Twistor Space for Rolling Bodies

On a natural circle bundle ${\mathbb{T}(M)}$ 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 ${\mathcal{D}}$ 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 ${\mathcal{D}}$ is (2,3,5) in ${\mathbb{T}(M)}$ . We show that if M is a Cartesian product of two Riemann surfaces (Σ ₁, g ₁) and (Σ ₂, g ₂), and if ${g = g_{1} \oplus (-g_2)}$ , then the circle bundle ${\mathbb{T}(\Sigma_1 \times \Sigma_2)}$ is just the configuration space for the physical system of two surfaces Σ ₁ and Σ ₂ 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 ${\mathcal{D}}$ on ${\mathbb{T}(\Sigma_1 \times \Sigma_2)}$ . We call ${\mathbb{T}(\Sigma_1 \times \Sigma_2)}$ the twistor space, and ${\mathcal{D}}$ 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 ${\mathcal{D}}$ ) have the simple Lie group G ₂ as the group of its symmetries?” Apart from the well known situation when the surfaces Σ ₁ and Σ ₂ 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 ${\mathcal{D}}$ with the symmetry group G ₂. Although we have found the differential equations for the curvatures of Σ ₁ and Σ ₂ that gives ${\mathcal{D}}$ with G ₂ symmetry, we are unable to solve them in full generality so far.