Twistor Space for Rolling Bodies

By Daniel An, P. Nurowski. 2012

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On a natural circle bundle $${\mathbb{T}(M)}$$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 $${\mathcal{D}}$$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 $${\mathcal{D}}$$D is (2,3,5) in $${\mathbb{T}(M)}$$T(M). We show that if M is a Cartesian product of two Riemann surfaces (Σ1, g1) and (Σ2, g2), and if $${g = g_{1} \oplus (-g_2)}$$g=g1⊕(-g2), then the circle bundle $${\mathbb{T}(\Sigma_1 \times \Sigma_2)}$$T(Σ1×Σ2) is just the configuration space for the physical system of two surfaces Σ1 and Σ2 rolling on each other. The condition for the two surfaces to roll on each other ‘without slipping or twisting’ identifies the restricted velocity space for such a system with the tautological distribution $${\mathcal{D}}$$D on $${\mathbb{T}(\Sigma_1 \times \Sigma_2)}$$T(Σ1×Σ2). We call $${\mathbb{T}(\Sigma_1 \times \Sigma_2)}$$T(Σ1×Σ2) the twistor space, and $${\mathcal{D}}$$D the twistor distribution for the rolling surfaces. Among others we address the following question: “For which pairs of surfaces does the restricted velocity distribution (which we identify with the twistor distribution $${\mathcal{D}}$$D) have the simple Lie group G2 as the group of its symmetries?” Apart from the well known situation when the surfaces Σ1 and Σ2 have constant curvatures whose ratio is 1:9, we unexpectedly find three different types of surfaces that when rolling ‘without slipping or twisting’ on a plane, have $${\mathcal{D}}$$D with the symmetry group G2. Although we have found the differential equations for the curvatures of Σ1 and Σ2 that gives $${\mathcal{D}}$$D with G2 symmetry, we are unable to solve them in full generality so far.

Subfields: Applied Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/d570ca15346d9eb2047bb7e3a4861c9a1b532e53