Twistor space for rolling bodies

TL;DR

This paper explores the geometric structure of rolling bodies using twistor space, revealing special symmetry conditions related to the exceptional Lie group G2, with implications for understanding constrained mechanical systems.

Contribution

It introduces a novel geometric framework linking twistor space to the rolling of bodies, identifying conditions for G2 symmetry in the velocity distribution.

Findings

Identifies conditions for G2 symmetry in rolling bodies.

Connects twistor geometry with mechanical constraints.

Finds specific surface types with G2 symmetry.

Abstract

On a natural circle bundle T(M) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null selfdual 2-planes, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. Over the open set where g is not antiselfdual, the distribution D is (2,3,5) in T(M). We show that if M is a Cartesian product of two Riemann surfaces (S1,g1) and (S2,g2), and if g=g1--g2, then the circle bundle T(S1 x S2) is just the configuration space for the physical system of two solid bodies B1 and B2, bounded by the surfaces S1 and S2 and rolling on each other. The condition for the two bodies to roll on each other `without slipping or twisting' identifies the restricted velocity space for such a system with the tautological distribution D on T(S1 x S2). We call T(S1 x S2) the twistor space, and D the twistor distribution for the rolling bodies. Among others we address the following question: "For which pairs of bodies does the restricted velocity distribution (which we identify with the twistor distribution D) have the simple Lie group G2 as its group of symmetries?" Apart from the well known situation when the boundaries S1 and S2 of the two bodies have constant curvatures whose ratio is 1:9, we unexpectedly find three different types of surfaces, which when bounding a body that rolls `without slipping or twisting' on a plane, have D with the symmetry group G2. Although we have found the differential equations for the curvatures of S1 and S2 that gives D with G2 symmetry, we are unable to solve them in full generality so far.