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.
we mean a subbundle of the tangent bundle of
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
This requirement is also known as Hörmander's condition.
A sub-Riemannian manifold is a triple
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
Horizontal curves can be taken either Lipschitz continuous, Absolutely continuous or in the Sobolev space
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.
A position of a car on the plane is determined by three parameters: two coordinates
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
spanned by left shifts of
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.
Solutions of the corresponding Hamilton–Jacobi equations for the sub-Riemannian Hamiltonian are called geodesics, and generalize Riemannian geodesics.