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Normal Forms for Submanifolds Under Group Actions

  • Peter J. OlverEmail author
Conference paper
  • 422 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 266)

Abstract

We describe computational algorithms for constructing the explicit power series expansions for normal forms of submanifolds under transformation groups. The procedure used to derive the coefficients relies on the recurrence formulae for differential invariants provided by the method of equivariant moving frames.

Keywords

Lie group Submanifold Normal form Moving frame Differential invariant Curvature Recurrence formula Invariantization 

References

  1. 1.
    Fels, M., Olver, P.J.: Moving coframes. II. regularization and theoretical foundations. Acta Appl. Math. 55, 127–208 (1999)Google Scholar
  2. 2.
    Olver, P.J., Pohjanpelto, J.: Moving frames for Lie pseudo-groups. Can. J. Math. 60, 1336–1386 (2008)Google Scholar
  3. 3.
    Olver, P.J., Pohjanpelto, J.: Differential invariant algebras of Lie pseudo-groups. Adv. Math. 222, 1746–1792 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bruce, J.W., Giblin, P.J.: Curves and Singularities. Cambridge University Press, Cambridge (1984)Google Scholar
  5. 5.
    Cipolla, R., Giblin, P.: Visual Motion of Curves and Surfaces. Cambridge University Press, Cambridge (2000)Google Scholar
  6. 6.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate texts in mathematics, 2nd edn., vol. 107. Springer, New York (1993)CrossRefGoogle Scholar
  7. 7.
    Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995)Google Scholar
  8. 8.
    Calabi, E., Olver, P.J., Shakiban, C., Tannenbaum, A., Haker, S.: Differential and numerically invariant signature curves applied to object recognition. Int. J. Comput. Vis. 26, 107–135 (1998)Google Scholar
  9. 9.
    Grim, A., O’Connor, T., Olver, P.J., Shakiban, C., Slechta, R., Thompson, R.: Automatic reassembly of three–dimensional jigsaw puzzles. Int. J. Image Graph. 16, 1650009 (2016)CrossRefGoogle Scholar
  10. 10.
    Grim, A., Shakiban, C.: Applications of signatures in diagnosing breast cancer. Minnesota J. Undergrad. Math. 1(1), 001 (2015)Google Scholar
  11. 11.
    Hoff, D., Olver, P.J.: Automatic solution of jigsaw puzzles. J. Math. Imaging Vis. 49, 234–250 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Marsland, S., McLachlan, R.I.: Möbius invariants of shapes and images. SIGMA: Symmetry Integr. Geom. Methods Appl. 12, 080 (2016)Google Scholar
  13. 13.
    Shakiban, C., Lloyd, P.: Signature curves statistics of DNA supercoils. In: Mladenov, I.M., Hirschfeld, A.C. (eds.) Geometry, Integrability and Quantization, vol. 5, pp. 203–210. Softex, Bulgaria (2004)Google Scholar
  14. 14.
    Hubert, E., Kogan, I.A.: Rational invariants of a group action. Construction and rewriting. J. Symb. Comput. 42, 203–217 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Guggenheimer, H.W.: Differential Geometry. McGraw-Hill, New York (1963)Google Scholar
  16. 16.
    Olver, P.J.: Joint invariant signatures. Found. Comput. Math. 1, 3–67 (2001)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Olver, P.J.: Lectures on moving frames. In: Levi, D., Olver, P., Thomova, Z., Winternitz, P. (eds.) Symmetries and Integrability of Difference Equations. London Math. Soc. Lecture Note Series, vol. 381, pp. 207–246. Cambridge University Press, Cambridge (2011)Google Scholar
  18. 18.
    Olver, P.J.: Moving frames and singularities of prolonged group actions. Selecta Math. 6, 41–77 (2000)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kim, P.: Invariantization of numerical schemes using moving frames. BIT 47, 525–546 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mansfield, E.L.: Algorithms for symmetric differential systems. Found. Comput. Math. 1, 335–383 (2001)Google Scholar
  21. 21.
    Olver, P.J.: Differential invariants of surfaces. Differ. Geom. Appl. 27, 230–239 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hubert, E., Olver, P.J.: Differential invariants of conformal and projective surfaces. SIGMA 3, 097 (2007)Google Scholar
  23. 23.
    Green, M.L.: The moving frame, differential invariants and rigidity theorems for curves in homogeneous spaces. Duke Math. J. 45, 735–779 (1978)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kogan, I.A., Olver, P.J.: Invariant Euler-Lagrange equations and the invariant variational bicomplex. Acta Appl. Math. 76, 137–193 (2003)Google Scholar
  25. 25.
    Gray, A., Abbena, E., Salamon, S.: Modern Differential Geometry of Curves and Surfaces with Mathematica, 3rd edn. Chapman & Hall/CRC, FL (2006)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA

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