Twistor Space for Rolling Bodies

Abstract

Abstract. 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.

Cartan’s incomplete classification and an explicit ambient metric of holonomy $$\mathrm{G}_2^*$$

“…First, only certain components g ab (ρ) have nonzero derivatives among them: Up to the symmetry g ab = g ba , only g 11 , g 13 , g 22 , g 25 , g 33 , g 34 are nonzero. 5 Second, the truncated Taylor series at ρ = 0 of g 25 , g 34 agree. Third, the truncated Taylor series at ρ = 0 of the coëfficients g 11 (ρ), g 13 (ρ), g 22 (ρ), g 33 (ρ) coincide with those of simple rational functions, namely, those in (6).…”

“…Some 103 years after Cartan's work, Doubrov and Govorov found a complex (2,3,5) distribution E that has 6-dimensional infinitesimal symmetry algebra but which is missing from Cartan's classification. Define a Lie algebra structure h on…”

“…Since this article was first uploaded to the arXiv, more examples were produced in [6], including some of the ambient metrics g f,h of conformal structures induced by (2,3,5) distributions D f,h given respectively in Monge normal form by the functions F f,h (x, y, p, q, z) . .…”

“…Suppose there are such f, h. We first show briefly that local equivalence of c E and c f,h implies local equivalence of E and D f,h : Theorem C of [29] states that if for two different (2,3,5) distributions D, D the induced conformal structures satisfy c D = c D then that conformal structure admits a so-called almost Einstein scale (see Section 2.5 of that reference), in which case Proposition A of that reference implies that the normal conformal holonomy of that conformal structure is a proper subgroup of G * 2 . On the other hand, Proposition 5.4 below shows that the metric holonomy Hol( g E ) is equal to G * 2 , and by [11,Corollary 1.2] it follows that the conformal holonomy Hol(c E ) of c E is the full group G * 2 , so c E does not admit an almost Einstein scale; hence E and D f,h are locally equivalent.…”

“…Such distributions that are maximally nonintegrable in the sense that This geometry has proved interesting for numerous reasons: It comprises a first class of distributions with continuous local invariants, it is connected to the geometry of surfaces rolling on one another without slipping or twisting [5,8,10] and hence to natural problems in control theory [2], it is intimately linked (as Cartan showed in the aforementioned article 1 ) to the exceptional simple complex Lie group of type G C 2 and then via the natural inclusion G * 2 → SO(3, 4) to conformal geometry [26], and it is the appropriate structure for naturally projectively compactifying strictly nearly (para-)Kähler structures in dimension 6 [21]. (Here, G * 2 denotes the automorphism group of the algebra of split octonions, one of the split real forms of the complex, connected simple Lie group G C 2 .)…”