# Geometry of isotypic Kronecker webs

Open Mathematics (2012)

• Volume: 10, Issue: 5, page 1872-1888
• ISSN: 2391-5455

top

## Abstract

top
An isotypic Kronecker web is a family of corank m foliations ${\left\{{ℱ}_{t}\right\}}_{t\in ℝ{P}^{1}}$ 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.

## How to cite

top

Wojciech Kryński. "Geometry of isotypic Kronecker webs." Open Mathematics 10.5 (2012): 1872-1888. <http://eudml.org/doc/269460>.

@article{WojciechKryński2012,
abstract = {An isotypic Kronecker web is a family of corank m foliations $\lbrace \mathcal \{F\}_t \rbrace _\{t \in \mathbb \{R\}P^1 \}$ 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.},
author = {Wojciech Kryński},
journal = {Open Mathematics},
keywords = {Kronecker web; Veronese web; Ordinary differential equation; ordinary differential equation; three-web},
language = {eng},
number = {5},
pages = {1872-1888},
title = {Geometry of isotypic Kronecker webs},
url = {http://eudml.org/doc/269460},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Wojciech Kryński
TI - Geometry of isotypic Kronecker webs
JO - Open Mathematics
PY - 2012
VL - 10
IS - 5
SP - 1872
EP - 1888
AB - An isotypic Kronecker web is a family of corank m foliations $\lbrace \mathcal {F}_t \rbrace _{t \in \mathbb {R}P^1 }$ 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.
LA - eng
KW - Kronecker web; Veronese web; Ordinary differential equation; ordinary differential equation; three-web
UR - http://eudml.org/doc/269460
ER -

## References

top
1. [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 Zbl0968.53001
2. [2] Chern S.-S., Sur la géométrie d’un système d’équations différentialles du second ordre, Bull. Sci. Math., 1939, 63, 206–212 Zbl65.1419.01
3. [3] Chern S., The geometry of higher path-spaces, J. Chinese Math. Soc., 1940, 2, 247–276 Zbl0063.00828
4. [4] Doubrov B., Komrakov B., Morimoto T., Equivalence of holonomic differential equations, Lobachevskii J. Math., 1999, 3, 39–71 Zbl0937.37051
5. [5] Dunajski M., Solitons, Instantons and Twistors, Oxf. Grad. Texts Math., 19, Oxford University Press, Oxford, 2010 Zbl1197.35209
6. [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 Zbl1096.53028
7. [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 Zbl1027.53080
8. [8] Gamkrelidze R.V., Ed., Geometry. I, Encyclopaedia Math. Sci., 28, Springer, Berlin, 1991
9. [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. [10] Jakubczyk B., Krynski W., Vector fields with distributions and invariants of ODEs, preprint available at http://www.impan.pl/Preprints/p728.pdf Zbl1271.34041
11. [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 Zbl05772555
12. [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. [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 Zbl0996.53053
14. [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 Zbl1042.53008
15. [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. [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. [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 Zbl0994.37034
18. [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

## 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.