A car as parabolic geometry

Abstract

We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type $(\mathbf{SO}(2,3),P_{12})$, where $P_{12}$ is a Borel parabolic subgroup in $\mathbf{SO}(2,3)$. We discuss the relations of this geometry of a car with the geometry of circles in the plane (a low dimensional Lie sphere geometry), the geometry of 3-dimensional conformal Minkowski spacetime, the geometry of 3-rd order ODEs, projective contact geometry in three dimensions, and the corresponding twistor fibrations. We indicate how all these classical geometries can be interpreted in terms of the nonholonomic kinematics of a car.

In this v.2 we replaced the wrong picture of the car's twistor diagram with the correct one, and corrected few typos related to the commutation relations in Section 4.3.1