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Sub-Riemannian manifold

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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 (see also distribution).

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 or bracket generating if for any we have that any tangent vector can be presented as a linear combination of Lie brackets of horizontal fields, i.e. vectors of the form where all vector fields are horizontal. This requirement is also known as Hörmander's condition.

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 (connected) sub-Riemannian manifold carries a natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as

where infimum is taken along all horizontal curves such that , . Horizontal curves can be taken either Lipschitz continuous, Absolutely continuous or in the Sobolev space producing the same metric in all cases.

The fact that the distance of two points is always finite (i.e. any two points are connected by an horizontal curve) is a consequence of Hörmander's condition known as Chow–Rashevskii theorem.



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. Therefore, the position of the car can be described by a point in a manifold

One can ask, what is the minimal distance one should drive to get from one position to another? This defines a Carnot–Carathéodory metric on the manifold

A closely related example of a sub-Riemannian metric can be constructed on a Heisenberg group: Take two elements and in the corresponding Lie algebra such that

spans the entire algebra. The horizontal distribution spanned by left shifts of and is completely non-integrable. Then choosing any smooth positive quadratic form on gives a sub-Riemannian metric on the group.



For every sub-Riemannian manifold, there exists a Hamiltonian, called the sub-Riemannian Hamiltonian, constructed out of the metric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub-Riemannian manifold.

Solutions of the corresponding Hamilton–Jacobi equations for the sub-Riemannian Hamiltonian are called geodesics, and generalize Riemannian geodesics.

See also



  • Agrachev, Andrei; Barilari, Davide; Boscain, Ugo, eds. (2019), Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Studies in Advanced Mathematics, Cambridge University Press, doi:10.1017/9781108677325, ISBN 9781108677325
  • Bellaïche, André; Risler, Jean-Jacques, eds. (1996), Sub-Riemannian geometry, Progress in Mathematics, vol. 144, Birkhäuser Verlag, ISBN 978-3-7643-5476-3, MR 1421821
  • Gromov, Mikhael (1996), "Carnot-Carathéodory spaces seen from within", in Bellaïche, André; Risler., Jean-Jacques (eds.), Sub-Riemannian geometry (PDF), Progr. Math., vol. 144, Basel, Boston, Berlin: Birkhäuser, pp. 79–323, ISBN 3-7643-5476-3, MR 1421823, archived from the original (PDF) on July 9, 2015
  • Le Donne, Enrico, Lecture notes on sub-Riemannian geometry (PDF)
  • Montgomery, Richard (2002), A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, ISBN 0-8218-1391-9