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 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.
Read more about this topic: Sub-Riemannian Manifold
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