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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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References:
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