By a distribution on ${\displaystyle M}$  we mean a subbundle of the tangent bundle of ${\displaystyle M}$  (see also distribution).

Given a distribution ${\displaystyle H(M)\subset T(M)}$  a vector field in ${\displaystyle H(M)}$  is called horizontal. A curve ${\displaystyle \gamma }$  on ${\displaystyle M}$  is called horizontal if ${\displaystyle {\dot {\gamma }}(t)\in H_{\gamma (t)}(M)}$  for any ${\displaystyle t}$ .

A distribution on ${\displaystyle H(M)}$  is called completely non-integrable or bracket generating if for any ${\displaystyle x\in M}$  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

A sub-Riemannian manifold is a triple ${\displaystyle (M,H,g)}$ , where ${\displaystyle M}$  is a differentiable manifold, ${\displaystyle H}$  is a completely non-integrable "horizontal" distribution and ${\displaystyle g}$  is a smooth section of positive-definite quadratic forms on ${\displaystyle H}$ .

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

${\displaystyle d(x,y)=\inf \int _{0}^{1}{\sqrt {g({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,dt,}$ 

where infimum is taken along all horizontal curves ${\displaystyle \gamma :[0,1]\to M}$  such that ${\displaystyle \gamma (0)=x}$ , ${\displaystyle \gamma (1)=y}$ . Horizontal curves can be taken either Lipschitz continuous, Absolutely continuous or in the Sobolev space ${\displaystyle H^{1}([0,1],M)}$  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 ${\displaystyle x}$  and ${\displaystyle y}$  for the location and an angle ${\displaystyle \alpha }$  which describes the orientation of the car. Therefore, the position of the car can be described by a point in a manifold

${\displaystyle \mathbb {R} ^{2}\times S^{1}.}$ 

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

${\displaystyle \mathbb {R} ^{2}\times S^{1}.}$ 

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

${\displaystyle \{\alpha ,\beta ,[\alpha ,\beta ]\}}$ 

spans the entire algebra. The horizontal distribution ${\displaystyle H}$  spanned by left shifts of ${\displaystyle \alpha }$  and ${\displaystyle \beta }$  is completely non-integrable. Then choosing any smooth positive quadratic form on ${\displaystyle H}$  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.

• Carnot group, a class of Lie groups that form sub-Riemannian manifolds.
• Distribution
• Hörmander's condition
• Optimal control

• Agrachev, Andrei; Barilari, Davide; Boscain, Ugo, eds. (2019), Comprehensive Introduction to Sub-Riemannian Geometry, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 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