Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 008, 28 pages      arXiv:1807.02734      https://doi.org/10.3842/SIGMA.2019.008

Homogeneous Real (2,3,5) Distributions with Isotropy

Travis Willse
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria

Received August 15, 2018, in final form January 26, 2019; Published online February 04, 2019

Abstract
We classify multiply transitive homogeneous real (2,3,5) distributions up to local diffeomorphism equivalence.

Key words: (2,3,5) distributions; generic distributions; homogeneous spaces; rolling distributions.

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References

  1. Agrachev A.A., Rolling balls and octonions, Proc. Steklov Inst. Math. 258 (2007), 13-22, arXiv:math.OC/0611812.
  2. An D., Nurowski P., Twistor space for rolling bodies, Comm. Math. Phys. 326 (2014), 393-414, arXiv:1210.3536.
  3. An D., Nurowski P., Symmetric (2,3,5) distributions, an interesting ODE of 7th order and Plebański metric, J. Geom. Phys. 126 (2018), 93-100, arXiv:1302.1910.
  4. Baez J.C., Huerta J., ${\rm G}_2$ and the rolling ball, Trans. Amer. Math. Soc. 366 (2014), 5257-5293, arXiv:1205.2447.
  5. Bor G., Montgomery R., ${\rm G}_2$ and the rolling distribution, Enseign. Math. 55 (2009), 157-196, arXiv:math.DG/0612469.
  6. Bryant R.L., Hsu L., Rigidity of integral curves of rank 2 distributions, Invent. Math. 114 (1993), 435-461.
  7. Čap A., Sagerschnig K., On Nurowski's conformal structure associated to a generic rank two distribution in dimension five, J. Geom. Phys. 59 (2009), 901-912, arXiv:0710.2208.
  8. Čap A., Slovák J., Parabolic geometries. I. Background and general theory, Mathematical Surveys and Monographs, Vol. 154, Amer. Math. Soc., Providence, RI, 2009.
  9. Cartan E., 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.
  10. Cartan E., Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes, Bull. Soc. Math. France 42 (1914), 12-48.
  11. Dave S., Haller S., On 5-manifolds admitting rank two distributions of Cartan type, Trans. Amer. Math. Soc. 371 (2019), 4911-4929, arXiv:1603.09700.
  12. Doubrov B., Govorov A., A new example of a generic 2-distribution on a 5-manifold with large symmetry algebra, arXiv:1305.7297.
  13. Doubrov B., Govorov A., Unpublished notes.
  14. Doubrov B., Holland J., Sparling G., Spacetime and ${\rm G}_2$, arXiv:0901.0543.
  15. Doubrov B., Kruglikov B., On the models of submaximal symmetric rank 2 distributions in 5D, Differential Geom. Appl. 35 (2014), 314-322, arXiv:1311.7057.
  16. Doubrov B., Medvedev A., The D., Homogeneous Levi non-degenerate hypersurfaces in ${\mathbb C}^3$, arXiv:1711.02389.
  17. Doubrov B., Zelenko I., Geometry of rank 2 distributions with nonzero Wilczynski invariants, J. Nonlinear Math. Phys. 21 (2014), 166-187, arXiv:1301.2797.
  18. Goursat E., Leçons sur le problème de Pfaff, Librairie Scientifique J. Hermann, Paris, 1922.
  19. Gover A.R., Panai R., Willse T., Nearly Kähler geometry and (2,3,5)-distributions via projective holonomy, Indiana Univ. Math. J. 66 (2017), 1351-1416, arXiv:1403.1959.
  20. Graham C.R., Willse T., Parallel tractor extension and ambient metrics of holonomy split $G_2$, J. Differential Geom. 92 (2012), 463-505, arXiv:1109.3504.
  21. Hammerl M., Sagerschnig K., Conformal structures associated to generic rank 2 distributions on 5-manifolds - characterization and Killing-field decomposition, SIGMA 5 (2009), 081, 29 pages, arXiv:0908.0483.
  22. Hammerl M., Sagerschnig K., The twistor spinors of generic 2- and 3-distributions, Ann. Global Anal. Geom. 39 (2011), 403-425, arXiv:1004.3632.
  23. Hilbert D., Über den begriff der klasse von differentialgleichungen, Math. Ann. 73 (1912), 95-108.
  24. Ishikawa G., Kitagawa Y., Tsuchida A., Yukuno W., Duality of (2,3,5)-distributions and Lagrangian cone structures, arXiv:1808.00149.
  25. Ishikawa G., Kitagawa Y., Yukuno W., Duality of singular paths for (2,3,5)-distributions, J. Dyn. Control Syst. 21 (2015), 155-171, arXiv:1308.2501.
  26. Kruglikov B., The D., The gap phenomenon in parabolic geometries, J. Reine Angew. Math. 723 (2017), 153-215, arXiv:1303.1307.
  27. Leistner T., Nurowski P., Ambient metrics with exceptional holonomy, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), 407-436, arXiv:0904.0186.
  28. Leistner T., Nurowski P., Sagerschnig K., New relations between ${\rm G}_2$ geometries in dimensions 5 and 7, Internat. J. Math. 28 (2017), 1750094, 46 pages, arXiv:1601.03979.
  29. Nurowski P., Differential equations and conformal structures, J. Geom. Phys. 55 (2005), 19-49, arXiv:math.DG/0406400.
  30. Randall M., Flat (2,3,5)-distributions and Chazy's equations, SIGMA 12 (2016), 029, 28 pages, arXiv:1506.02473.
  31. Sagerschnig K., Split octonions and generic rank two distributions in dimension five, Arch. Math. (Brno) 42 (2006), 329-339.
  32. Sagerschnig K., Weyl structures for generic rank two distributions in dimension five, Ph.D. Thesis, Universität Wien, 2008.
  33. Sagerschnig K., Willse T., The almost Einstein operator for (2,3,5) distributions, Arch. Math. (Brno) 53 (2017), 347-370, arXiv:1705.00996.
  34. Sagerschnig K., Willse T., The geometry of almost Einstein (2,3,5) distributions, SIGMA 13 (2017), 004, 56 pages, arXiv:1606.01069.
  35. Strazzullo F., Symmetry analysis of general rank-3 Pfaffian systems in five variables, Ph.D. Thesis, Utah State University, 2009, available at http://digitalcommons.usu.edu/etd/449/.
  36. Willse T., Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy, Differential Geom. Appl. 33 (2014), 81-111, arXiv:1302.7163.
  37. Willse T., Cartan's incomplete classification and an explicit ambient metric of holonomy ${\rm G}_2^*$, Eur. J. Math. 4 (2018), 622-638, arXiv:1411.7172.

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