相关论文: G_2 and the "Rolling Distribution"

Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as…

In 1910, \'{E}lie Cartan famously realized the split real form of the exceptional Lie group $G_2$ as the symmetry group of the maximally symmetric rank 2 distribution on a 5-dimensional manifold with the small growth vector (2,3,5). In this…

``Rubber'' coated rolling bodies satisfy a no-twist in addition to the no slip satisfied by ``marble'' coated bodies. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the…

In his 1910 paper, \'Elie Cartan gave a tour-de-force solution to the (local) equivalence problem for generic rank 2 distributions on 5-manifolds, i.e. $(2,3,5)$-distributions. From a modern perspective, these structures admit equivalent…

Using a complex parametrisation of $su(2)$, we show a change of coordinates that maps the maximally symmetric rolling $(2,3,5)$-distribution to the flat Cartan distribution. This establishes the local equivalence between the maximally…

Using a parametrisation of $sl_2$ given by the second prolongation of the group action of unimodular fractional linear transformations as presented in an article of Clarkson and Olver, we find a Monge normal form describing the rolling of…

In the present paper, we study the infinitesimal symmetries of the model of two Riemannian manifolds $(M,g)$ and $(\hat M,\hat g)$ rolling without twisting or slipping. We show that, under certain genericity hypotheses, the natural bundle…

Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space $G / H$ with reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$, we consider rollings of $\mathfrak{m}$ over…

We discover a new example of a generic rank 2-distribution on a 5-manifold with a 6-dimensional transitive symmetry algebra, which is not present in Cartan's classical five variables paper. It corresponds to the Monge equation z' = y +…

We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic…

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with…

We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real simple Lie group in the rank 2 case. We prove that such representations are described by a conformal structure…

There are two well-known parabolic split $G_2$-geometries in dimension five, $(2,3,5)$-distributions and $G_2$-contact structures. Here we link these two geometries with yet another $G_2$-related contact structure, which lives on a…

We give a description of Nurowski's conformal structure for some examples of bracket-generating rank 2 distributions in dimension 5, aka $(2,3,5)$-distributions, namely the An-Nurowski circle twistor distribution for pairs of surfaces of…

Let $X$ be an $n$-dimensional Riemannian manifold with "large positive" scalar curvature. In this paper, we prove in a variety of cases that if $X$ "spreads" in $(n-2)$ directions {\it "distance-wise"}, then it {\it can't} much "spread" in…

Random operators constitute fundamental building blocks of models of complex systems yet are far from fully understood. Here, we explain an asymmetry emerging upon repeating identical isotropic (uniformly random) operations. Specifically,…

We investigate global solvability, in the framework of smooth functions and Schwartz distributions, of certain sums of squares of vector fields defined on a product of compact Riemannian manifolds $T \times G$, where $G$ is further assumed…

The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers…

We consider the maximally symmetric $(2,3,5)$-distribution given by the An-Nurowski circle twistor bundle over the product of an An-Nurowski surface and the plane. This circle twistor distribution encodes the configuration space of an…

Motivated by relating the representation theory of the split real and $p$-adic forms of a connected reductive algebraic group $G$, we describe a subset of $2^r$ orbits on the complex flag variety for a certain symmetric subgroup. (Here $r$…