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{{short description|Type of generalization of a Riemannian manifold}}
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 space]]s 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.

==Definitions==

By a ''distribution'' on <math>M</math> we mean a [[subbundle]] of the [[tangent bundle]] of <math>M</math> (see also [[Distribution (differential geometry)|distribution]]).

Given a distribution <math>H(M)\subset T(M)</math> a vector field in <math>H(M)</math> is called ''horizontal''. A curve <math>\gamma</math> on <math>M</math> is called horizontal if <math>\dot\gamma(t)\in H_{\gamma(t)}(M)</math> for any
<math>t</math>.

A distribution <math>H(M)</math> is called ''completely non-integrable'' or ''bracket generating'' if for any <math>x\in M</math> we have that any tangent vector can be presented as a [[linear combination]] of [[Lie bracket of vector fields|Lie brackets]] of horizontal fields, i.e. vectors of the form <math display="block">A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),\dotsc\in T_x(M)</math> where all vector fields <math>A,B,C,D, \dots</math> are horizontal. This requirement is also known as [[Hörmander's condition]].

A sub-Riemannian manifold is a triple <math>(M, H, g)</math>, where <math>M</math> is a differentiable [[manifold]], <math>H</math> is a completely non-integrable "horizontal" distribution and <math>g</math> is a smooth section of positive-definite [[quadratic form]]s on <math>H</math>.

Any (connected) sub-Riemannian manifold carries a natural [[intrinsic metric]], called the metric of Carnot–Carathéodory, defined as
:<math>d(x, y) = \inf\int_0^1 \sqrt{g(\dot\gamma(t),\dot\gamma(t))} \, dt,</math>
where infimum is taken along all ''horizontal curves'' <math>\gamma: [0, 1] \to M</math> such that <math>\gamma(0)=x</math>, <math>\gamma(1)=y</math>.
Horizontal curves can be taken either [[Lipschitz continuous]], [[Absolutely continuous]] or in the [[Sobolev space]] <math> H^1([0,1],M) </math> 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]].

==Examples==

A position of a car on the plane is determined by three parameters: two coordinates <math>x</math> and <math>y</math> for the location and an angle <math>\alpha</math> which describes the orientation of the car. Therefore, the position of the car can be described by a point in a manifold
:<math>\mathbb R^2\times S^1.</math>

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
:<math>\mathbb R^2\times S^1.</math>

A closely related example of a sub-Riemannian metric can be constructed on a [[Heisenberg group]]: Take two elements <math>\alpha</math> and <math>\beta</math> in the corresponding Lie algebra such that
:<math>\{ \alpha,\beta,[\alpha,\beta]\}</math>

spans the entire algebra. The distribution <math>H</math> spanned by left shifts of <math>\alpha</math> and <math>\beta</math> is ''completely non-integrable''. Then choosing any smooth positive quadratic form on <math>H</math> gives a sub-Riemannian metric on the group.

==Properties==

For every sub-Riemannian manifold, there exists a [[Hamiltonian mechanics|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 equation]]s for the sub-Riemannian Hamiltonian are called geodesics, and generalize [[Geodesic|Riemannian geodesics]].

==See also==
*[[Carnot group]], a class of [[Lie group]]s that form sub-Riemannian manifolds.
*[[Distribution (differential geometry)|Distribution]]
*[[Hörmander's condition]]
*[[Optimal control]]

==References==
*{{Citation | editor1-last=Agrachev | editor1-first=Andrei | editor2-last=Barilari| editor2-first=Davide | editor3-last=Boscain| editor3-first=Ugo | title=Comprehensive Introduction to Sub-Riemannian Geometry | doi=10.1017/9781108677325 | publisher=Cambridge University Press | series= Cambridge Studies in Advanced Mathematics | isbn=9781108677325 | year=2019 | url=https://hal.archives-ouvertes.fr/hal-02019181/file/ABB-v2-hal.pdf }}
*{{Citation | editor1-last=Bellaïche | editor1-first=André | editor2-last=Risler | editor2-first=Jean-Jacques | title=Sub-Riemannian geometry | url=https://books.google.com/books?id=7Z7IMze7pDwC | publisher=Birkhäuser Verlag | series=Progress in Mathematics | isbn=978-3-7643-5476-3 | mr=1421821 | year=1996 | volume=144}}
*{{Citation | last1=Gromov | first1=Mikhael | editor1-last=Bellaïche | editor1-first=André | editor2-last=Risler. | editor2-first=Jean-Jacques | title=Sub-Riemannian geometry | url=https://www.ihes.fr/~gromov/PDF/carnot_caratheodory.pdf | archive-url=https://web.archive.org/web/20150709072037/https://www.ihes.fr/~gromov/PDF/carnot_caratheodory.pdf | archive-date=July 9, 2015 | publisher=Birkhäuser | location=Basel, Boston, Berlin | series=Progr. Math. | mr=1421823 | year=1996 | volume=144 | chapter=Carnot-Carathéodory spaces seen from within | pages=79–323 | isbn=3-7643-5476-3}}
*{{citation|url=https://cvgmt.sns.it/media/doc/paper/5339/sub-Riem_notes.pdf|title=Lecture notes on sub-Riemannian geometry|first=Enrico|last= Le Donne}}
*{{Citation |first=Richard |last=Montgomery |title=A Tour of Subriemannian Geometries, Their Geodesics and Applications |series=Mathematical Surveys and Monographs |volume=91 |year=2002 |publisher=American Mathematical Society |isbn=0-8218-1391-9}}

{{Manifolds}}
{{Riemannian geometry}}

[[Category:Metric geometry]]
[[Category:Riemannian geometry]]
[[Category:Riemannian manifolds]]

Sub-Riemannian manifold - Revision history

imported>Llsalcedo at 08:40, 9 September 2025