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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 .

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. 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. The existence of geodesics of the corresponding Hamilton–Jacobi equations for the sub-Riemannian Hamiltonian is given by the Chow–Rashevskii theorem.

- Carnot group, a class of Lie groups that form sub-Riemannian manifolds
- Distribution

- 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 - 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