Skip to main content
SpringerLink
Log in

Schwarz reflection geometry I: Continuous iteration of reflection

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract.

Differential equations are derived for a continous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry.

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. Calini, A., Langer, J.: Schwarz reflection geometry II. In preparation

  2. Cecil, T.E.: Lie Sphere Geometry. Springer, 1992

  3. Cohn, H.: Conformal mapping on Riemann surfaces. Dover Publications, Inc. 1967

  4. Davis, P.J.: The Schwarz Function and its Applications. The Carus Mathematical Monographs, No. 17, The Mathematical Association of America, 1974

  5. Guieu, L., Ovsienko, V.: Structures symplectiques sur les espaces de courbes projectives et affines. J. Geom. Phys. 16(2), 120–148 (1995)

    Article  MATH  Google Scholar 

  6. Kasner, E.: Geometry of conformal symmetry (Schwarzian reflection). Annals Math. 38, 873–879 (1937)

    MATH  Google Scholar 

  7. Kirillov, A.A.: Geometric Approach to discrete series unirreps for Vir. Journal de Mathémathiques pure et appliquees 77, 735–746 (1998)

    Google Scholar 

  8. Langer, J., Singer, D.A.: A decomposition theorem for diffeomorphisms of the circle and geodesic fields on Riemann surfaces of genus one. Invent. Math. 69, 229–242 (1982)

    MathSciNet  MATH  Google Scholar 

  9. Loos, O.: Symmetric Spaces I: General Theory. W. A. Benjamin, Inc. 1969

  10. Michor, P.W., Ratiu, T.S.: On the geometry of the Virasoro-Bott group. J. Lie Theory 8(2), 293–309 (1998)

    MATH  Google Scholar 

  11. Muciño-Raymundo, J.: Complex structures adapted to smooth vector fields. Math. Ann. 322, 229–265 (2002)

    Article  Google Scholar 

  12. Neretin, Y.A.: A complex semigroup that contains the group of diffeomorphisms of the circle. Funkt. Anal. i Prilozh. 21(2), 82–83 (1987)

    MATH  Google Scholar 

  13. Pressley, A., Segal, G.: Loop Groups, Oxford Science Publications, 1986

  14. Segal, G.: The definition of conformal field theory. Differential geometric methods in theoretical physics. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 250, Kluwer Acad. Publ., Dordrecht 1988, pp. 165–171

  15. Shapiro, H.S.: The Schwarz Function and its Generalization to Higher Dimensions. University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9, John Wiley and Sons, 1992

  16. Smale, S.: Regular curves on Riemannian manifolds. Trans. Am. Math. Soc. 87(2), 492–512 (1958)

    MATH  Google Scholar 

  17. Springer, G.: Introduction to Riemann surfaces. Addison-Wesley, 1957

  18. Strebel, K.: Quadratic Differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 5, Springer, 1984

  19. Wiegmann, P.B., Zabrodin, A.: Conformal maps and dispersionless integrable hierarchies. Commun. Math. Phys. 213, 523–538 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Charleston, Charleston, SC, 29424, USA

    Annalisa Calini

  2. Department of Mathematics, Case Western Reserve University, Cleveland, OH, 44106, USA

    Joel Langer

Authors
  1. Annalisa Calini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joel Langer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Annalisa Calini.

Additional information

Mathematics Subject Classification (1991): 53C35, 53A30, 30D05.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Calini, A., Langer, J. Schwarz reflection geometry I: Continuous iteration of reflection. Math. Z. 244, 775–804 (2003). https://doi.org/10.1007/s00209-003-0523-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-003-0523-1

Keywords

Search

Navigation