An intrinsic approach to the control of rolling bodies

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

“…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.…”

“…For a general conformal structure c and choice g ∈ c of representative metric consider the Taylor series expansion (2). The first equation in [18, (3.18)] gives that (in dimension 5) the coëfficient μ (2) ab of the quadratic term is −B ab + P a c P bc , where (again specializing to dimension 5) P is the Schouten tensor of g, P ab = (R ab − g cd R cd g ab /8)/3, B is the Bach tensor of g, B ab = P b[c,a] c + P cd W cabd , and W is the Weyl tensor of g. Using the conformal invariance of W and the conformal transformation laws for B and P [18, pp.…”

“…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 .)…”