• Corpus ID: 50818244

G2 and the Rolling Ball

@article{Baez2012G2AT,
  title={G2 and the Rolling Ball},
  author={John C. Baez and John Huerta},
  journal={arXiv: Differential Geometry},
  year={2012}
}
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 the symmetries of a ball rolling on a larger ball, but only when the ratio of radii is 1:3. Using the split octonions, we devise a similar, but more global, picture of G2: it acts as the symmetries of a 'spinorial ball rolling on a projective plane', again… 
View PDF on arXiv
35 Citations
Highly Influential Citations
1
Background Citations
19
Methods Citations
1
Results Citations
1

35 Citations

Citation Type

Citation Type

Local equivalence of some maximally symmetric rolling distributions and $SU(2)$ Pfaffian systems
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…
Singularities of tangent surfaces in Cartan's split G2-geometry
In the split G2-geometry, we study the correspondence found by E. Cartan between the Cartan distribution and the contact distribution with Monge structure on spaces of five variables. Then the…
Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy ☆
The Geodesics of Rolling Ball Systems
A well-known and interesting family of sub-Riemannian space are the systems involving two balls rolling against each other without slipping or twisting. In this note, we show how the sub-Riemannian…
Duality on Geodesics of Cartan Distributions and Sub-Riemannian Pseudo-Product Structures
Abstract Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown in (15), that, if the…
The double Cayley Grassmannian
We study the smooth projective symmetric variety of Picard number one that compactifies the exceptional complex Lie group G2, by describing it in terms of vector bundles on the spinor variety of…
Normal Form for Nonsemisimple Singularities
The theory of versal normal form has been playing a role in normal form since the introduction of the concept by V.I. Arnol’d in [1, 2]. But there has been no systematic use of it that is in line…
Local equivalence of some maximally symmetric $(2,3,5)$-distributions
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…
New relations between $G_2$-geometries in dimensions 5 and 7
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…
Nearly Kähler Geometry and $(2,3,5)$-Distributions via Projective Holonomy
We show that any dimension 6 nearly K\"ahler (or nearly para-K\"ahler) geometry arises as a projective manifold equipped with a $\mathrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we…
...
...

References

SHOWING 1-10 OF 31 REFERENCES
G_2 and the "Rolling Distribution"
Associated to the problem of rolling one surface along another there is a five-manifold M with a rank two distribution. If the two surfaces are spheres then M is the product of the rotation group…
Quaternions and octonions in Mechanics
In fact, this group is Spin(3), the 2-fold cover of SO(3), the group of rotations of R. This has been known for quite some time and is perhaps the simplest realization of Hamilton’s expectations…
Buildings and shadows
This is an expository paper describing geometries associated to the groups of Lie type, following Jacques Tits' theory of buildings and his earlier theory of shadows. The geometry of shadows in the…
Split octonions and generic rank two distributions in dimension five
In his famous five variables paper Elie Cartan showed that one can canonically associate to a generic rank 2 distribution on a 5 dimensional manifold a Cartan geometry modeled on the homogeneous…
On variational approach to differential invariants of rank two distributions
The Octonions
The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics.…
Buildings of Spherical Type and Finite BN-Pairs
These notes are a slightly revised and extended version of mim- graphed notes written on the occasion of a seminar on buildings and BN-pairs held at Oberwolfach in April 1968. Their main purpose is…
Élie Cartan and geometric duality
It is a great honor for me to be asked to give a lecture about the work of Élie Cartan at the institute that was founded in his name. When I was asked to do this, I was immediately beset by doubts as…
Geometric Quantization
We review the definition of geometric quantization, which begins with defining a mathematical framework for the algebra of observables that holds equally well for classical and quantum mechanics. We…
Buildings: Theory and Applications
TLDR
This book treats Jacques Tits's beautiful theory of buildings, making that theory accessible to readers with minimal background, and focuses on all three approachs to buildings, “old-fashioned," combinatorial (chamber systems), and metric so that the reader can learn all three or focus on only one.
...
...