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

SIGMA 19 (2023), 058, 29 pages      arXiv:2301.09364
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class

Johnson Allen Kessy and Dennis The
Department of Mathematics and Statistics, UiT The Arctic University of Norway, 9037 Tromsø, Norway

Received April 07, 2023, in final form August 01, 2023; Published online August 10, 2023

The fundamental invariants for vector ODEs of order $\ge 3$ considered up to point transformations consist of generalized Wilczynski invariants and C-class invariants. An ODE of C-class is characterized by the vanishing of the former. For any fixed C-class invariant ${\mathcal U}$, we give a local (point) classification for all submaximally symmetric ODEs of C-class with ${\mathcal U} \not \equiv 0$ and all remaining C-class invariants vanishing identically. Our results yield generalizations of a well-known classical result for scalar ODEs due to Sophus Lie. Fundamental invariants correspond to the harmonic curvature of the associated Cartan geometry. A key new ingredient underlying our classification results is an advance concerning the harmonic theory associated with the structure of vector ODEs of C-class. Namely, for each irreducible C-class module, we provide an explicit identification of a lowest weight vector as a harmonic 2-cochain.

Key words: submaximal symmetry; system of ODEs; C-class equations; Cartan geometry.

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  1. Bryant R.L., Two exotic holonomies in dimension four, path geometries, and twistor theory, in Complex Geometry and Lie Theory (Sundance, UT, 1989), Proc. Sympos. Pure Math., Vol. 53, American Mathematical Society, Providence, 1991, 33-88.
  2. Čap A., On canonical Cartan connections associated to filtered G-structures, arXiv:1707.05627.
  3. Čap A., Doubrov B., The D., On C-class equations, Comm. Anal. Geom., to appear, arXiv:1709.01130.
  4. Čap A., Slovák J., Parabolic geometries I. Background and general theory, Math. Surveys Monogr., Vol. 154, American Mathematical Society, Providence, 2009.
  5. Cartan É., Les espaces généralisés et l'intégration de certaines classes d'équations différentielles, C. R. Hebd. Séances Acad. Sci. 206 (1938), 1689-1693.
  6. Casey S., Dunajski M., Tod P., Twistor geometry of a pair of second order ODEs, Comm. Math. Phys. 321 (2013), 681-701, arXiv:1203.4158.
  7. Doubrov B., Three-dimensional homogeneous spaces with non-solvable transformation groups, arXiv:1704.04393.
  8. Doubrov B., Contact trivialization of ordinary differential equations, in Differential Geometry and Its Applications (Opava, 2001), Math. Publ., Vol. 3, Silesian University Opava, Opava, 2001, 73-84.
  9. Doubrov B., Generalized Wilczynski invariants for non-linear ordinary differential equations, in Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 25-40, arXiv:math.DG/0702251.
  10. Doubrov B., Komrakov B., Morimoto T., Equivalence of holonomic differential equations, Lobachevskii J. Math. 3 (1999), 39-71.
  11. Doubrov B., Medvedev A., Fundamental invariants of systems of ODEs of higher order, Differential Geom. Appl. 35 (2014), suppl., 291-313, arXiv:1312.0574.
  12. Dunajski M., Tod P., Paraconformal geometry of $n$th-order ODEs, and exotic holonomy in dimension four, J. Geom. Phys. 56 (2006), 1790-1809, arXiv:math.DG/0502524.
  13. Godlinski M., Nurowski P., Geometry of third-order ODEs, arXiv:0902.4129.
  14. Godlinski M., Nurowski P., $\mathrm{GL}(2,\mathbb{R})$ geometry of ODE's, J. Geom. Phys. 60 (2010), 991-1027, arXiv:0710.0297.
  15. Grossman D.A., Torsion-free path geometries and integrable second order ODE systems, Selecta Math. (N.S.) 6 (2000), 399-442.
  16. Kessy J.A., The D., Symmetry gaps for higher order ordinary differential equations, J. Math. Anal. Appl. 516 (2022), 126475, 23 pages, arXiv:2110.03954.
  17. Kostant B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74 (1961), 329-387.
  18. Kruglikov B., The D., The gap phenomenon in parabolic geometries, J. Reine Angew. Math. 723 (2017), 153-215, arXiv:1303.1307.
  19. Kryński W., Paraconformal structures, ordinary differential equations and totally geodesic manifolds, J. Geom. Phys. 103 (2016), 1-19, arXiv:1310.6855.
  20. Lie S., Classification und integration von gewöhnlichen differentialgleichungen zwischen $x$, $y$, die eine Gruppe von Transformationen gestatten, Math. Ann. 32 (1888), 213-281.
  21. Lie S., Vorlesungen über continuirliche Gruppen mit geometrischen und anderen Anwendungen, Teubner, Leipzig, 1893.
  22. Medvedev A., Geometry of third order ODE systems, Arch. Math. (Brno) 46 (2010), 351-361.
  23. Medvedev A., Third order ODEs systems and its characteristic connections, SIGMA 7 (2011), 076, 15 pages, arXiv:1104.0965.
  24. Olver P.J., Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
  25. Schneider E., Projectable Lie algebras of vector fields in 3D, J. Geom. Phys. 132 (2018), 222-229, arXiv:1803.08878.
  26. Se-ashi Y., On differential invariants of integrable finite type linear differential equations, Hokkaido Math. J. 17 (1988), 151-195.
  27. Shah S.W., Mahomed F.M., Azad H., Symmetry classification of scalar $n$th order ordinary differential equations, arXiv:2208.10395.
  28. The D., On uniqueness of submaximally symmetric parabolic geometries, arXiv:2107.10500.
  29. The D., A Cartan-theoretic classification of multiply-transitive $(2,3,5)$-distributions, arXiv:2205.03387.
  30. Winther H., Minimal projective orbits of semi-simple Lie groups, arXiv:2302.12138.

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