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2012 | 10 | 5 | 1872-1888
Tytuł artykułu

Geometry of isotypic Kronecker webs

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Języki publikacji
EN
Abstrakty
EN
An isotypic Kronecker web is a family of corank m foliations $\{ \mathcal{F}_t \} _{t \in \mathbb{R}P^1 } $ such that the curve of annihilators t ↦ (T x F t)⊥ ∈ Grm(T x* M) is a rational normal curve in the Grassmannian Grm(T x*M) at any point x ∈ M. For m = 1 we get Veronese webs introduced by I. Gelfand and I. Zakharevich [Gelfand I.M., Zakharevich I., Webs, Veronese curves, and bi-Hamiltonian systems, J. Funct. Anal., 1991, 99(1), 150–178]. In the present paper, we consider the problem of local classification of isotypic Kronecker webs and for a given web we construct a canonical connection. We compute the curvature of the connection in the case of webs of equal rank and corank. We also show the correspondence between Kronecker webs and systems of ODEs for which certain sets of differential invariants vanish. The equations are given up to contact transformations preserving independent variable. As a particular case, with m = 1 we obtain the correspondence between Veronese webs and ODEs.
Wydawca
Czasopismo
Rocznik
Tom
10
Numer
5
Strony
1872-1888
Opis fizyczny
Daty
wydano
2012-10-01
online
2012-07-24
Twórcy
Bibliografia
  • [1] Akivis M.A., Goldberg V.V., Differential geometry of webs, In: Handbook of Differential Geometry, Vol. I, North-Holland, Amsterdam, 2000, 1-152, Chapter 1
  • [2] Chern S.-S., Sur la géométrie d’un système d’équations différentialles du second ordre, Bull. Sci. Math., 1939, 63, 206–212
  • [3] Chern S., The geometry of higher path-spaces, J. Chinese Math. Soc., 1940, 2, 247–276
  • [4] Doubrov B., Komrakov B., Morimoto T., Equivalence of holonomic differential equations, Lobachevskii J. Math., 1999, 3, 39–71
  • [5] Dunajski M., Solitons, Instantons and Twistors, Oxf. Grad. Texts Math., 19, Oxford University Press, Oxford, 2010
  • [6] Dunajski M., Tod P., Paraconformal geometry of nth-order ODEs, and exotic holonomy in dimension four, J. Geom. Phys., 2006, 56(9), 1790–1809 http://dx.doi.org/10.1016/j.geomphys.2005.10.007
  • [7] Frittelli S., Kozameh C., Newman E.T., Differential geometry from differential equations, Commun. Math. Phys., 2001, 223(2), 383–408 http://dx.doi.org/10.1007/s002200100548
  • [8] Gamkrelidze R.V., Ed., Geometry. I, Encyclopaedia Math. Sci., 28, Springer, Berlin, 1991
  • [9] Gelfand I.M., Zakharevich I., Webs, Veronese curves, and bi-Hamiltonian systems, J. Funct. Anal., 1991, 99(1), 150–178 http://dx.doi.org/10.1016/0022-1236(91)90057-C
  • [10] Jakubczyk B., Krynski W., Vector fields with distributions and invariants of ODEs, preprint available at http://www.impan.pl/Preprints/p728.pdf
  • [11] Krynski W., Paraconformal structures and differential equations, Differential Geom. Appl., 2010, 28(5), 523–531 http://dx.doi.org/10.1016/j.difgeo.2010.05.003
  • [12] Nagy P.T., Webs and curvature, In: Web Theory and Related Topics, Toulouse, December, 1996, World Scientific Publishing, River Edge, 2001, 48–91 http://dx.doi.org/10.1142/9789812794581_0003
  • [13] Panasyuk A., Veronese webs for bi-Hamiltonian structures of higher corank, In: Poisson Geometry, Warsaw, August 3–15, 1998, Banach Center Publ., 51, Polish Academy of Sciences, Warsaw, 2000, 251–261
  • [14] Panasyuk A., On integrability of generalized Veronese curves of distributions, Rep. Math. Phys., 2002, 50(3), 291–297 http://dx.doi.org/10.1016/S0034-4877(02)80059-3
  • [15] Turiel F.-J., C ∞-équivalence entre tissus de Veronese et structures bihamiltoniennes, C. R. Acad. Sci. Paris Sér. I Math., 1999, 328(10), 891–894 http://dx.doi.org/10.1016/S0764-4442(99)80292-4
  • [16] Turiel F.-J., C ∞-classification des germes de tissus de Veronese, C. R. Acad. Sci. Paris Sér. I Math., 1999, 329(5), 425–428 http://dx.doi.org/10.1016/S0764-4442(00)88618-8
  • [17] Zakharevich I., Kronecker webs, bihamiltonian structures, and the method of argument translation, Transform. Groups, 2001, 6(3), 267–300 http://dx.doi.org/10.1007/BF01263093
  • [18] Zakharevich I., Nonlinear wave equation, nonlinear Riemann problem and the twistor transform of Veronese webs, preprint available at http://arxiv.org/abs/math-ph/0006001
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-012-0081-z
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