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 [math]\displaystyle{ M }[/math] we mean a subbundle of the tangent bundle of [math]\displaystyle{ M }[/math] (see also distribution).
Given a distribution [math]\displaystyle{ H(M)\subset T(M) }[/math] a vector field in [math]\displaystyle{ H(M) }[/math] is called horizontal. A curve [math]\displaystyle{ \gamma }[/math] on [math]\displaystyle{ M }[/math] is called horizontal if [math]\displaystyle{ \dot\gamma(t)\in H_{\gamma(t)}(M) }[/math] for any [math]\displaystyle{ t }[/math].
A distribution on [math]\displaystyle{ H(M) }[/math] is called completely non-integrable or bracket generating if for any [math]\displaystyle{ x\in M }[/math] 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 [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ (M, H, g) }[/math], where [math]\displaystyle{ M }[/math] is a differentiable manifold, [math]\displaystyle{ H }[/math] is a completely non-integrable "horizontal" distribution and [math]\displaystyle{ g }[/math] is a smooth section of positive-definite quadratic forms on [math]\displaystyle{ H }[/math].
Any (connected) sub-Riemannian manifold carries a natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as
- [math]\displaystyle{ 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]\displaystyle{ \gamma: [0, 1] \to M }[/math] such that [math]\displaystyle{ \gamma(0)=x }[/math], [math]\displaystyle{ \gamma(1)=y }[/math]. Horizontal curves can be taken either Lipschitz continuous, Absolutely continuous or in the Sobolev space [math]\displaystyle{ 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]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] for the location and an angle [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] in the corresponding Lie algebra such that
- [math]\displaystyle{ \{ \alpha,\beta,[\alpha,\beta]\} }[/math]
spans the entire algebra. The horizontal distribution [math]\displaystyle{ H }[/math] spanned by left shifts of [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] is completely non-integrable. Then choosing any smooth positive quadratic form on [math]\displaystyle{ H }[/math] 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
- Bellaïche, André; Risler, Jean-Jacques, eds. (1996), Sub-Riemannian geometry, Progress in Mathematics, 144, Birkhäuser Verlag, ISBN 978-3-7643-5476-3, https://books.google.com/books?id=7Z7IMze7pDwC
- Gromov, Mikhael (1996), "Carnot-Carathéodory spaces seen from within", in Bellaïche, André; Risler., Jean-Jacques, Sub-Riemannian geometry, Progr. Math., 144, Basel, Boston, Berlin: Birkhäuser, pp. 79–323, ISBN 3-7643-5476-3, https://www.ihes.fr/~gromov/PDF/carnot_caratheodory.pdf
- Le Donne, Enrico, Lecture notes on sub-Riemannian geometry, https://cvgmt.sns.it/media/doc/paper/5339/sub-Riem_notes.pdf
- Montgomery, Richard (2002), A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, 91, American Mathematical Society, ISBN 0-8218-1391-9
 |  |
