Sub-Riemannian manifold

Inmathematics,asub-Riemannian manifoldis 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 metriccalled themetric of Carnot–Carathéodory.TheHausdorff dimensionof suchmetric spacesis always anintegerand 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 phasemay be understood in the language of sub-Riemannian geometry. TheHeisenberg group,important toquantum mechanics,carries a natural sub-Riemannian structure.

Definitions

By adistributionon$M$we mean asubbundleof thetangent bundleof$M$(see alsodistribution).

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

A distribution on$H(M)$is calledcompletely non-integrableorbracket generatingif for any$x\in M$we have that any tangent vector can be presented as alinear combinationofLie bracketsof horizontal fields, i.e. vectors of the form$$A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),\dotsc \in T_{x}(M)$$where all vector fields$A,B,C,D,\dots$are horizontal. This requirement is also known asHörmander's condition.

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

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

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

where infimum is taken along allhorizontal curves$\gamma:[0,1]\to M$such that$\gamma (0)=x$,$\gamma (1)=y$. Horizontal curves can be taken eitherLipschitz continuous,Absolutely continuousor in theSobolev space$H^{1}([0,1],M)$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.

Examples

A position of a car on the plane is determined by three parameters: two coordinates$x$and$y$for the location and an angle$\alpha$which describes the orientation of the car. Therefore, the position of the car can be described by a point in a manifold

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

One can ask, what is the minimal distance one should drive to get from one position to another? This defines aCarnot–Carathéodory metricon the manifold

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

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

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

spans the entire algebra. The distribution$H$spanned by left shifts of$\alpha$and$\beta$iscompletely non-integrable.Then choosing any smooth positive quadratic form on$H$gives a sub-Riemannian metric on the group.

Properties

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 equationsfor the sub-Riemannian Hamiltonian are called geodesics, and generalizeRiemannian geodesics.

See also

• Carnot group,a class ofLie groupsthat form sub-Riemannian manifolds.
• Distribution
• Hörmander's condition
• Optimal control

References

• Agrachev, Andrei; Barilari, Davide; Boscain, Ugo, eds. (2019),Comprehensive Introduction to Sub-Riemannian Geometry,Cambridge Studies in Advanced Mathematics, Cambridge University Press,doi:10.1017/9781108677325,ISBN9781108677325
• Bellaïche, André; Risler, Jean-Jacques, eds. (1996),Sub-Riemannian geometry,Progress in Mathematics, vol. 144, Birkhäuser Verlag,ISBN978-3-7643-5476-3,MR1421821
• 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,ISBN3-7643-5476-3,MR1421823,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,ISBN0-8218-1391-9