Inmathematics, asub-Riemannian manifold is a certain type of generalization of aRiemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-calledhorizontal subspaces.

Sub-Riemannian manifolds (and so,a fortiori, Riemannian manifolds) carry a naturalintrinsic metric called themetric of Carnot–Carathéodory. TheHausdorff dimension of suchmetric spaces is always aninteger and larger than itstopological dimension (unless it is actually a Riemannian manifold).

Sub-Riemannian manifolds often occur in the study of constrained systems inclassical 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 theBerry phase may be understood in the language of sub-Riemannian geometry. TheHeisenberg group, important toquantum mechanics, carries a natural sub-Riemannian structure.

By adistribution on${\displaystyle M}$ we mean asubbundle of thetangent bundle of${\displaystyle M}$ (see alsodistribution).

Given a distribution${\displaystyle H(M)\subset T(M)}$ a vector field in${\displaystyle H(M)}$ is calledhorizontal. 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${\displaystyle H(M)}$ is calledcompletely non-integrable orbracket generating if for any${\displaystyle x\in M}$ we have that any tangent vector can be presented as alinear combination ofLie brackets of horizontal fields, i.e. vectors of the form$${\displaystyle A(x),[A,B](x),[A,[B,C]](x),[A,[B,[C,D]]](x),\dots \in T_x(M)}$$ where all vector fields${\displaystyle A,B,C,D,\dots }$ are horizontal. This requirement is also known asHörmander's condition.

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

Any (connected) sub-Riemannian manifold carries a naturalintrinsic 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 allhorizontal curves${\displaystyle \gamma :[0,1]\to M}$ such that${\displaystyle \gamma (0)=x}$,${\displaystyle \gamma (1)=y}$.Horizontal curves can be taken eitherLipschitz continuous,Absolutely continuous or in theSobolev 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 asChow–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 aCarnot–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 aHeisenberg 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 distribution${\displaystyle H}$ spanned by left shifts of${\displaystyle \alpha }$ and${\displaystyle \beta }$ iscompletely 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 aHamiltonian, called thesub-Riemannian Hamiltonian, constructed out of the metric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub-Riemannian manifold.

Solutions of the correspondingHamilton–Jacobi equations for the sub-Riemannian Hamiltonian are called geodesics, and generalizeRiemannian geodesics.

• Carnot group, a class ofLie 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(PDF), 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 fromthe 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

• Metric geometry
• Riemannian geometry
• Riemannian manifolds

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