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
Sub-Riemannian manifolds often occur in the study of constrained systems in
Definitions
By a distribution on we mean a subbundle of the tangent bundle of (see also distribution).
Given a distribution a vector field in is called horizontal. A curve on is called horizontal if for any .
A distribution on is called completely non-integrable or bracket generating if for any 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
A sub-Riemannian manifold is a triple , where is a differentiable manifold, is a completely non-integrable "horizontal" distribution and is a smooth section of positive-definite quadratic forms on .
Any (connected) sub-Riemannian manifold carries a natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as
where infimum is taken along all horizontal curves such that , . Horizontal curves can be taken either
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 and for the location and an angle which describes the orientation of the car. Therefore, the position of the car can be described by a point in a manifold
One can ask, what is the minimal distance one should drive to get from one position to another? This defines a
A closely related example of a sub-Riemannian metric can be constructed on a Heisenberg group: Take two elements and in the corresponding Lie algebra such that
spans the entire algebra. The horizontal distribution spanned by left shifts of and is completely non-integrable. Then choosing any smooth positive quadratic form on 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, ISBN 9781108677325
- Bellaïche, André; Risler, Jean-Jacques, eds. (1996), Sub-Riemannian geometry, Progress in Mathematics, vol. 144, Birkhäuser Verlag, 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, 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