannian approximation techniques and intrinsic sub-Riemannian geometry, delivering explicit, computable formulas for curvature that enable the classification of special surfaces in important model groups.