Sub-Riemannian Geometry

How does a blind person 'see' corners? How does a neuron process orientations? How does a car park? All these problems are sub-Riemannian: motion with constrained degrees of freedom.

• **Robotics:** Motion planning for cars, manipulators, and drones, sub-Riemannian optimization with control constraints.
• **Neuroscience:** Hoffman-Petitot models of visual cortex V1: neurons process orientations via sub-Riemannian geometry on the Heisenberg group.
• **Quantum Control:** Quantum gates = sub-Riemannian geodesics on unitary groups SU(2ⁿ).

Предварительные знания

• Geodesics
• Smooth Manifolds
• Connections and Covariant Derivative

Horizontal Distributions

Mars rover Perseverance (NASA, 2021) uses sub-Riemannian geometry for optimal path planning over rough terrain: 200 meters/day. In neuroscience, axon growth follows sub-Riemannian geometry: the NGF concentration gradient is constrained to a 2D growth cone, making directed growth 10x more efficient than random search. A **sub-Riemannian structure** on a manifold M is a pair (D, g), where D ⊂ TM is a smooth distribution (a sub-bundle of tangent vectors) and g is a metric on D. Admissible curves are only those whose velocity lies in D.

A car on the plane is a sub-Riemannian system. The state is (x, y, θ) ∈ ℝ² × S¹. Only forward motion and steering are allowed, this is the horizontal distribution. Parallel parking is finding a horizontal curve between two configurations.

Hörmander's Condition: Connectivity via Brackets

**Hörmander's condition (bracket-generating):** The distribution D generates all of TM via iterated Lie brackets. When this holds, any two points can be connected by a horizontal curve (Chow-Rashevsky theorem), the sub-Riemannian distance is finite.

If only 2 of 3 directions are allowed, how do this leads to the third? Through brackets! [X₁,X₂] = X₁X₂ − X₂X₁ is the net displacement of a tiny 'rectangle' of horizontal steps. The cumulative displacement points in the bracket direction. That is how a car parks sideways.

Pontryagin's Maximum Principle

**Pontryagin's Maximum Principle (PMP)** is a necessary condition for optimality in optimal control problems. Sub-Riemannian geodesics are trajectories of a Hamiltonian system on T*M restricted to horizontal directions.

The sub-Riemannian problem: find the shortest admissible curve between two points. PMP reformulates this as an optimal control problem: choose the control vector u(t) ∈ ℝᵏ (velocity components in D) to minimize length. The solution comes from a Hamiltonian on the cotangent bundle.

Sub-Riemannian Balls and the Carnot-Carathéodory Metric

The **Carnot-Carathéodory metric** d_CC(p,q) is the infimum of lengths of horizontal curves from p to q. Sub-Riemannian balls have an anisotropic shape: distances in vertical directions shrink like ε^(1/2) compared to horizontal ε.

Hörmander's hypoelliptic operators L = Σᵢ Xᵢ², 'sub-Riemannian Laplacians'. The bracket-generating condition guarantees hypoellipticity of L. This is used in the theory of singular integrals, the heat equation, and diffusion processes.

Key Ideas

• **Sub-Riemannian structure** (D,g), metric on horizontal distribution D ⊂ TM
• **Hörmander's condition**, D bracket-generating ⟹ any point is reachable (Chow-Rashevsky theorem)
• **Pontryagin's principle**: geodesics = extremals of a Hamiltonian system on T*M (+ abnormal ones!)
• **Carnot-Carathéodory metric**: anisotropic; Hausdorff dimension > topological dimension

Related Topics

Sub-Riemannian geometry unifies control theory, PDEs, and geometry:

• Geodesics — Sub-Riemannian geodesics generalize Riemannian ones; normal extremals → PMP → Hamiltonian system
• Symplectic Geometry — PMP is formulated on the cotangent bundle T*M; extremals are Hamiltonian trajectories
• Connections and Covariant Derivative — Horizontal distributions on fiber bundles are the foundation of connection theory; sub-Riemannian geometry generalizes this

Вопросы для размышления

• Why do abnormal geodesics have no analogue in Riemannian geometry? What is their topological significance?
• Parallel parking is an example of a sub-Riemannian geodesic. Why does the optimal parking strategy look the way it does?
• How does Hörmander's condition for PDEs relate to controllability in control theory? Is there a physical intuition?

Связанные уроки

• calc-01-sequences