A car as parabolic geometry

2019/08/03 by Hill, C. Denson, Nurowski, Paweł
#Differential Geometry (math.DG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1908.01169

Abstract

We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type (\bf SO(2,3),P₁₂), where P₁₂ is a Borel parabolic subgroup in \bf 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.

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