11.3 Rectifiability and currents in sub-Riemannian geometry
7 min read•august 14, 2024
Sub-Riemannian geometry adds a twist to rectifiability and . We're dealing with spaces where movement is restricted, like a car that can't move sideways. This affects how we measure things and define smooth surfaces.
Rectifiable sets and currents help us study these funky spaces. They let us work with shapes that aren't necessarily smooth, opening doors to tackle tricky problems in robotics, biology, and more. It's like giving us new tools to explore a weird, bendy world.
Rectifiable sets in sub-Riemannian geometry
Definition and properties
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- Rectifiable sets in sub-Riemannian geometry are sets that can be covered by a countable union of Lipschitz images of subsets of Euclidean space, up to a set of Hausdorff measure zero
- The Hausdorff measure in sub-Riemannian geometry is defined using the Carnot-Carathéodory distance, which is the infimum of the lengths of all horizontal curves connecting two points (Heisenberg group, Grushin plane)
- The Hausdorff measure allows for the measurement of the size of sets in sub-Riemannian spaces, taking into account the geometry of the horizontal distribution
- Rectifiable sets have a well-defined notion of tangent space and normal vector at almost every point with respect to the Hausdorff measure
- The tangent space at a point of a is a subgroup of the tangent bundle of the sub-Riemannian manifold, compatible with the horizontal distribution
- The normal vector at a point of a rectifiable set is a horizontal vector orthogonal to the tangent space, which can be used to define the notion of perimeter and mean curvature
Existence and characterization
- The existence of rectifiable sets in sub-Riemannian spaces can be proven using the notion of a Lipschitz graph over a Euclidean space
- A set in a sub-Riemannian manifold is rectifiable if and only if it is the countable union of Lipschitz graphs, up to a set of Hausdorff measure zero (Heisenberg group, Grushin plane)
- Lipschitz graphs provide a way to locally parametrize rectifiable sets using Lipschitz functions from Euclidean space to the sub-Riemannian manifold
- The proof of the existence of rectifiable sets relies on the Pansu-Rademacher theorem, which states that Lipschitz functions in sub-Riemannian geometry are differentiable almost everywhere with respect to the Hausdorff measure
- The differential of a Lipschitz function at a point is a group homomorphism from the tangent space of the domain to the tangent space of the codomain, compatible with the horizontal distribution
- The area formula for Lipschitz maps in sub-Riemannian geometry relates the Hausdorff measure of a rectifiable set to the Jacobian determinant of the Lipschitz parametrization, generalizing the classical area formula in Euclidean geometry
Currents on sub-Riemannian manifolds
Definition and properties
- Currents in sub-Riemannian geometry are continuous linear functionals on the space of smooth, compactly supported differential forms
- The space of currents on a sub-Riemannian manifold is the dual of the space of smooth, compactly supported differential forms, equipped with the appropriate topology
- Currents provide a way to generalize the notion of integration of differential forms over submanifolds to more general objects, such as rectifiable sets and fractal sets
- The of a current in sub-Riemannian geometry is defined using the comass norm, which is the supremum of the current evaluated on forms of unit norm
- The mass of a current measures its size and regularity, and provides a way to define a distance between currents
- The mass of a current is lower semicontinuous with respect to the weak topology on the space of currents, which is important in variational problems and the study of minimal surfaces
- The boundary of a current in sub-Riemannian geometry is defined using the exterior derivative of differential forms, and satisfies the property that the boundary of a boundary is zero
- The on currents is a continuous linear operator with respect to the mass norm, and satisfies the Stokes theorem relating the integral of a form over the boundary of a current to the integral of its exterior derivative over the current itself
- The compactness theorem for currents in sub-Riemannian geometry states that a sequence of currents with uniformly bounded mass has a subsequence that converges weakly to a current, which is important in the study of minimizing currents and the existence of minimal surfaces
Calculus and intersection theory
- The theory of currents on sub-Riemannian manifolds provides a powerful tool for developing a calculus of variations and studying geometric variational problems
- The first variation formula for currents relates the derivative of the mass of a current along a deformation to the integral of the deformation vector field against the generalized mean curvature of the current
- The second variation formula for currents provides a way to study the stability and regularity of minimizing currents, and is related to the notion of Jacobi fields and conjugate points in sub-Riemannian geometry
- The intersection theory of currents in sub-Riemannian geometry allows for the study of the intersection of rectifiable sets and the intersection product of currents
- The intersection of two rectifiable sets can be defined using the wedge product of their associated currents, and satisfies the expected properties such as commutativity and associativity
- The intersection product of two currents can be defined using the Hodge star operator and the exterior product of differential forms, and satisfies the Leibniz rule with respect to the boundary operator
- The intersection theory of currents has applications in geometric measure theory, such as the study of the slice topology and the isoperimetric inequality, and in geometric topology, such as the study of linking numbers and the intersection of cycles
Rectifiability and currents for sub-Riemannian problems
Perimeter and minimal surfaces
- The concepts of rectifiability and currents can be used to define the notion of perimeter and minimal surfaces in sub-Riemannian geometry
- The perimeter of a set in a sub-Riemannian manifold can be defined as the mass of the current associated to its boundary, which generalizes the classical notion of perimeter in Euclidean geometry
- A minimal surface in a sub-Riemannian manifold is a rectifiable set that locally minimizes the perimeter among all sets with the same boundary, and can be characterized using the first variation formula for currents
- The study of perimeter and minimal surfaces in sub-Riemannian geometry is important in understanding the geometry and topology of sub-Riemannian spaces
- The isoperimetric inequality in sub-Riemannian geometry relates the perimeter of a set to its Hausdorff measure, and provides a way to estimate the size of a set from the size of its boundary (Heisenberg group, Grushin plane)
- The Plateau problem in sub-Riemannian geometry asks for the existence and regularity of minimal surfaces with a given boundary curve, and can be studied using the direct method of the calculus of variations and the compactness theorem for currents
Geodesics and optimal control
- The theory of currents can be used to develop a calculus of variations in sub-Riemannian geometry, which is useful in the study of geodesics and optimal control problems
- A geodesic in a sub-Riemannian manifold is a horizontal curve that locally minimizes the sub-Riemannian distance between its endpoints, and can be characterized using the Euler-Lagrange equations for the sub-Riemannian energy functional
- An optimal control problem in sub-Riemannian geometry consists of finding a horizontal curve that minimizes a given cost functional, such as the sub-Riemannian energy or the time of travel, subject to certain constraints on the control inputs
- The study of geodesics and optimal control problems in sub-Riemannian geometry has applications in robotics, control theory, and the study of human movement
- The sub-Riemannian geodesic equations provide a way to model the motion of robots and vehicles with nonholonomic constraints, such as wheeled robots and underwater vehicles
- The sub-Riemannian Pontryagin maximum principle provides a necessary condition for optimality in sub-Riemannian optimal control problems, and can be used to derive efficient numerical algorithms for computing optimal trajectories
- The sub-Riemannian geometry of the visual cortex and the motor cortex has been used to model the perception of contours and the control of hand movements in humans, providing insight into the neural mechanisms underlying vision and motor control
Regularity and singularities
- The theory of rectifiable sets and currents can be used to study the regularity of sub-Riemannian minimizers and the existence of singular minimizers in sub-Riemannian geometry
- A sub-Riemannian minimizer is a rectifiable set that locally minimizes the sub-Riemannian perimeter or the sub-Riemannian energy, and can be studied using the tools of geometric measure theory and the calculus of variations
- The regularity theory for sub-Riemannian minimizers aims to show that minimizers are smooth outside a small set of singularities, and to classify the possible singularities that can occur (Heisenberg group, Grushin plane)
- The study of regularity and singularities of sub-Riemannian minimizers is important in understanding the structure of sub-Riemannian spaces and the behavior of optimal curves and surfaces
- The Bernstein problem in sub-Riemannian geometry asks whether entire minimal graphs in the Heisenberg group are necessarily vertical planes, and has been solved in the affirmative using the tools of geometric measure theory and the theory of currents
- The existence of singular minimizers in sub-Riemannian geometry, such as the Martinet surface in the Martinet sub-Riemannian structure, shows that the regularity theory for sub-Riemannian minimizers is more delicate than in the Euclidean case, and requires a careful analysis of the horizontal distribution and the curvature of the sub-Riemannian metric
- The study of sub-Riemannian geodesics with singularities, such as the abnormal extremals in the Pontryagin maximum principle, is important in understanding the global geometry of sub-Riemannian spaces and the structure of the cut locus and the conjugate locus in sub-Riemannian geometry
Key Terms to Review (18)
A. f. monge: A. F. Monge refers to a mathematical concept relating to the theory of optimal transport, particularly focusing on the geometric properties of mappings that minimize transportation costs between different distributions. This idea connects deeply with the study of rectifiability and currents in sub-Riemannian geometry, where Monge's work has significant implications for understanding how shapes and measures interact within constrained environments.
Boundary Operator: The boundary operator is a mathematical tool used in geometric measure theory to define the boundary of chains and currents. It transforms a chain or current into its boundary, effectively capturing how the geometry of a space is structured and changing as one considers its boundaries. This operator plays a crucial role in various concepts, including the relationship between flat chains and cochains, the slicing and projection of currents, and the formulation of approximation theorems.
Caccioppoli's Theorem: Caccioppoli's Theorem is a fundamental result in Geometric Measure Theory that establishes conditions under which a set can be approximated by smooth surfaces, specifically in the context of minimal surfaces. It provides a crucial link between geometric properties of sets and their measure-theoretic characteristics, playing a significant role in various applications, including image processing and sub-Riemannian geometry.
Compactness arguments: Compactness arguments refer to a set of techniques used in mathematics, particularly in analysis and topology, that leverage the properties of compact sets to establish the existence of solutions or to prove various mathematical statements. These arguments often utilize the fact that every open cover of a compact set has a finite subcover, which is crucial for demonstrating the completeness and boundedness of functions or spaces.
Covering arguments: Covering arguments are techniques used in geometric measure theory to analyze and understand the properties of sets and measures, particularly in relation to rectifiability and currents. These arguments help to establish whether a given set can be approximated by simpler geometric structures, such as smooth manifolds or rectifiable sets, by covering them with simpler shapes that behave nicely under certain conditions.
Currents: Currents are generalized objects in geometric measure theory that extend the notion of integration to include non-smooth and irregular spaces, often represented as multi-dimensional generalizations of measures. They play a crucial role in analyzing rectifiable sets, variational problems, and singularities in minimizers, thereby linking geometric properties to analytical methods.
Federer-Fleming Theorem: The Federer-Fleming Theorem is a fundamental result in geometric measure theory that provides a criterion for the rectifiability of subsets of Euclidean space. It connects the concepts of currents and rectifiable sets, establishing that a set is rectifiable if it can be approximated by Lipschitz mappings and has finite perimeter. This theorem plays a crucial role in understanding geometric properties of sets in various contexts, particularly in sub-Riemannian geometry, where the notion of rectifiability is essential for analyzing geometric structures.
Flat current: A flat current is a mathematical object used in geometric measure theory that generalizes the concept of oriented surfaces. It captures the idea of integrating differential forms over these surfaces in a manner that respects their flatness, meaning that they locally resemble Euclidean spaces. This concept is particularly important when analyzing rectifiability and various geometric structures in spaces that may not have smooth properties.
Geometric Integration Theory: Geometric integration theory studies the integration of geometric objects, particularly in spaces with non-standard structures, like sub-Riemannian manifolds. It connects the concepts of geometry and analysis, focusing on how to define measures and integrals in these geometric settings, allowing for the exploration of geometric properties of measures, currents, and rectifiable sets.
H. Federer: H. Federer is a prominent mathematician known for his contributions to geometric measure theory, particularly in the study of rectifiability and currents. His work laid the foundation for understanding geometric structures and their measures in various contexts, including sub-Riemannian geometry and harmonic analysis, which are crucial for analyzing shapes, sizes, and their properties in higher dimensions.
Integral Current: An integral current is a generalized notion of a geometric object that allows for the representation of singular and flat chains in a way that measures oriented area or volume with integer coefficients. Integral currents can be seen as the mathematical objects that arise when working with various geometric structures, helping to connect concepts like slicing and projection, as well as rectifiability in more complex geometric settings such as sub-Riemannian manifolds.
Lipschitz Image: A Lipschitz image is the result of a mapping where the distance between points in the image does not exceed a constant multiple of the distance between their pre-images. This concept is crucial for understanding how geometric properties are preserved under certain types of transformations, especially in the context of rectifiability and currents in sub-Riemannian geometry, where the behavior of curves and surfaces can be complex and varied.
Mass: In the context of geometric measure theory, mass refers to a generalized notion of size that extends the concept of measure to higher-dimensional spaces, particularly in relation to currents. Mass plays a critical role in understanding properties of normal and rectifiable currents, where it quantifies how much 'weight' a current carries and allows for comparisons between different currents and their behaviors under various operations.
Rectifiable set: A rectifiable set is a subset of a Euclidean space that can be approximated well by a countable union of Lipschitz images of compact subsets, allowing us to assign a finite measure to its 'length' or 'area.' This concept is essential for understanding geometric properties and integrating over complex shapes, and it connects closely to various aspects of geometric measure theory.
Regularity Results: Regularity results refer to theorems and findings in geometric measure theory that guarantee certain smoothness properties of sets or functions, particularly in contexts where they are typically singular or irregular. These results play a crucial role in understanding the structure of rectifiable sets and currents, especially in spaces that may lack traditional smooth structures, such as sub-Riemannian manifolds.
Slicing Theorem: The Slicing Theorem is a fundamental concept in geometric measure theory that establishes how to analyze the structure of sets and measures by intersecting them with lower-dimensional slices. This theorem helps in understanding boundary rectifiability and the properties of currents, making it crucial for studying complex geometric structures and their behaviors when reduced to lower dimensions.
Varifolds: Varifolds are generalizations of smooth surfaces used in geometric measure theory, allowing for a broader framework to study geometric objects with singularities or varying dimensions. They provide a way to analyze and represent sets that may not be rectifiable, making them essential for understanding more complex geometric structures.
Whitney's Theorem: Whitney's Theorem is a fundamental result in differential geometry that characterizes the rectifiability of sets in terms of their tangential behavior. It provides conditions under which a set can be considered rectifiable, linking geometric properties with measure-theoretic aspects in various settings, particularly in sub-Riemannian geometry. This theorem is pivotal for understanding how currents can be represented and manipulated within these geometrical frameworks.