📏Geometric Measure Theory Unit 11 – Sub-Riemannian Geometric Measure Theory

Key Concepts and Definitions

• Sub-Riemannian geometry studies smooth manifolds equipped with a distribution (subbundle of the tangent bundle) and a metric on this distribution
• Horizontal curves are absolutely continuous curves on the manifold whose tangent vectors belong to the distribution almost everywhere
• Carnot-Carathéodory distance between two points is defined as the infimum of lengths of horizontal curves connecting these points
• Hausdorff dimension of a sub-Riemannian manifold is always greater than or equal to its topological dimension
  • Strict inequality occurs in non-equiregular cases
• Horizontal gradient of a function at a point is the unique horizontal vector such that its scalar product with any horizontal vector equals the derivative of the function along this vector
• Sub-Laplacian is a second-order differential operator defined via the horizontal gradient, playing a role analogous to the Laplacian in Riemannian geometry
• Popp's measure is a natural smooth measure on sub-Riemannian manifolds, constructed using the volume form induced by a specific inner product on the horizontal distribution

Historical Context and Development

• Sub-Riemannian geometry emerged in the 1970s as a generalization of Riemannian geometry, motivated by problems in control theory and analysis of hypoelliptic operators
• Early contributions were made by Hörmander, Strichartz, Rothschild, and Stein in the context of hypoelliptic PDEs and nilpotent Lie groups
• Gromov's seminal work in the 1980s on Carnot groups and their metric structures laid the foundation for the geometric study of sub-Riemannian manifolds
• In the 1990s, Montgomery's book "A Tour of Subriemannian Geometries, Their Geodesics and Applications" provided a comprehensive introduction to the field
  • Popularized sub-Riemannian geometry among mathematicians and highlighted its connections to various areas of mathematics and physics
• Recent developments include the study of curvature-dimension conditions, optimal transport, and geometric measure theory in sub-Riemannian settings
• Applications have expanded to include robotics, neuroscience (models of visual cortex), and quantum control theory

Sub-Riemannian Structures

• A sub-Riemannian manifold is a triple $(M, D, g)$ where $M$ is a smooth manifold, $D$ is a subbundle of the tangent bundle $TM$, and $g$ is a smooth metric on $D$
  • $D$ is often called the horizontal distribution and its sections are called horizontal vector fields
• Bracket-generating condition: the Lie algebra generated by the horizontal vector fields spans the entire tangent space at each point
  • Ensures the existence of horizontal curves connecting any two points (Chow-Rashevskii theorem)
• Regular sub-Riemannian structures have a constant-rank distribution, while non-regular structures may have varying ranks
• Equiregular sub-Riemannian structures have a constant growth vector of the distribution under the Lie bracket operation
  • Simplest non-trivial example is the Heisenberg group, a step-2 Carnot group
• Contact structures are a special class of sub-Riemannian structures where the distribution is codimension-1 and non-integrable
• Carnot groups are nilpotent Lie groups equipped with a left-invariant sub-Riemannian structure, serving as local models for general sub-Riemannian manifolds

Measure Theory Basics

• Measure theory provides a rigorous foundation for integration and the study of various notions of volume and area in sub-Riemannian spaces
• Hausdorff measure is a fundamental concept that generalizes the notion of volume to arbitrary metric spaces, including sub-Riemannian manifolds
  • Defined using coverings by small sets, taking the infimum of the sum of the diameters of these sets raised to a given power (the dimension)
• Spherical Hausdorff measure is a variant of Hausdorff measure that uses coverings by metric balls instead of arbitrary sets
  • Often more convenient for geometric arguments and coincides with the Hausdorff measure up to a positive constant factor
• Perimeter of a measurable set in a sub-Riemannian space is defined as the Hausdorff measure of the boundary of the set
  • Plays a crucial role in isoperimetric inequalities and the study of minimal surfaces
• Coarea formula relates the integral of a function on a manifold to the integrals of its restriction to level sets, using the perimeter of these level sets
  • Fundamental tool for proving geometric and functional inequalities in sub-Riemannian spaces
• Tangent measures capture the local behavior of a measure at a point, obtained as weak limits of rescaled measures
  • Provide insight into the geometry of the support of the measure and its relation to the sub-Riemannian structure

Geometric Properties in Sub-Riemannian Spaces

• Geodesics are locally length-minimizing horizontal curves, generalizing the concept of straight lines in Euclidean spaces
  • Characterized by the Pontryagin Maximum Principle from optimal control theory
• Abnormal geodesics are geodesics that do not arise as projections of solutions to the Hamiltonian system associated with the sub-Riemannian structure
  • Their existence is a distinctive feature of sub-Riemannian geometry, absent in Riemannian settings
• Cut locus of a point is the set of endpoints of maximal geodesics starting from that point, beyond which geodesics cease to be globally minimizing
• Sphere is the set of points at a fixed Carnot-Carathéodory distance from a given point
  • May have a non-smooth structure, unlike in Riemannian geometry
• Hausdorff dimension of a sub-Riemannian sphere is always less than the Hausdorff dimension of the ambient space
  • Related to the presence of abnormal geodesics and the non-Riemannian nature of the geometry
• Isoperimetric inequality relates the volume of a set to the perimeter of its boundary, with the optimal shape being a metric ball
  • Sub-Riemannian isoperimetric inequalities involve the Hausdorff measures of appropriate dimensions
• Poincaré inequality bounds the norm of a function by the norm of its horizontal gradient, capturing the interplay between the metric and the differential structure

Analysis Techniques and Tools

• Hypoelliptic operators are linear differential operators that exhibit better regularity properties than elliptic operators in Riemannian geometry
  • Sub-Laplacian is a prime example, related to the heat equation and diffusion processes in sub-Riemannian spaces
• Functional inequalities, such as Sobolev and Poincaré inequalities, control the norms of functions and their derivatives in sub-Riemannian spaces
  • Play a crucial role in the study of PDEs, geometric analysis, and probability theory
• Potential theory investigates the properties of harmonic functions (solutions to the sub-Laplacian equation) and their relations to the geometry of the space
  • Capacity of a set measures its size from the perspective of potential theory and is used to study fine properties of functions and sets
• Heat kernel is the fundamental solution to the heat equation associated with the sub-Laplacian, describing the diffusion of heat in sub-Riemannian spaces
  • Its asymptotic behavior reveals geometric properties such as the Hausdorff dimension and the volume growth of balls
• Optimal transport studies the problem of moving mass from one distribution to another while minimizing a given cost function, typically related to the sub-Riemannian distance
  • Wasserstein distances quantify the optimal transport cost and provide a metric structure on the space of probability measures
• Curvature-dimension conditions, such as the generalized Ricci curvature lower bound, extend the notion of Ricci curvature to sub-Riemannian spaces
  • Control geometric and functional inequalities, heat kernel estimates, and the behavior of geodesics

Applications and Real-World Examples

• Control theory: sub-Riemannian geometry naturally arises in the study of controllable systems with nonholonomic constraints
  • Optimal control problems, such as finding the shortest path between two configurations subject to constraints on the velocity, lead to sub-Riemannian geodesics
• Robotics: motion planning for robots with wheels or legs can be modeled using sub-Riemannian geometry
  • Example: parallel parking a car with front-wheel drive is a sub-Riemannian optimal control problem on the Heisenberg group
• Neuroscience: the primary visual cortex (V1) is modeled as a contact structure, with the distribution representing the preferred local orientations of neurons
  • Sub-Riemannian geodesics in V1 correspond to the perceived completion of interrupted or occluded contours (Citti & Sarti model)
• Quantum control: sub-Riemannian geometry appears in the control of quantum systems, such as manipulating the state of a quantum bit (qubit) in quantum computing
  • Minimum time required to steer a quantum system from one state to another is related to the sub-Riemannian distance between the states
• Image processing: sub-Riemannian techniques are used for image inpainting, the process of reconstructing missing or damaged parts of an image
  • Minimizing the sub-Riemannian perimeter of the inpainting region leads to visually pleasing and coherent reconstructions
• Thermodynamics: sub-Riemannian geometry provides a framework for modeling thermodynamic systems with constraints, such as adiabatic processes
  • Optimal control theory in sub-Riemannian settings is used to study the efficiency and limitations of such processes

Challenges and Open Problems

• Regularity of sub-Riemannian distance: conditions for the Carnot-Carathéodory distance to be smooth away from the diagonal and the structure of the cut locus
• Isoperimetric problem: characterizing the shapes that minimize the perimeter for a given volume in sub-Riemannian spaces
  • Known in some cases (Heisenberg group, Grushin plane) but open in general
• Curvature and topology: understanding the relations between curvature invariants (such as generalized Ricci curvature) and the topology of sub-Riemannian manifolds
  • Analogues of classical results like the Bonnet-Myers theorem and the Gauss-Bonnet theorem
• Minimal surfaces: extending the theory of minimal surfaces (surfaces with zero mean curvature) to sub-Riemannian settings
  • Study their regularity, stability, and geometric properties
• Spectral theory of sub-Laplacian: understanding the spectrum and eigenfunctions of the sub-Laplacian on compact sub-Riemannian manifolds
  • Relation to the geometry of the manifold and potential applications to quantum mechanics and spectral geometry
• Optimal transport and synthetic curvature: developing a complete theory of optimal transport in sub-Riemannian spaces and its connections to generalized notions of curvature
  • Extending results from metric measure spaces to sub-Riemannian settings
• Sub-Riemannian limit of Riemannian geometry: studying the behavior of Riemannian manifolds as they converge to a sub-Riemannian structure
  • Understanding the convergence of geodesics, curvature, and geometric invariants in this limit