Geometry of isotypic Kronecker webs

Wojciech Kryński

Open Mathematics (2012)

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

Abstract

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An isotypic Kronecker web is a family of corank m foliations { ℱ t } t ∈ 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

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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

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