Skip to main content
SpringerLink
Log in

Discretization of multidimensional submanifolds associated with spin-valued spectral problems

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We present a large family of Spin(p, q)-valued discrete spectral problems. The associated discrete nets generated by the so-called Sym-Tafel formula are circular nets (i.e., all elementary quadrilaterals are inscribed into circles). These nets are discrete analogues of smooth multidimensional immersions in ℝm including isothermic surfaces, Guichard nets, and some other families of orthogonal nets.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. J. Ablowitz, R. Beals, and K. Tenenblat, “On the solution of the generalized wave and generalized sine-Gordon equations,” Stud. Appl. Math., 74, 177–203 (1986).

    MATH  MathSciNet  Google Scholar 

  2. R. Abłamowicz and G. Sobczyk, eds., Lectures on Clifford (Geometric) Algebras and Applications, Birkhauser, Boston (2004).

    MATH  Google Scholar 

  3. L. V. Ahlfors, Complex Analysis, McGraw-Hill, New York (1953).

    MATH  Google Scholar 

  4. Yu. A. Aminov, “On immersions of regions of the n-dimensional Lobachevsky space into (2n − 1)-dimensional Euclidean space,” Dokl. Akad. Nauk SSSR, 236, 521–524 (1977).

    MathSciNet  Google Scholar 

  5. Yu. A. Aminov, “Isometric immersions of regions of the n-dimensional Lobachevsky space into (2n − 1)-dimensional Euclidean space,” Math. USSR Sb., 39, 359–386 (1981).

    Article  MATH  Google Scholar 

  6. W. Biernacki and J. L. Cieśliński, “A compact form of the Darboux-Bäcklund transformation for some spectral problems in Clifford algebras,” Phys. Lett. A, 288, 167–172 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Bobenko, “Discrete conformal maps and surfaces,” in: P. A. Clarkson and F. Nijhoff, eds., Symmetries and Integrability of Difference Equations, Cambridge Univ. Press, Cambridge (1999), pp. 97–108.

    Google Scholar 

  8. A. I. Bobenko, D. Matthes, and Yu. B. Suris, “Discrete and smooth orthogonal systems: C ∞-approximation,” Internat. Math. Res. Notices, No. 45, 2415–2459 (2003).

  9. A. I. Bobenko and U. Pinkall, “Discrete isothermic surfaces,” J. Reine Angew. Math., 475, 187–208 (1996).

    MATH  MathSciNet  Google Scholar 

  10. A. I. Bobenko and U. Pinkall, “Discrete surfaces with constant negative Gaussian curvature and the Hirota equation,” J. Differential Geom., 43, 527–611 (1996).

    MATH  MathSciNet  Google Scholar 

  11. A. I. Bobenko and U. Pinkall, “Discretization of surfaces and integrable systems,” in: A. I. Bobenko and R. Seiler, eds., Discrete Integrable Geometry and Physics, Oxford Univ. Press, Oxford (1999), pp. 3–58.

    Google Scholar 

  12. M. Brück, X. Du, J. Park, and C. L. Terng, The Submanifold Geometries Associated to Grassmanian Systems, arXiv:math.DG/0006216 (2000).

  13. J. Cieśliński, “A generalized formula for integrable classes of surfaces in Lie algebras,” J. Math. Phys., 38, 4255–4272 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Cieśliński, “The cross ratio and Clifford algebras,” Adv. Appl. Clifford Algebras, 7, 133–139 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Cieśliński, “The spectral interpretation of n-spaces of constant negative curvature immersed in ℝ2n−1,” Phys. Lett. A, 236, 425–430 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Cieśliński, “The Backlund transformation for discrete isothermic surfaces,” in: P. A. Clarkson and F. Nijhoff, eds., Symmetries and Integrability of Difference Equations, Cambridge Univ. Press., Cambridge (1999), pp. 109–121.

    Google Scholar 

  17. J. L. Cieśliński, “A class of linear spectral problems in Clifford algebras,” Phys. Lett. A, 267, 251–255 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  18. J. L. Cieśliński, “How isothermic surfaces helped to understand other integrable systems,” Rend. Sem. Mat. Messina, supplement Atti del Congresso Internazionale in onore di Pasquale Calapso, proceedings of a Messina conference of 1998, pp. 135–147 (2000).

  19. J. L. Cieśliński, “Geometry of submanifolds derived from Spin-valued spectral problems,” Theor. Math. Phys., 137, 1396–1405 (2003).

    Article  Google Scholar 

  20. J. Cieśliński, A. Doliwa, and P. M. Santini, “The integrable discrete analogues of orthogonal coordinate systems are multidimensional circular lattices,” Phys. Lett. A, 235, 480–488 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Doliwa, “Integrable multidimensional discrete geometry,” in: H. Aratyn and A. S. Sorin, eds., Integrable Hierarchies and Modern Physical Theories, Kluwer Academic (2001), pp. 355–389.

  22. A. Doliwa and P. M. Santini, “Multidimensional quadrilateral lattices are integrable,” Phys. Lett. A, 233, 365–372 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Doliwa, P. M. Santini, and M. Mañas, “Transformations of quadrilateral lattices,” J. Math. Phys., 41, 944–990 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  24. L. P. Eisenhart, Transformations of Surfaces, Princeton Univ. Press (1923).

  25. L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York (1960).

    MATH  Google Scholar 

  26. D. Ferus and F. Pedit, “Curved flats in symmetric spaces,” Manuscripta Math., 91, 445–454 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  27. U. Hertrich-Jeromin, “On conformally flat hypersurfaces and Guichard’s nets,” Beitr. Algebra Geom., 35, 315–331 (1994).

    MATH  MathSciNet  Google Scholar 

  28. P. Lounesto, Clifford Algebras and Spinors, 2nd edition, Cambridge Univ. Press, Cambridge (2001).

    MATH  Google Scholar 

  29. S. P. Novikov, S. V. Manakov, L. P. Pitaievsky, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Plenum, New York (1984).

    Google Scholar 

  30. C. Rogers and W. K. Schief, Backlund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, Cambridge Univ. Press, Cambridge (2002).

    Google Scholar 

  31. R. Sauer, Differenzengeometrie [in German], Springer, Berlin (1970).

    MATH  Google Scholar 

  32. W. K. Schief, “Isothermic surfaces in spaces of arbitrary dimension: Integrability, discretization and Backlund transformations. A discrete Calapso equation,” Stud. Appl. Math., 106, 85–137 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  33. A. Sym, “Soliton surfaces,” Lett. Nuovo Cimento, 33, 394–400 (1982).

    Article  MathSciNet  Google Scholar 

  34. A. Sym, “Soliton surfaces and their application. Soliton geometry from spectral problems,” in: R. Martini, ed., Geometric Aspects of the Einstein Equations and Integrable Systems, Lect. Notes Phys., Vol. 239, Springer, Berlin (1985), pp. 154–231.

    Google Scholar 

  35. K. Tenenblat and C. L. Terng, “Bäcklund theorem for n-dimensional submanifolds of ℝ 2n−1,” Ann. Math., 111, 477–490 (1980).

    Article  MathSciNet  Google Scholar 

  36. W. Wunderlich, “Zur Differenzengeometrie der Flächen konstanter negativer Krümmung,” Sitzungsber. Ak. Wiss., 160, 39–77 (1951).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instytut Fizyki Teoretycznej, Uniwersytet w Białymstoku, 15-424 Białystok, ul. Lipowa 41, Poland

    J. L. Cieśliński

Authors
  1. J. L. Cieśliński
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. L. Cieśliński.

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 253–262, 2006.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cieśliński, J.L. Discretization of multidimensional submanifolds associated with spin-valued spectral problems. J Math Sci 149, 1032–1038 (2008). https://doi.org/10.1007/s10958-008-0042-z

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-0042-z

Keywords

Search

Navigation