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On integrability in Grassmann geometries: integrable systems associated with fourfolds in $$\mathbf{Gr}(3,5)$$. (English) Zbl 1404.37090
Let $$\mathrm{Gr}(3,5)$$ be the Grassmannian of $$3$$-dimensional linear subspaces of a $$5$$-dimensional vector space $$V$$. The authors study an interesting relation between submanifolds $$X$$ of $$\mathrm{Gr}(3,5)$$ and certain systems $$\Sigma(X)$$ of partial differential equations. Examples include the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation and Plebańsky’s heavenly equations. Integrable properties of $$\Sigma(X)$$ are proved to be equivalent to differential geometric properties of the submanifold $$X$$.
From the point of view of integrable systems, the paper is strictly related to the more general results obtained by E. V. Ferapontov and K. R. Khusnutdinova [Commun. Math. Phys. 248, No. 1, 187–206 (2004; Zbl 1070.37047)], but the specialization to the above class of systems allows a more in-deep description of the geometric properties. Indeed, within this approach and for the class of equations considered, the authors are able to provide a geometric characterization of integrable systems and to prove the equivalence of the following four different methods of integrability: the method of hydrodynamic reductions, the method of dispersionless Lax pair, the method of integrability on solutions (based on Einstein-Weyl geometry) and a method of integrability based on the intrinsic geometry of the submanifold $$X$$ (and related to the twistor approach to integrability).

##### MSC:
 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 53A30 Conformal differential geometry (MSC2010) 53A40 Other special differential geometries 53B15 Other connections 53B25 Local submanifolds 53B50 Applications of local differential geometry to the sciences 53Z05 Applications of differential geometry to physics
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