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Sub-Riemannian manifold, the Glossary

In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold.^[1]

30 relations: Absolute continuity, Carnot group, Chow–Rashevskii theorem, Classical mechanics, Distribution (differential geometry), Geodesic, Geometric phase, Hamilton–Jacobi equation, Hamiltonian mechanics, Hausdorff dimension, Hörmander's condition, Heisenberg group, Integer, Intrinsic metric, Lebesgue covering dimension, Lie bracket of vector fields, Lie group, Linear combination, Lipschitz continuity, Manifold, Mathematics, Metric space, Optimal control, Quadratic form, Quantum mechanics, Riemannian manifold, Sobolev space, Sub-Riemannian manifold, Subbundle, Tangent bundle.
Riemannian manifolds

Absolute continuity

In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.

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Carnot group

In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra.

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In sub-Riemannian geometry, the Chow–Rashevskii theorem (also known as Chow's theorem) asserts that any two points of a connected sub-Riemannian manifold, endowed with a bracket generating distribution, are connected by a horizontal path in the manifold. Sub-Riemannian manifold and Chow–Rashevskii theorem are metric geometry.

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Classical mechanics

Classical mechanics is a physical theory describing the motion of objects such as projectiles, parts of machinery, spacecraft, planets, stars, and galaxies.

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Distribution (differential geometry)

In differential geometry, a discipline within mathematics, a distribution on a manifold M is an assignment x \mapsto \Delta_x \subseteq T_x M of vector subspaces satisfying certain properties.

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Geodesic

In geometry, a geodesic is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold.

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Geometric phase

In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian.

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Hamilton–Jacobi equation

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.

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Hamiltonian mechanics

In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833.

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Hausdorff dimension

In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. Sub-Riemannian manifold and Hausdorff dimension are metric geometry.

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Hörmander's condition

In mathematics, Hörmander's condition is a property of vector fields that, if satisfied, has many useful consequences in the theory of partial and stochastic differential equations.

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Heisenberg group

In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form \end under the operation of matrix multiplication.

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Integer

An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.

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Intrinsic metric

In the mathematical study of metric spaces, one can consider the arclength of paths in the space. Sub-Riemannian manifold and Intrinsic metric are metric geometry.

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Lebesgue covering dimension

In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way.

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Lie bracket of vector fields

In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted. Sub-Riemannian manifold and Lie bracket of vector fields are Riemannian geometry.

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Lie group

In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

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Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Mathematics

Mathematics is a field of study that discovers and organizes abstract objects, methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

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Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points.

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Optimal control

Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized.

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Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial).

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Quantum mechanics

Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms.

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Riemannian manifold

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Sub-Riemannian manifold and Riemannian manifold are Riemannian geometry and Riemannian manifolds.

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References

[1] https://en.wikipedia.org/wiki/Sub-Riemannian_manifold

Also known as Carnot-Carathéodory metric, Sub Riemannian geometry, Sub-Riemannian geometry, Sub-Riemannian metric.