Twistor Space for Rolling Bodies

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 ( Σ 1 , g 1 ) and ( Σ 2 , g 2 ), and if g = g 1 ⊕ ( - g 2 ) , then the circle bundle T ( Σ 1 × Σ 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 D on T ( Σ 1 × Σ 2 ) . We call T ( Σ 1 × Σ 2 ) the twistor space , and 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 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 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 D with G 2 symmetry, we are unable to solve them in full generality so far.