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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''.
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| 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).
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| 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.
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| ==Definitions==
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| By a ''distribution'' on <math>M</math> we mean a [[subbundle]] of the [[tangent bundle]] of <math>M</math>.
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| Given a distribution <math>H(M)\subset T(M)</math> a vector field in <math>H(M)\subset T(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
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| <math>t</math>.
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| A distribution on <math>H(M)</math> is called '''completely non-integrable''' if for any <math>x\in M</math> we have that any tangent vector can be presented as a [[linear combination]] of vectors of the following types <math>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.
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| 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>.
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| Any '''sub-Riemannian manifold''' carries the natural [[intrinsic metric]], called the '''metric of Carnot–Carathéodory''', defined as
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| :<math>d(x, y) = \inf\int_0^1 \sqrt{g(\dot\gamma(t),\dot\gamma(t))} \, dt,</math>
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| 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>.
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| ==Examples==
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| 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 car can be described by a point in a manifold
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| :<math>\mathbb R^2\times S^1.</math>
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| 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
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| :<math>\mathbb R^2\times S^1.</math>
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| 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
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| :<math>\{ \alpha,\beta,[\alpha,\beta]\}</math>
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| spans the entire algebra. The horizontal 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.
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| ==Properties==
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| 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. The existence of geodesics of the corresponding [[Hamilton–Jacobi equation]]s for the sub-Riemannian Hamiltonian is given by the [[Chow–Rashevskii theorem]].
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| ==See also==
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| *[[Carnot group]], a class of [[Lie group]]s that form sub-Riemannian manifolds
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| ==References==
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| *{{Citation | editor1-last=Bellaïche | editor1-first=André | editor2-last=Risler | editor2-first=Jean-Jacques | title=Sub-Riemannian geometry | url=http://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}}
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| *{{Citation | last1=Gromov | first1=Mikhael | editor1-last=Bellaïche | editor1-first=André | editor2-last=Risler. | editor2-first=Jean-Jacques | title=Sub-Riemannian geometry | url=http://www.ihes.fr/~gromov/PDF/carnot_caratheodory.pdf | 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}}
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| *{{citation|url=http://www.math.ethz.ch/~ledonnee/sub-Riem_notes.pdf |title=Lecture notes on sub-Riemannian geometry|first=Enrico|last= Le Donne}}
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| * Richard Montgomery, ''A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91)'', (2002) American Mathematical Society, ISBN 0-8218-1391-9.
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| [[Category:Metric geometry]]
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| [[Category:Riemannian manifolds]]
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