Abstract
We show that classical Wilczynski-Se-ashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of non-linear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invariants and establish relationship of such equations with deformation theory of rational curves on complex algebraic surfaces.
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References
R. Bryant, Two exotic holonomies in dimension four, path geometries, and twistor theory, Proc. Symp. Pure Math. (1991), 53: 33–88.
E. Cartan, Sur les variétés à tconnexion projective, Bull. Soc. Math. France (1924), 52: 205–241.
S.-S. Chern, The geometry of the differential equation y‴=F(x, y, y′, y″, Sci. Rep. Nat. Tsing Hua Univ. (1950), 4: 97–111.
B. Doubrov, B. Komrakov, AND T. Morimoto, Equivalence of holonomic differential equations, Lobachevskij Journal of Mathematics (1999), 3: 39–71.
B. Doubrov, Contact trivialization of ordinary differential equations, Differential Geometry and Its applications, 2001, pp. 73–84.
M. Dunajski AND P. Todd, Paraconformal geometry of n-th order ODEs, and exotic holonomy in dimension four, J. Geom. Phys. (2006), 56: 1790–1809.
M. Fels, The equivalence problem for systems of second order ordinary differential equations, Proc. London Math. Soc. (1995), 71(1): 221–240.
M. Fels, The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations, Trans. Amer. Math. Soc. (1996),348: 5007–5029.
D. Grossman, Torsion-free path geometries and integrable second order ODE systems, Sel. Math., New Ser. (2000), 6: 399–442.
G.H. Halphen, Sur les invariants differentiels des courbes ganches, Journ. de l’Ecole Polytechnique (1880), 28: 1–25.
S. Lie, Klassifikation und Integration von gewönlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten, I-IV, Gesamelte Abhandlungen, Vol. 5, Leipzig-Teubner, 1924, S. 240–310, 362–427, 432–448.
P.J. Olver, Symmetry, invariants, and equivalence, New York, Springer Verlag, 1995.
H. Sato AND A.Y. Yoshikawa, Third order ordinary differential equations and Legendre connections, J. Math. Soc. Japan (1998), 50: 993–1013.
Yu. Se-ashi, On differential invariants of integrable finite type linear differential equations, Hokkaido Math. J. (1988), 17: 151–195.
Yu. Se-ashi, A geometric construction of Laguerre-Forsyth’s canonical forms of linear ordinary differential equations, Adv. Studies in Pure Math. (1993), 22: 265–297.
N. Tanaka, Geometric theory of ordinary differential equations, Report of Grant-in-Aid for Scientific Research MESC Japan, 1989.
E.J. Wilczynski, Projective differential geometry of curves and ruled surfaces, Leipzig, Teubner, 1905.
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Doubrov, B. (2008). Generalized Wilczynski Invariants for Non-Linear Ordinary Differential Equations. In: Eastwood, M., Miller, W. (eds) Symmetries and Overdetermined Systems of Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 144. Springer, New York, NY. https://doi.org/10.1007/978-0-387-73831-4_2
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DOI: https://doi.org/10.1007/978-0-387-73831-4_2
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