Dancing polygons, rolling balls, and the Cartan-Engel distribution. (English) Zbl 07734501
Two seemingly unrelated themes are related in this paper. The first one refers to dancing pairs of polygons in the real projective plane, and the second one refers to spherical polygons with trivial rolling monodromy. A pair of planar polygons is called dancing if one is inscribed in the other and they satisfy a certain cross-ratio relation at each vertex of the circumscribing polygon. Dancing pairs correspond to trajectories of a non-holonomic mechanical system, consisting of a ball rolling, without slipping and twisting, along a polygon drawn on the surface of a ball three times larger than the rolling ball. Three models of the Cartan-Engel distribution and their interrelations are used in this paper: dancing pairs, rolling balls and split octonions. A diagram shows a diffeomorphism and more maps relating all these elements and it is proved that all the maps preserve the Cartan-Engel distributions on the respective spaces.
Reviewer: Gabriela Cristescu (Arad)
MSC:
| 53A20 | Projective differential geometry |
| 53A40 | Other special differential geometries |
| 53A55 | Differential invariants (local theory), geometric objects |
| 37J60 | Nonholonomic dynamical systems |
Keywords:
(2, 3, 5)-distribution; simple group \(G_2\); projective polygon pairs; rolling distributionReferences:
| [1] | Agrachev, Andrei A. Rolling balls and octonions.Proc. Steklov Inst. Math.258 (2007), no. 1, 13-22. ISBN:978-5-02-036672-5, ISBN:978-5-02-035888-1.MR2400520,Zbl 1153.70010,arXiv:math/0611812, doi:10.1134/S0081543807030030.985 · Zbl 1153.70010 |
| [2] | An, Daniel; Nurowski, Paweł. Twistor space for rolling bodies.Comm. Math. Phys.326(2014), no. 2, 393-414.MR3165459,Zbl 1296.53100,arXiv:1210.3536, doi:10.1007/s00220-013-1839-2.985 · Zbl 1296.53100 |
| [3] | Baez, John C.; Huerta, John.𝐺2and the rolling ball.Trans. Amer. Math. Soc.366(2014), no. 10, 5257-5293.MR3240924,Zbl 1354.20031,arXiv:1205.2447, doi:10.1090/S0002-9947-2014-05977-1.982,985,991,994,1008 · Zbl 1354.20031 |
| [4] | Bor, Gil; Hernández-Lamoneda, Luis; Nurowski, Paweł. The dancing metric,𝐺2symmetry and projective rolling.Trans. Amer. Math. Soc.370(2018), no. 6, 4433-4481. MR3811534,Zbl 1387.53021,arXiv:1506.00104, doi:10.1090/tran/7277.982,983,989, 990,992,1000,1008,1009 · Zbl 1387.53021 |
| [5] | Bor, Gil; Montgomery, Richard.𝐺2and the rolling distribution.Enseign. Math. (2)55(2009), no. 1-2, 157-196.MR2541507,Zbl 1251.70008,arXiv:math/0612469, doi:10.4171/LEM/55-1-8.982,985,989,991,1008,1012,1013 · Zbl 1251.70008 |
| [6] | Bryant, Robert L. Élie Cartan and geometric duality.Journées Élie Cartan 1998 et 1999 16(2000), 5-20.https://scholars.duke.edu/display/pub1139888.989 · Zbl 1263.53015 |
| [7] | Bryant, Robert L.; Hsu, Lucas. Rigidity of integral curves of rank2distributions.Invent. Math.114(1993), no. 2, 435-461.MR1240644,Zbl 0807.58007, doi:10.1007/BF01232676.982,985,989,990,991 · Zbl 0807.58007 |
| [8] | Cartan, Élie. Sur la structure des groupes finis et continus.C. R. Math. Acad. Sci. Paris 116(1893), 962-964.JFM 25,0637.02.989 · JFM 25.0637.02 |
| [9] | Cartan, Élie. Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre.Ann. Sci. École Norm. Sup. (3)27(1910), 109-192.MR1509120, JFM 41.0417.01.989 · JFM 41.0417.01 |
| [10] | Engel, Friedrich. Sur un groupe simple à quatorze paramètres.C. R. Math. Acad. Sci. Paris116(1893), 786-788.JFM 25.0631.01.989 · JFM 25.0631.01 |
| [11] | Jacobson, Nathan. Lie Algebras.Dover Publications, Inc., New York, 1979. ix+331 pp. ISBN: 0-486-63832-4.MR0559927.992,1009 · Zbl 0121.27504 |
| [12] | Kaplansky, Irving. Linear algebra and geometry. A second course.Allyn and Bacon, Inc.,Boston, Mass., 1969. xii+139 pp.MR0249444,Zbl 0184.24201.996 · Zbl 0184.24201 |
| [13] | Murphy,Bernard.Rollingcircles.AnanimationusingGeoGebra. https://www.geogebra.org/m/sehp2VmH.986 |
| [14] | Sagerschnig, Katja. Split octonions and generic rank two distributions in dimension five.Arch. Math. (Brno)42(2006), 329-339.MR2322419,Zbl 1164.53362.992 · Zbl 1164.53362 |
| [15] | Spherical geometry.Wikipedia.https://en.wikipedia.org/wiki/Spherical_geometry.999 |
| [16] | Zorn, Max A. Theorie der alternativen ringe.Abh. Math. Sem. Univ. Hamburg8(1931), no. 1, 123-147.MR3069547,Zbl 56.0140.01, doi:10.1007/BF02940993 · JFM 56.0140.01 |
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