Selecta Mathematica

, Volume 24, Issue 5, pp 4749–4780 | Cite as

A cohomological approach to immersed submanifolds via integrable systems

  • J. de LucasEmail author
  • A. M. Grundland
Open Access


A geometric approach to immersion formulas for soliton surfaces is provided through new cohomologies on spaces of special types of \({\mathfrak {g}}\)-valued differential forms. We introduce Poincaré-type lemmas for these cohomologies, which appropriately describe the integrability conditions of Lax pairs associated with systems of PDEs. Our methods clarify the structure and properties of the deformations and soliton surfaces for the aforesaid Lax pairs. Our findings allow for the generalization of the theory of soliton surfaces in Lie algebras to general soliton submanifolds. Techniques from the theory of infinite-dimensional jet manifolds and diffieties enable us to justify certain common assumptions of the theory of soliton surfaces. Theoretical results are illustrated through \({\mathbb {C}}P^{N-1}\) sigma models.


Cohomology \({\mathbb {C}}P^{N-1}\) sigma model Generalized symmetries \({\mathfrak {g}}\)-valued differential forms \({\mathfrak {g}}\)-valued de Rham cohomology Integrable systems Immersion formulas Soliton surfaces 

Mathematics Subject Classification

Primary 35Q53 Secondary 35Q58 53A05 



A.M. Grundland was partially supported by the research Grant ANR-11LABX-0056-LMHLabEX LMH (France) and from the NSERC (Canada). J. de Lucas and A.M. Grundland acknowledge partial support from Project MAESTRO DEC-2012/06/A/ST1/00256 of the National Science Center (Poland). This work was partially accomplished during the stay of A.M. Grundland and J. de Lucas at the École Normale Superieure de Cachan (CMLA). The authors would also like to thank CMLA for its hospitality and attention during their stay. Finally, we thank an anonymous referee for valuable comments to improve the paper.


  1. 1.
    Ablowitz, M.J.: Nonlinear Phenomena. Springer, Berlin (1982)Google Scholar
  2. 2.
    Babelon, O., Bernard, D., Talon, M.: Introduction to Classical Integrable Systems (Cambridge Monographs on Mathematical Physics). Cambridge University Press, Cambridge (2006)Google Scholar
  3. 3.
    Biernacki, W., Cieslinski, J.L.: A compact form of the Darboux-Backlund transformation for some spectral problems in Clifford algebras. Phys. Lett. A 288, 167–172 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bobenko, A.I.: Surfaces in terms of 2 by 2 matrices. Old and new integrable cases. In: Fordy, A.P., Wood, J.C. (eds.) Harmonic Maps and Integrable Systems. Aspects of Mathematics, vol. E 23. Vieweg+Teubner Verlag, Wiesbaden (1994)Google Scholar
  5. 5.
    Bobenko, A., Eitner, U.: Painlevé Equations in the Differential Geometry of Surfaces. Lecture Notes in Mathematics, vol. 1753. Springer, Berlin (2000)CrossRefGoogle Scholar
  6. 6.
    Bolton, J., Jensen, G.R., Rigoli, M., Woodward, L.M.: On conformal minimal immersions of \(S^2\) into \({\mathbb{C}}P^n\). Math. Ann. 279, 599–620 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cartan, E.: Sur la structure des groupes infinis de transformation in Les systèmes différentiels en Involution. Gauthier-Villars, Paris (1953)Google Scholar
  8. 8.
    Chavolin, J., Joanny, J.F., Zinn-Justin, J.: Liquids at Interfaces. Elsevier, Amsterdam (1989)Google Scholar
  9. 9.
    Chen, F.F.: Introduction to Plasma Physics and Controlled Fusion. Plasma Physics, vol. 1. Plenum Press, New York (1983)Google Scholar
  10. 10.
    Cieśliński, J.: A generalized formula for integrable classes of surfaces in Lie algebras. J. Math. Phys. 38, 4255–4272 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cieśliński, J.L.: Geometry of submanifolds derived from spin-valued spectral problem. J. Theor. Math. Phys. 137, 1396–1405 (2003)CrossRefGoogle Scholar
  12. 12.
    Cieśliński, J.L.: Pseudospherical surfaces on time scales: a geometric deformation and the spectral approach. J. Phys. A 40, 12525–12538 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    David, F., Ginsparg, P., Zinn-Justin, J.: Fluctuating Geometries in Statistical Mechanics and Field Theory. North-Holland, Amsterdam (1996)Google Scholar
  14. 14.
    Davydov, A.S.: Solitons in Molecular Systems. Kluwer, New York (1991)CrossRefGoogle Scholar
  15. 15.
    Dillen, F.J.E., Verstraelen, L.C.A.: Handbook of Differential Geometry. North-Holland, Amsterdam (2000)zbMATHGoogle Scholar
  16. 16.
    Din, A.M., Horváth, Z., Zakrzewski, W.J.: The Riemann-Hilbert problem and finite action \({\mathbb{C}}P^{N-1}\) solutions. Nucl. Phys. B 233, 269–288 (1984)CrossRefGoogle Scholar
  17. 17.
    Din, A.M., Zakrzewski, W.: General class of solutions in the \({\mathbb{C}}P^{N-1}\) model. Nucl. Phys. B 174, 397–406 (1980)CrossRefGoogle Scholar
  18. 18.
    Doliwa, A., Sym, A.: Constant mean curvature surfaces in \(E^3\) as an example of soliton surfaces. In: Nonlinear Evolution Equations and Dynamical Systems. World Scientific, River Edge, pp. 111–117 (1992)Google Scholar
  19. 19.
    Eichenherr, H.: \(SU(N)\) invariant nonlinear \(\sigma \) models. Nucl. Phys. B 146, 215–223 (1978)CrossRefGoogle Scholar
  20. 20.
    Fokas, A.S., Gel’fand, I.M.: Surfaces on Lie groups, on Lie algebras, and their integrability. Commun. Math. Phys. 177, 203–220 (1996)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Fokas, A.S., Gel’fand, I.M., Finkel, F., Liu, Q.M.: A formula for constructing infinitely many surfaces on Lie algebras and integrable equations. Sel. Math. 6, 347–375 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Giachetta, G., Mangiarotti, L., Sardanashvily, G.: Cohomology of the infinite-order jet space and the inverse problem. J. Math. Phys. 42, 4272–4282 (2001)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Goldstein, P.P., Grundland, A.M.: Invariant recurrence relations for \(CP^{N-1}\) models. J. Phys. A 43, 265206 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Golo, V.L., Perelomov, A.M.: Solution of the duality equations for the two-dimensional \(SU(N)\)-invariant chiral model. Phys. Lett. B 79, 112–113 (1978)CrossRefGoogle Scholar
  25. 25.
    Gross, D.J., Piran, T., Weinberg, S.: Two-Dimensional Quantum Gravity and Random Surfaces. World Scientific, Singapore (1992)Google Scholar
  26. 26.
    Grundland, A.M.: Soliton surfaces in the generalized symmetry approach. Theor. Math. Phys. 188, 1322–1333 (2016)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Grundland, A.M., Levi, D., Martina, L.: On immersion formulas for soliton surfaces. Acta Polytech. 56, 180–192 (2016)CrossRefGoogle Scholar
  28. 28.
    Grundland, A.M., Post, S.: Soliton surfaces associated with generalized symmetries of integrable equations. J. Phys. A 44, 165203 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Grundland, A.M., Post, S.: Surfaces immersed in Lie algebras associated with elliptic integrals. J. Phys. A 45, 015204 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Grundland, A.M., Post, S., Riglioni, D.: Soliton surfaces and generalized symmetries of integrable systems. J. Phys. A 47, 015201 (2014)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Grundland, A.M., Strasburger, A., Dziewa–Dawidczyk, D.: \({\mathbb{C}}P^N\) sigma models via the \(SU(2)\) coherent states approach, Banach Center Publications, Polish Academy of Sciences, 50th seminar ‘Sophus Lie’ 113 (2018)Google Scholar
  32. 32.
    Grundland, A.M., Strasburger, A., Zakrzewski, W.J.: Surfaces immersed in \({{\mathfrak{s}}}{{\mathfrak{u}}}(N+1)\) Lie algebras obtained from the \({\mathbb{C}}P^N\) sigma models. J. Phys. A 39, 9187 (2006)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Grundland, A.M., Yurduşen, I.: On analytic descriptions of two-dimensional surfaces associated with the \({\mathbb{C}}P^{N-1}\) sigma model. J. Phys. A 42, 172001 (2009)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Guo, X.R.: Three new \((2+1)-\)dimensional integrable systems and some related Darboux transformations. Commun. Theor. Phys. 65, 735–742 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Hélein, F.: Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2001)CrossRefGoogle Scholar
  36. 36.
    Hopf, H.: Über die Abbildungen der Dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104, 637–665 (1931)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Igonin, S., Krasilshchik, J.: On one-parametric families of Bcklund transformations. Adv. Stud. Pure Math. 37, 99–114 (2002)Google Scholar
  38. 38.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. (Wiley Classics Library). Wiley, New York (1996)Google Scholar
  39. 39.
    Konopelchenko, B.G., Landolfi, G.: Generalized Weierstrass representation for surfaces in multi-dimensional Riemann spaces. J. Geom. Phys. 29, 319–333 (1999)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Konopelchenko, B.G.: Induced surfaces and their integrable dynamics. Stud. Appl. Math. 96, 9–51 (1996)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Krasil’shchik, J., Verbovetsky, A.: Geometry of jet spaces and integrable systems. J. Geom. Phys. 61, 1633–1674 (2011)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Kruglikov, B., Lychagin, V.V.: Geometry of differential equations. In: Krupka, D., Saunders, D. (eds.) Handbook of Global Analysis, vol. 1214, pp. 725–771. Elsevier, Amsterdam (2008)CrossRefGoogle Scholar
  43. 43.
    Landolfi, G.: New results on the Canham–Helfrich membrane model via the generalized Weierstrass representation. J. Phys. A 36, 11937–11954 (2003)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Manakov, S.V., Santini, P.M.: Inverse scattering problem for vectors fields and the Cauchy problem for the heavenly equations. Phys. Lett. A 359, 613–619 (2006)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Manton, N., Sutcliffe, P.: Topological Solitons (Cambridge Monographs on Mathematical Physics). Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  46. 46.
    Marvan, M.: On the horizontal gauge cohomology and nonremovability of the spectral parameter. Acta Appl. Math. 72, 51–65 (2002)MathSciNetCrossRefGoogle Scholar
  47. 47.
    May, J.P.: A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1999)Google Scholar
  48. 48.
    Mikhailov, A.V.: Integrable Magnetic Models Soliton. Modern Problems in Condensed Matter, vol. 17, pp. 623–690. North-Holland, Amsterdam (1986)Google Scholar
  49. 49.
    Mikhailov, A.V., Shabat, A.B., Sokolov, V.V.: The symmetry approach to classification of integrable equations. What is integrability? In: Zakharov, V.E. (ed.) Nonlinear Dynamics, pp. 115–184. Springer, Berlin (1991)Google Scholar
  50. 50.
    Nelson, D., Piran, T., Weinberg, S.: Statistical Mechanics of Membranes and Surfaces. World Scientific, Singapore (1992)zbMATHGoogle Scholar
  51. 51.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)CrossRefGoogle Scholar
  52. 52.
    Ou-Yang, Z., Liu, J., Xie, Y.: Geometric Methods in Elastic Theory of Membranes in Liquid Crystal Phases. World Scientific, Singapore (1999)CrossRefGoogle Scholar
  53. 53.
    Polchinski, J., Strominger, A.: Effective string theory. Phys. Rev. Lett. 67, 1681–1684 (1991)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Rogers, C., Schief, W.K.: Bäcklund and Darboux Transformations. Geometry and Modern Applications in Soliton Theory. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2000)Google Scholar
  55. 55.
    Safran, S.A.: Statistical Thermodynamics of Surfaces, Interfaces, and Membranes. Frontiers of Physics, vol. 90. Westview Press, Boulder (2003)zbMATHGoogle Scholar
  56. 56.
    Sasaki, J.R.: General class of solutions of the complex Grassmannian and \({\mathbb{C}}P^{N-1}\) model. Phys. Lett. B 130, 69–72 (1983)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Sommerfeld, A.: Lectures on Theoretical Physics. Academic Press, New York (1952)zbMATHGoogle Scholar
  58. 58.
    Sym, A.: Soliton surfaces. Lett. Nuovo Cimento 33, 394–400 (1982)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Sym, A.: Soliton surfaces and their applications (soliton geometry from spectral problems). In: Geometric Aspect of the Einstein Equation and Integrable Systems. Lectures Notes in Physics, vol. 239, pp. 154–231. Springer, Berlin (1995)Google Scholar
  60. 60.
    Tafel, J.: Surfaces in \({\mathbb{R}}^3\) with prescribed curvature. J. Geom. Phys. 17, 381–390 (1995)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Urbantke, H.K.: The Hopf fibration-seven times in physics. J. Geom. Phys. 46, 125–150 (2003)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Vinogradov, A.M.: Cohomological Analysis of Partial Differential Equations and Secondary Calculus. Translations of Mathematical Monographs, vol. 204. American Mathematical Society, Providence (2001)CrossRefGoogle Scholar
  63. 63.
    Vinogradov, A.M., Krasil’shchik, I.S.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Translations of Mathematical Monographs, vol. 182. American Mathematical Society, Providence (1999)Google Scholar
  64. 64.
    Zakharov, V.E.: Dispersionless limit of integrable systems in 2+1 dimensions. In: Ercolani, N.M., Gabitov, I.R., Levermore, C.D., Serre, D. (eds.) Singular Limits of Dispersive Waves. NATO Advanced Study Institute, Series B: Physics, vol. 320. Plenum, New York (1994)Google Scholar
  65. 65.
    Zakharov, V.E., Mikhailov, A.V.: Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method. Sov. Phys. JETP 74, 1953–1973 (1978)MathSciNetGoogle Scholar
  66. 66.
    Zakrzewski, W.J.: Low-Dimensional Sigma Models. Adam Hilger, Bristol (1989)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematical Methods in PhysicsUniversity of WarsawWarsawPoland
  2. 2.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada
  3. 3.Department of Mathematics and Computer ScienceUniversité du Québec à Trois-RivièresTrois-RivièresCanada

Personalised recommendations