Concept

Sub-Riemannian manifold

Résumé
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces. Sub-Riemannian manifolds (and so, a fortiori, Riemannian manifolds) carry a natural intrinsic metric called the metric of Carnot–Carathéodory. The Hausdorff dimension of such metric spaces is always an integer and larger than its topological dimension (unless it is actually a Riemannian manifold). Sub-Riemannian manifolds often occur in the study of constrained systems in classical mechanics, such as the motion of vehicles on a surface, the motion of robot arms, and the orbital dynamics of satellites. Geometric quantities such as the Berry phase may be understood in the language of sub-Riemannian geometry. The Heisenberg group, important to quantum mechanics, carries a natural sub-Riemannian structure. By a distribution on we mean a subbundle of the tangent bundle of . Given a distribution a vector field in is called horizontal. A curve on is called horizontal if for any A distribution on is called completely non-integrable if for any we have that any tangent vector can be presented as a linear combination of vectors of the following types where all vector fields are horizontal. A sub-Riemannian manifold is a triple , where is a differentiable manifold, is a completely non-integrable "horizontal" distribution and is a smooth section of positive-definite quadratic forms on . Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as where infimum is taken along all horizontal curves such that , . A position of a car on the plane is determined by three parameters: two coordinates and for the location and an angle which describes the orientation of the car.
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Publications associées

Chargement

Personnes associées

Chargement

Unités associées

Chargement

Concepts associés

Chargement

Cours associés

Chargement

Séances de cours associées

Chargement

MOOCs associés

Chargement