Sub-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 spaces 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 ${\displaystyle M}$ 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 $${\displaystyle A(x),\ [A,B](x),\ [A,[B,C]](x),\ [A,[B,[C,D]]](x),\dotsc \in T_{x}(M)}$$ where all vector fields ${\displaystyle A,B,C,D,\dots }$ are horizontal. This requirement is also known as Hörmander's condition.

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.

Examples

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 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.

Properties

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.

See also

• Carnot group, a class of Lie groups that 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, 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