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.
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 if for any we have that any tangent vector can be presented as a linear combination of vectors of the following types where all vector fields are horizontal.
Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as
where infimum is taken along all horizontal curves such that , .
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 Carnot–Carathéodory metric on the manifold
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.
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. The existence of geodesics of the corresponding Hamilton–Jacobi equations for the sub-Riemannian Hamiltonian is given by the Chow–Rashevskii theorem.
- Bellaïche, André; Risler, Jean-Jacques, eds. (1996), Sub-Riemannian geometry, Progress in Mathematics, 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., 144, Basel, Boston, Berlin: Birkhäuser, pp. 79–323, ISBN 3-7643-5476-3, MR 1421823
- Le Donne, Enrico, Lecture notes on sub-Riemannian geometry (PDF)
- Richard Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91), (2002) American Mathematical Society, ISBN 0-8218-1391-9.