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# Sub-Riemannian geometry

Sub-Riemannian geometry stands as a cornerstone of modern mathematical research, bridging the gap between pure and applied mathematics with its unique focus on the properties of spaces that are not fully Euclidean. This fascinating field explores the behaviour of curves, surfaces, and spaces that lack a conventional metric, opening up a plethora of applications in areas such as quantum physics, control theory, and robotics. To remember its essence, think of Sub-Riemannian geometry as the study of geometric structures that navigate the complexities of non-Euclidean spaces, providing a mathematical toolkit for understanding the intricacies of motion and shape beyond traditional Euclidean confines.

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## What is Sub-Riemannian Geometry?

Sub-Riemannian geometry is a branch of mathematics that generalises the notions of Riemannian geometry to spaces where the metric might only be partially defined. It has applications across various fields, including physics, robotics, and control theory. Understanding this complex topic requires delving into its foundational concepts and recognising how it differs from traditional Riemannian geometry.

### Defining the Basics of Sub-Riemannian Geometry

To grasp the essence of Sub-Riemannian geometry, it's crucial to understand its structure and foundational principles. In simplest terms, this branch of geometry deals with the study of smooth manifolds equipped with a smoothly varying distribution of tangent subspaces, constrained in such a way that length and distance can be defined, but not in all directions. This constraint creates a geometry that is rich and complex, with paths between points being curves that are tangent to these subspaces.

Sub-Riemannian Geometry: A field of differential geometry that studies manifolds endowed with a smoothly varying family of tangent subspaces, constraining paths to those tangent to these subspaces, leading to novel definitions of length and distance.

Consider a city in which you can only walk along certain paths or sidewalks but not directly through buildings or parks. The shortest path between two points in such a city, following the allowable paths, exemplifies the principle behind Sub-Riemannian geometry. Each sidewalk represents a tangent subspace, and the entirety of paths embodies the manifold's constraints.

### Key Differences Between Riemannian and Sub-Riemannian Geometry

While both Riemannian and Sub-Riemannian geometry deal with surfaces and curves, several key differences distinguish them. The primary difference lies in how they define distances and the constraints on the paths between points. These differences not only highlight the uniqueness of Sub-Riemannian geometry but also its applications and the mathematical challenges it presents.

• Definition of Distance: In Riemannian geometry, the distance between points is defined in all directions on a manifold. In contrast, Sub-Riemannian geometry only defines distances along specific directions constrained by the tangent subspaces.
• Constraints on Paths: Riemannian geometry allows for movement in any direction, representing an 'all-access' situation. Sub-Riemannian geometry, however, restricts paths to those that are tangent to the subspaces, limiting the possible directions of movement.
• Applications: The constraints of Sub-Riemannian geometry make it particularly suited for problems in control theory and robotics, where the motion is often restricted to specific paths, unlike the broader applications of Riemannian geometry.

Sub-Riemannian geometry's unique constraints are reminiscent of navigating a maze: While the goal is to reach the end, one must follow specific paths that the maze allows.

## Sub-Riemannian Geometry General Theory and Examples

Sub-Riemannian geometry stands as an intriguing branch of mathematics, intertwining complex analysis and real-world applications. By diving deep into its concepts, one can uncover the beauty and intricacies of its structure. This section aims to illuminate the fundamental aspects of Sub-Riemannian geometry, followed by tangible examples from the real world that showcase its relevance and applicability.

### Fundamental Concepts in Sub-Riemannian Geometry

The core of Sub-Riemannian geometry lies in its unique approach to measuring distances and defining paths on manifolds. Unlike traditional geometries, where metrics and paths are unrestricted, Sub-Riemannian geometry imposes constraints that generate a rich and complex mathematical structure.

In essence, Sub-Riemannian geometry is concerned with spaces where only certain directions of movement are allowed at any point. This specificity renders the standard concept of a straight line inadequate, replacing it with the notion of 'admissible paths'. These paths must be tangent to a selected distribution of directions at each point.

Admissible Path: In Sub-Riemannian geometry, an admissible path is a curve whose tangent at each point lies within a preselected subset of directions or a distribution. This concept is crucial for defining distance within constrained spaces.

An example of admissible paths can be found in the control of robotic arms within a confined space. These arms can only move in certain directions due to physical constraints. Here, the arm's movement trajectory, restricted to specific angles and rotations, resembles an admissible path within Sub-Riemannian geometry.

At the heart of understanding Sub-Riemannian geometry is the Carnot-Caratheodory metric, defined primarily on admissible paths. The metric enables measurement of distances by considering the infimum of the lengths of admissible paths connecting two points. Mathematically, if we consider two points, A and B, on a manifold, their distance, d(A, B), is given by:

egin{equation}d(A, B) = \inf \left\{ length(\gamma) : \gamma \text{ is an admissible path from } A \text{ to } B \right\}.

This definition emphasises the geometry's intrinsic connection to optimal control theory, where finding the shortest or least costly path between two states is often the goal.

### Real-world Examples Illustrating Sub-Riemannian Geometry

Sub-Riemannian geometry finds its applications in several fields, underscoring its versatility and practicality. From robotics to visual perception, the principles of Sub-Riemannian geometry help in modelling and solving complex real-world problems.

One notable application is in robotics, where the movement of robots in environments with obstacles requires planning paths that are not only possible but optimal. These constraints closely model admissible paths in Sub-Riemannian spaces, where the robot’s movements must align with predetermined permissible directions.

Another fascinating application is in the field of vision and image processing. The human visual system can be modelled using Sub-Riemannian geometry to understand how we perceive curves and edges in our environment. This modelling is crucial for developing algorithms that enable computers to process and interpret visual information similarly to humans.

Sub-Riemannian geometry offers a framework not only for understanding complex mathematical structures but also for tackling real-world problems in innovative and efficient ways.

## Geodesics in Sub-Riemannian Geometry

Geodesics in Sub-Riemannian Geometry highlight the most direct paths within a constrained space, illuminating the optimal routes that can occur under specific rules of movement. This concept is paramount for understanding the inherent structures and potential strategies within these geometrical frameworks.

### Understanding Geodesics in Sub-Riemannian Geometry

Geodesics in Sub-Riemannian geometry are akin to the 'straight lines' in Euclidean space but are defined within the confines of the geometry's unique constraints. These paths represent the shortest or most efficient trajectory between two points, under the limitations imposed by the geometry's structure.

The computation of geodesics involves complex mathematical formulations, relying on understanding the geometry's underlying distribution and the application of calculus of variations to find paths that minimise the distance travelled, fitting within the allowed movements.

Geodesic: A curve that is locally a distance-minimising path between points, defined within a geometric space. In Sub-Riemannian geometry, geodesics adhere to the constraints of movement allowed by the geometry’s structure.

Imagine navigating a park where pathways are laid out in a specific pattern, and one can only walk along these paths and not cut across the grass. If trying to get from a picnic area to a pond as quickly as possible, the shortest path following the pathways represents a geodesic in the context of the park's geometry.

The mathematical representation of a geodesic in Sub-Riemannian geometry involves solving the Hamilton-Jacobi equation, a fundamental equation in classical mechanics and calculus of variations. This equation helps in describing the evolution over time of a dynamical system, and in the context of Sub-Riemannian geometry, it helps in identifying the shortest paths that comply with the geometry’s constraints. The Hamilton-Jacobi equation is given by:

egin{equation}H(q, \frac{\partial S}{\partial q}, t) = 0 d{equation}

where H is the Hamiltonian, q represents the coordinates in the configuration space, and S is the action integral as a function of coordinates and time. Solving this equation for Sub-Riemannian geometries requires sophisticated techniques and is at the heart of understanding how geodesics behave in these spaces.

### The Importance of Geodesics in Mathematical Analysis

The study of geodesics in Sub-Riemannian geometry is not only fascinating for its mathematical beauty but also for its practical implications. Geodesics provide insights into the optimal paths for movement within constrained environments, akin to finding the most efficient routes in a road network that limits the directions of travel.

Moreover, the analysis of geodesics aids in understanding the intrinsic curvature and topology of the spaces considered. This, in turn, impacts several fields, including physics, where concepts of geodesics underpin theories of spacetime and gravity, and robotics, where navigating robots through restricted paths efficiently is crucial.

In a Sub-Riemannian manifold, thinking of geodesics as the 'straightest' possible paths within the given constraints can provide intuitional understanding, despite the complex mathematics involved.

## Advanced Topics in Sub-Riemannian Geometry

Sub-Riemannian geometry, while intricate, opens numerous avenues for exploring advanced mathematical concepts and applications. By understanding its relationship with Lie groups, the role of abnormal minimisers, and connections to optimal transport, you delve deeper into the subject, uncovering the elegant complexity and utility of Sub-Riemannian geometry.

### Sub-Riemannian Geometry and Lie Groups

In the intersection between Sub-Riemannian geometry and Lie groups, mathematical structures become both more complex and more fascinating. Lie groups play a central role, serving as the backbone for understanding the symmetries and structures inherent in Sub-Riemannian spaces.

Lie groups, comprising sets of continuous transformation groups with smooth operations, offer a powerful tool for studying the continuous symmetries of Sub-Riemannian manifolds. This connection is crucial for the analysis of geometric properties and the classification of possible geometries within the Sub-Riemannian framework.

Lie Group: A group that is also a differentiable manifold, where the group operations of multiplication and inversion are smooth functions. Lie groups are pivotal in studying symmetries in mathematical physics and differential geometry.

Consider the group of rotations in three-dimensional space, which forms a Lie group known as SO(3). This group plays a significant role in Sub-Riemannian geometry as it describes the symmetries of a sphere, allowing for the exploration of paths and distances on the sphere's surface that respect these rotational symmetries.

### On the Role of Abnormal Minimisers in Sub-Riemannian Geometry

Abnormal minimisers represent a peculiar aspect of Sub-Riemannian geometry, underlying the geometry's inherent complexity. These paths, which may not rely on the traditional understanding of shortest or most efficient paths, highlight the rich structure and unforeseen complications that can arise in constrained spaces.

Abnormal Minimiser: A path between two points in a Sub-Riemannian space that is locally shortest, not due to variations in length or energy but because it adheres to the geometry's intrinsic constraints, sometimes defying intuition.

Imagine navigating a maze with walls so tall they cannot be seen over. An abnormal minimiser might be a path that, counterintuitively, initially moves away from the final destination because direct routes are blocked, utilizing the maze's layout to reach the end point efficiently.

### Sub-Riemannian Geometry and Optimal Transport

The application of Sub-Riemannian geometry to optimal transport problems embodies the fusion of abstract mathematical theory with practical problem-solving. Optimal transport, a concept originating in economics, seeks the most efficient ways to move goods or resources from one place to another. Within Sub-Riemannian manifolds, optimal transport problems gain an additional layer of complexity, as the 'shortest paths' must navigate the manifold's constrained geometry. This challenge provides a rich ground for applying Sub-Riemannian methods to find not just any solution, but the most efficient one given the manifold's unique structure.

The Monge-Kantorovich problem, a foundational problem in optimal transport, becomes particularly intricate within the framework of Sub-Riemannian geometry, revealing new depths to both fields.

## Sub-Riemannian geometry - Key takeaways

• Sub-Riemannian Geometry: A branch of mathematics dealing with manifolds and tangent subspaces, where length and distance are defined only along certain constrained directions.
• Admissible Paths: Curves in Sub-Riemannian geometry where each tangent lies within a constrained subset of directions at every point, crucial for defining distances.
• Geodesics: In Sub-Riemannian geometry, the most direct or efficient trajectories between two points that adhere to the geometry's constraints.
• Lie Groups: Continuous transformation groups that provide a structure for studying the symmetries of Sub-Riemannian manifolds, important for geometric analysis.
• Abnormal Minimisers: Locally shortest paths in Sub-Riemannian spaces, which may not align with traditional conceptions of efficiency due to the geometry's intrinsic constraints.
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What is the primary difference between Riemannian and sub-Riemannian geometry?
The primary difference between Riemannian and sub-Riemannian geometry lies in the constraint on allowable paths: sub-Riemannian geometry restricts paths to those tangent to a chosen distribution of planes within the tangent bundle, while Riemannian geometry allows paths in any direction.
What are some common applications of sub-Riemannian geometry in engineering and physics?
Sub-Riemannian geometry finds applications in robotics for motion planning and control, in quantum physics for modelling atom-photon interactions, in signal processing for image and data analysis, and in control theory for the design of optimal and robust control systems.
How does one define the notion of distance in sub-Riemannian geometry?
In sub-Riemannian geometry, distance between two points is defined as the infimum length of curves that are tangent to a given distribution of planes and connect these points, based on a specific metric derived from the geometry's constraints and structure.
What are the essential elements required to construct a sub-Riemannian manifold?
To construct a sub-Riemannian manifold, one needs a manifold equipped with a smoothly varying distribution of tangent subspaces (a smooth bracket-generating distribution) and a smoothly varying inner product defined on these subspaces.
How do control theory and sub-Riemannian geometry intersect?
In control theory, the controllability of systems can often be modelled through geometric paths, for which sub-Riemannian geometry provides the mathematical framework to describe the shortest paths or geodesics under given constraints, mirroring the optimization problems encountered in control systems.

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