Local equivalence of some maximally symmetric ( 2 , 3 , 5 ) -distributions II

Matthew Randall

Archivum Mathematicum (2025)

  • Issue: 1, page 9-41
  • ISSN: 0044-8753

Abstract

top
We show the change of coordinates that maps the maximally symmetric ( 2 , 3 , 5 ) -distribution given by solutions to the k = 2 3 and k = 3 2 generalised Chazy equation to the flat Cartan distribution. This establishes the local equivalence between the maximally symmetric k = 2 3 and k = 3 2 generalised Chazy distribution and the flat Cartan or Hilbert-Cartan distribution. We give the set of vector fields parametrised by solutions to the k = 2 3 and k = 3 2 generalised Chazy equation and the corresponding Ricci-flat conformal scale that bracket-generate to give the split real form of 𝔤 2 .

How to cite

top

Randall, Matthew. "Local equivalence of some maximally symmetric $(2,3,5)$-distributions II." Archivum Mathematicum (2025): 9-41. <http://eudml.org/doc/299899>.

@article{Randall2025,
abstract = {We show the change of coordinates that maps the maximally symmetric $(2,3,5)$-distribution given by solutions to the $k=\frac\{2\}\{3\}$ and $k=\frac\{3\}\{2\}$ generalised Chazy equation to the flat Cartan distribution. This establishes the local equivalence between the maximally symmetric $k=\frac\{2\}\{3\}$ and $k=\frac\{3\}\{2\}$ generalised Chazy distribution and the flat Cartan or Hilbert-Cartan distribution. We give the set of vector fields parametrised by solutions to the $k=\frac\{2\}\{3\}$ and $k=\frac\{3\}\{2\}$ generalised Chazy equation and the corresponding Ricci-flat conformal scale that bracket-generate to give the split real form of $\mathfrak \{g\}_2$.},
author = {Randall, Matthew},
journal = {Archivum Mathematicum},
keywords = {(2; 3; 5)-distributions; generalised Chazy equation},
language = {eng},
number = {1},
pages = {9-41},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Local equivalence of some maximally symmetric $(2,3,5)$-distributions II},
url = {http://eudml.org/doc/299899},
year = {2025},
}

TY - JOUR
AU - Randall, Matthew
TI - Local equivalence of some maximally symmetric $(2,3,5)$-distributions II
JO - Archivum Mathematicum
PY - 2025
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
IS - 1
SP - 9
EP - 41
AB - We show the change of coordinates that maps the maximally symmetric $(2,3,5)$-distribution given by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation to the flat Cartan distribution. This establishes the local equivalence between the maximally symmetric $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy distribution and the flat Cartan or Hilbert-Cartan distribution. We give the set of vector fields parametrised by solutions to the $k=\frac{2}{3}$ and $k=\frac{3}{2}$ generalised Chazy equation and the corresponding Ricci-flat conformal scale that bracket-generate to give the split real form of $\mathfrak {g}_2$.
LA - eng
KW - (2; 3; 5)-distributions; generalised Chazy equation
UR - http://eudml.org/doc/299899
ER -

References

top
  1. Ablowitz, M.J., Chakravarty, S., Halburd, R., 10.4310/AJM.1998.v2.n4.a1, Asian J. Math. 2 (1998), 1–6. (1998) MR1734122DOI10.4310/AJM.1998.v2.n4.a1
  2. Ablowitz, M.J., Chakravarty, S., Halburd, R., 10.1111/1467-9590.00121, Stud. Appl. Math. 103 (1999), 75–88. (1999) MR1697568DOI10.1111/1467-9590.00121
  3. An, D., Nurowski, P., 10.1007/s00220-013-1839-2, Comm. Math. Phys. 326 (2) (2014), 393–414. (2014) Zbl1296.53100MR3165459DOI10.1007/s00220-013-1839-2
  4. An, D., Nurowski, P., 10.1016/j.geomphys.2018.01.009, J. Geom. Phys. 126 (2018), 93–100. (2018) MR3766469DOI10.1016/j.geomphys.2018.01.009
  5. Baez, J.C., Huerta, J., 10.1090/S0002-9947-2014-05977-1, Trans. Amer. Math. Soc. 366 (10) (2014), 5257–5293. (2014) MR3240924DOI10.1090/S0002-9947-2014-05977-1
  6. Bor, G., Montgomery, R., 10.4171/lem/55-1-8, Enseign. Math. (2) 55 (1–2) (2009), 157–196. (2009) Zbl1251.70008MR2541507DOI10.4171/lem/55-1-8
  7. Cartan, E., Sur la structure des groupes simples finis et continus, C.R. Acad. Sci. Paris 116 (1893), 784–786. (1893) 
  8. Cartan, E., 10.24033/asens.618, Ann. Sci. Éc. Norm. Supér. (3) 27 (1910), 109–192. (1910) MR1509120DOI10.24033/asens.618
  9. Chazy, J., Sur les équations différentielles dont l’intégrale générale est uniforme et admet des singularités essentielles mobiles, C.R. Acad. Sci. Paris 149 (1909), 563–565. (1909) 
  10. Chazy, J., 10.1007/BF02393131, Acta Math. 34 (1911), 317–385. (1911) MR1555070DOI10.1007/BF02393131
  11. Clarkson, P.A., Olver, P.J., 10.1006/jdeq.1996.0008, J. Differential Equations 124 (1) (1996), 225–246. (1996) MR1368067DOI10.1006/jdeq.1996.0008
  12. Doubrov, B., Kruglikov, B., 10.1016/j.difgeo.2014.06.008, Differential Geom. Appl. 35 (suppl.) (2014), 314–322. (2014) MR3254311DOI10.1016/j.difgeo.2014.06.008
  13. Engel, F., Sur un group simple à quatorze paramètres, C.R. Acad. Sci. Paris 116 (1893), 786–788. (1893) 
  14. Engel, F., Zwei merkwürdige Gruppen des Raums von fünf Dimensionen, Jahresbericht der Deutschen Mathematiker-Vereinigung 8 (1900), 196–198. (1900) 
  15. Halphen, G.H., 10.1007/BF02422456, Acta Math. 3 (1883), 325–380, doi:10.1007/BF02422456. (1883) MR1554628DOI10.1007/BF02422456
  16. Krichever, I.M., Generalized elliptic genera and Baker-Akhiezer functions, Mat. Zametki 47 (2) (1990), 34–45 (Transl. Math. Notes 47, no. 1–2, (1990), 132–142). (1990) MR1048541
  17. Maier, R.S., Lamé polynomials, hyperelliptic reductions and Lamé band structure, Philos. Trans. Roy. Soc. A 366 (2008), 1115–1153. (2008) MR2377687
  18. Nurowski, P., Differential equations and conformal structures, J. Geom. Phys. 55 (2005), 19–49. (2005) Zbl1082.53024MR2157414
  19. Randall, M., Local equivalence of some maximally symmetric ( 2 , 3 , 5 ) -distributions I, arxiv:2108.04599. 
  20. Randall, M., Flat (2,3,5)-distributions and Chazy’s equations, SIGMA 12 (2016), 029. (2016) MR3475620
  21. Randall, M., Automorphisms and transformations of solutions to the generalised Chazy equation for various parameters, J. Differential Equations 268 (12) (2020), 7998–8025. (2020) MR4079026
  22. Randall, M., Schwarz triangle functions and duality for certain parameters of the generalised Chazy equation, New Zealand J. Math. 50 (2020), 181–205. (2020) MR4216442
  23. Randall, M., Automorphism of solutions to Ramanujan’s differential equations and other results, Kyushu J. Math. 75 (1) (2021), 77–94. (2021) MR4272372
  24. Randall, M., Nurowski’s conformal class of a maximally symmetric (2,3,5)-distribution and its Ricci-flat representatives, J. Nonlinear Math. Phys. 28 (1) (2021), 1–13. (2021) MR4250810
  25. Randall, M., A Monge normal form for the rolling distribution, Proc. Amer. Math. Soc. 151 (2) (2023), 853–863. (2023) MR4520032
  26. Randall, M., Local equivalence of some maximally symmetric rolling distributions and SU(2) Pfaffian systems, Hokkaido Math. J. 52 (1) (2023), 97–128. (2023) MR4612637
  27. Strazzullo, F., Symmetry Analysis of General Rank-3 Pfaffian Systems in Five Variables, Ph.D. thesis, Utah State University, 2009. (2009) MR2717829
  28. Whittaker, E.T., Watson, G.N., A Course of Modern Analysis, 4th ed., Cambridge Mathematical Library, Cambridge University Press, 1996. (1996) Zbl0951.30002MR1424469
  29. Willse, T., Highly symmetric 2-plane fields on 5-manifolds and Heisenberg 5-group holonomy, Differential Geom. Appl. 33 (suppl.) (2014), 81–111. (2014) MR3159952
  30. Willse, T., Cartan’s incomplete classification and an explicit ambient metric of holonomy G 2 * , Eur. J. Math. 4 (2) (2018), 622–638. (2018) MR3799160

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.