Letters in Mathematical Physics

, Volume 89, Issue 2, pp 131–139 | Cite as

Integrable Discrete Nets in Grassmannians

  • Vsevolod Eduardovich Adler
  • Alexander Ivanovich Bobenko
  • Yuri Borisovich Suris


We consider discrete nets in Grassmannians \({\mathbb{G}^{d}_{r}}\), which generalize Q-nets (maps \({\mathbb{Z}^N\to\mathbb{P}^d}\) with planar elementary quadrilaterals) and Darboux nets (\({\mathbb{P}^d}\)-valued maps defined on the edges of \({\mathbb{Z}^N}\) such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.

Mathematics Subject Classification (2000)

15A03 37K25 


discrete differential geometry multidimensional consistency Grassmannian noncommutative Darboux system 


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Copyright information

© Springer 2009

Authors and Affiliations

  • Vsevolod Eduardovich Adler
    • 1
    • 2
  • Alexander Ivanovich Bobenko
    • 2
  • Yuri Borisovich Suris
    • 3
  1. 1.L.D. Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Institut für MathematikTechnische Universität BerlinBerlinGermany

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