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Twistor space for rolling bodies. (English) Zbl 1296.53100
Authors’ abstract: 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 \((\Sigma_1,g_1)\) and \((\Sigma_2,g_2)\), 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 \(\Sigma_1\) and \(\Sigma_2\) 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 \(\Sigma_1\) and \(\Sigma_2\) 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 \(\Sigma_1\) and \(\Sigma_2\) that gives \(\mathcal D\) with \(G_2\) symmetry, we are unable to solve them in full generality so far.

MSC:
53C28 Twistor methods in differential geometry
70E18 Motion of a rigid body in contact with a solid surface
53C80 Applications of global differential geometry to the sciences
58A30 Vector distributions (subbundles of the tangent bundles)
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