Skip to main content
SpringerLink
Log in

Notes on Transformations in Integrable Geometry

  • Chapter
  • First Online:
Special Metrics and Group Actions in Geometry

Part of the book series: Springer INdAM Series ((SINDAMS,volume 23))

Abstract

We describe the gauge-theoretic approach to transformations in integrable geometry through discussion of two classical examples: surface of constant negative Gauss curvature and isothermic surfaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    See [10, 14] for modern treatments.

References

  1. A.-V. Bäcklund, Om ytor med kostant negativ krökning, Lunds Universitets Årsskrift XIX (1883)

    Google Scholar 

  2. L. Bianchi, Ricerche sulle superficie elicoidali e sulle superficie a curvatura costante. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2, 285–341 (1879). MR1556595

    MathSciNet  Google Scholar 

  3. L. Bianchi, Sulle trasformazione di bäcklund per le superficie pseudosferiche. Rend. Lincei 5, 3–12 (1892)

    MATH  Google Scholar 

  4. L. Bianchi, Ricerche sulle superficie isoterme e sulla deformazione delle quadriche. Ann. di Mat. 11, 93–157 (1905)

    Article  MATH  Google Scholar 

  5. L. Bianchi, Complementi alle ricerche sulle superficie isoterme. Ann. di Mat. 12, 19–54 (1905)

    Article  MATH  Google Scholar 

  6. A. Bobenko, U. Pinkall, Discrete isothermic surfaces. J. Reine Angew. Math. 475, 187–208 (1996). MR1396732 (97f:53004)

    Google Scholar 

  7. A.I. Bobenko, W.K. Schief, Discrete indefinite affine spheres, in Discrete Integrable Geometry and Physics (Vienna, 1996) (1999), pp. 113–138. MR1676596 (2001e:53012)

    Google Scholar 

  8. E. Bour, Théorie de la déformation des surfaces, J. L’École Impériale Polytechnique, XXXIX, 1–148 (1862)

    Google Scholar 

  9. F.E. Burstall, Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems, in Integrable Systems, Geometry, and Topology (2006), pp. 1–82. MR2222512 (2008b:53006)

    Google Scholar 

  10. F.E. Burstall, D.M.J. Calderbank, Conformal submanifold geometry I–III (2010). Preprint, arXiv:1006.5700 [math.DG]

    Google Scholar 

  11. F.E. Burstall, D.M.J. Calderbank, Conformal submanifold geometry iv–v (in preparation)

    Google Scholar 

  12. F. Burstall, U. Hertrich-Jeromin, Harmonic maps in unfashionable geometries. Manuscripta Math. 108 (2), 171–189 (2002). MR1918585 (2003f:53114)

    Google Scholar 

  13. F.E. Burstall, Á.C. Quintino, Dressing transformations of constrained Willmore surfaces. Commun. Anal. Geom. 22 (3), 469–518 (2014). MR3228303

    Google Scholar 

  14. F. Burstall, F. Pedit, U. Pinkall, Schwarzian derivatives and flows of surfaces, in Differential Geometry and Integrable Systems (Tokyo, 2000) (2002), pp. 39–61. MR1955628 (2004f:53010)

    Google Scholar 

  15. F.E. Burstall, N.M. Donaldson, F. Pedit, U. Pinkall, Isothermic submanifolds of symmetric R-spaces. J. Reine Angew. Math. 660, 191–243 (2011). MR2855825

    MathSciNet  MATH  Google Scholar 

  16. P. Calapso, Sulle superficie a linee di curvatura isoterme. Rendiconti Circolo Matematico di Palermo 17, 275–286 (1903)

    Article  MATH  Google Scholar 

  17. E. Christoffel, Ueber einige allgemeine Eigenshaften der Minimumsflächen. Crelle’s J. 67, 218–228 (1867)

    Article  MathSciNet  Google Scholar 

  18. D.J. Clarke, Integrability in submanifold geometry, Ph.D. Thesis, 2012

    Google Scholar 

  19. L. Crane, Action of the loop group on the self-dual Yang-Mills equation. Commun. Math. Phys. 110 (3), 391–414 (1987). MR891944 (88i:58167)

    Google Scholar 

  20. G. Darboux, Leçcons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. Parts 1 and 2, Gauthier-Villars, Paris, 1887

    Google Scholar 

  21. G. Darboux, Sur les surfaces isothermiques. C.R. Acad. Sci. Paris 128, 1299–1305, 1538 (1899)

    Google Scholar 

  22. A. Demoulin, Sur les systèmes et les congruences K. C. R. Acad. Sci. Paris Sér. I Math. 150, 150, 156–159, 310–312 (1910)

    Google Scholar 

  23. E.V. Ferapontov, W.K. Schief, Surfaces of Demoulin: differential geometry, Bäcklund transformation and integrability. J. Geom. Phys. 30(4), 343–363 (1999). MR1700564 (2001i:53018)

    Google Scholar 

  24. S. Lie, Ueber flächen, deren krümmungsradien durch eine relation verknüpft sind. Archiv for Mathematik og Naturvidenskab IV, 507–512 (1879)

    Google Scholar 

  25. M. Pember, Special surface classes, Ph.D. Thesis, 2015

    Google Scholar 

  26. W.K. Schief, Isothermic surfaces in spaces of arbitrary dimension: integrability, discretization, and Bäcklund transformations—a discrete Calapso equation. Stud. Appl. Math. 106(1), 85–137 (2001). MR1805487 (2002k:37140)

    Google Scholar 

  27. A. Sym, Soliton surfaces and their applications (soliton geometry from spectral problems), in Geometric Aspects of the Einstein Equations and Integrable Systems (Scheveningen, 1984) (1985), pp. 154–231. MR828048 (87g:58056)

    Google Scholar 

  28. C.-L. Terng, Geometric transformations and soliton equations, in Handbook of Geometric Analysis, vol. 2 (Int. Press, Somerville, 2010), pp. 301–358. MR2743444

    Google Scholar 

  29. C.-L. Terng, K. Uhlenbeck, Bäcklund transformations and loop group actions. Commun. Pure Appl. Math. 53(1), 1–75 (2000). MR1715533 (2000k:37116)

    Google Scholar 

  30. K. Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model. J. Differ. Geom. 30(1), 1–50 (1989). MR1001271 (90g:58028)

    Google Scholar 

  31. R.S. Ward, Integrable and solvable systems, and relations among them. Philos. Trans. R. Soc. London Ser. A 315(1533), 451–457 (1985). With discussion, New developments in the theory and application of solitons. MR836745 (87e:58105)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Fran Burstall

Authors
  1. Fran Burstall
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fran Burstall .

Editor information

Editors and Affiliations

  1. GMA - IME, Universidade Federal Fluminense GMA - IME, Niteroi, Rio de Janeiro, Brazil

    Simon G. Chiossi

  2. Dipartimento di Mathematica "G. Peano", Università di Torino, Torino, Italy

    Anna Fino

  3. Dipartimento di Matematica e Informatica, Università di Firenze, Firenze, Firenze, Italy

    Emilio Musso

  4. Dipartimento di Scienze Matematiche, Politecnico di Torino, Torino, Italy

    Fabio Podestà

  5. Dipart. di Matematica, Giuseppe Peano, Università di Torino, Torino, Italy

    Luigi Vezzoni

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Burstall, F. (2017). Notes on Transformations in Integrable Geometry. In: Chiossi, S., Fino, A., Musso, E., Podestà, F., Vezzoni, L. (eds) Special Metrics and Group Actions in Geometry. Springer INdAM Series, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-67519-0_3

Download citation

Publish with us

Policies and ethics

Search

Navigation