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2012 | 10 | 5 | 1872-1888
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Geometry of isotypic Kronecker webs

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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.
Opis fizyczny
  • [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
  • [7] Frittelli S., Kozameh C., Newman E.T., Differential geometry from differential equations, Commun. Math. Phys., 2001, 223(2), 383–408
  • [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
  • [10] Jakubczyk B., Krynski W., Vector fields with distributions and invariants of ODEs, preprint available at
  • [11] Krynski W., Paraconformal structures and differential equations, Differential Geom. Appl., 2010, 28(5), 523–531
  • [12] Nagy P.T., Webs and curvature, In: Web Theory and Related Topics, Toulouse, December, 1996, World Scientific Publishing, River Edge, 2001, 48–91
  • [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
  • [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
  • [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
  • [17] Zakharevich I., Kronecker webs, bihamiltonian structures, and the method of argument translation, Transform. Groups, 2001, 6(3), 267–300
  • [18] Zakharevich I., Nonlinear wave equation, nonlinear Riemann problem and the twistor transform of Veronese webs, preprint available at
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