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Webs and the Plebański equation

Published online by Cambridge University Press:  16 May 2016

WOJCIECH KRYŃSKI
WOJCIECH KRYŃSKI*
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, Śniadeckich 8, 00-959 Warszawa, Poland. e-mail: krynski@impan.pl
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Abstract

We consider 3-webs, hyper-para-complex structures and integrable Segre structures on manifolds of even dimension and generalise the second heavenly Plebański equation in the context of higher-dimensional hyper-para-complex structures. We also characterise the Segre structures admitting a compatible hyper-para-complex structure in terms of systems of ordinary differential equations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 3 , November 2016 , pp. 455 - 468
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Akivis, M. A. and Goldberg, V. V. Semiintegrable almost Grassmann structures. Differential Geom. Appl. 10 (3) (1999), 257–294.CrossRefGoogle Scholar
[2] Akivis, M. A. and Goldberg, V. V. Differential geometry of webs. In: Handbook of Differential Geometry, Vol. I (North-Holland, Amsterdam, 2000), 1–152, Chapter 1.Google Scholar
[3] Alekseevsky, D., Medori, C. and Tomassini, A. Homogeneous para-Kähler Einstein manifolds. Russ. Math. Surv. 64, No. 1 (2009).CrossRefGoogle Scholar
[4] Boyer, C. A note on hyperhermitian four-manifolds. Proc. Amer. Math. Soc. 102 (1988), 157–164.Google Scholar
[5] Calderbank, D. Integrable background geometries. SIGMA 10 (2014).Google Scholar
[6] Casey, S., Dunajski, M. and Tod, P. Twistor geometry of a pair of second order ODEs. Commun. Math Phys. 321 (2013), 681–701.CrossRefGoogle Scholar
[7] Chern, S.-S.. Sur la géométrie d'un systéme d'équations différentialles du second ordre. Bull. Sci. Math. 63 (1939), 206–212.Google Scholar
[8] Derdzinski, A. Connections with skew-symmetric Ricci tensor on surfaces. Results Math. 52, No. 3–4 (2008), 223–245.CrossRefGoogle Scholar
[9] Dunajski, M. A class of Einstein-Weyl spaces associated to an integrable system of hydrodynamic type. J. Geom. Phys. 51 (2004), 126–137.CrossRefGoogle Scholar
[10] Dunajski, M. and Kryński, W. Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs. Math. Proc. Camb. Phil. Soc. 157, Issue 01 (2014), 139–150.CrossRefGoogle Scholar
[11] Dunajski, M. and Kryński, W. Point invariants of third-order ODEs and hyper-CR Einstein-Weyl structures. J. Geom. Phys. 86 (2014), 296–302.CrossRefGoogle Scholar
[12] Dunajski, M. and West, S. Anti-self-dual conformal structures with null Killing vectors from projective structures. Commun. Math. Phys. 272 (2007), 85–118.CrossRefGoogle Scholar
[13] Dunajski, M. and West, S. Anti-self-dual conformal structures in neutral signature. Recent developments in pseudo-Riemannian geometry, 113148, ESI Lect. Math. Phys., Eur. Math. Soc., Zurich, 2008.Google Scholar
[14] Ferapontov, E. and Kruglikov, B. Dispersionless integrable systems in 3D and Einstein–Weyl geometry. J. Differential Geom. 97, No. 2 (2014), 215–254.CrossRefGoogle Scholar
[15] Finley, J. and Plebański, J. Further heavenly metrics and their symmetries. J. Math. Phys. 17, No. 4 (1976), 585–596.CrossRefGoogle Scholar
[16] Gelfand, I. M. and Zakharevich, I. Webs, Veronese curves, and bi-Hamiltonian systems. J. Funct. Anal. 99, no. 1 (1991), 150–178.CrossRefGoogle Scholar
[17] Grossman, D. A. Torsion-free path geometries and integrable second order ODE systems. Selecta Mathematica. 6, Issue 4, (2000), 399–342.CrossRefGoogle Scholar
[18] Jakubczyk, B. and Kryński, W. Vector fields with distributions and invariants of ODEs. J. Geometric Mechanics. 5, No. 1 (2013).Google Scholar
[19] Kruglikov, B. and Morozov, O. SDiff(2) and uniqueness of the Plebański equation. J. Mathematical Physics. 53, No. 8 (2012).CrossRefGoogle Scholar
[20] Kryński, W. Geometry of isotypic Kronecker webs. Central European J. Mathe. 10 (2012) 1872–1888.CrossRefGoogle Scholar
[21] Kryński, W. Webs and projective structures on a plane. Differential Geom. Appl. 37 (2014) 133–140.CrossRefGoogle Scholar
[22] Kryński, W. Paraconformal structures, ordinary differential equations and totally geodesic manifolds. J. Geom. Phys. 103 (2016) doi:10.1016/j.geomphys.2016.01.003.CrossRefGoogle Scholar
[23] Mettler, T. Reduction of β-integrable 2-Segre structures. Comm. Anal. Geom. 21, No. 2 (2013), 331–353.CrossRefGoogle Scholar
[24] Nagy, P. Webs and curvature. In Web Theory and Related Topics. (Toulouse, December, 1996) (World Scientific Publishing, River Edge, 2001), 48–91.CrossRefGoogle Scholar
[25] Panasyuk, A. Veronese webs for bi-Hamiltonian structures of higher corank. In Poisson Geometry (Warsaw, August 315, 1998), Banach Center Publ. 51 (2000) 251–261.Google Scholar
[26] Penrose, R. Nonlinear gravitons and curved twistor theory. Gen. Relativity a Gravitation. 7 (1976), 31–52.CrossRefGoogle Scholar
[27] Plebański, J. Some solutions of complex Einstein equations. J. Math. Phys. 16, No. 12 (1975), 2395–2402.CrossRefGoogle Scholar
[28] Randall, M. Local obstructions to projective surfaces admitting skew-symmetric Ricci tensor. J. Geom. Phys. 76 (2014), 192–199.CrossRefGoogle Scholar
[29] Zakharevich, I. Nonlinear wave equation, nonlinear Riemann problem, and the twistor transform of Veronese webs. arXiv:math-ph/0006001 (2000).Google Scholar
[30] Zakharevich, I. Kronecker webs, bihamiltonian structures and the method of argument translation. Transform. Groups. 6, No. 3 (2001), 267–300.CrossRefGoogle Scholar