Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-25T01:33:05.133Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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).Google 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.Google 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.Google 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.CrossRefGoogle 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.Google 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.Google 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.Google 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.Google Scholar
[27] Plebański, J. Some solutions of complex Einstein equations. J. Math. Phys. 16, No. 12 (1975), 2395–2402.Google 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.Google Scholar