Automorphisms of Klein Surfaces of Algebraic Genus One

Year 2006, Volume: 1 Issue: 5, 71 - 78, 01.04.2006

Adnan Melekoğlu

Abstract

Keywords

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Automorphisms of Klein Surfaces of Algebraic Genus One

Abstract

Cebirsel cinsi bir olan Klein yüzeyleri; Möbius şeridi, silindir ve Klein şişesidir. Bu çalışmada bu yüzeylerin otomorfizmaları belirlenmiştir.Let X be a compact Riemann surface of genusg ≥ 1 . An automorphism of X is a conformal or anti-conformal homeomorphism f : X → X . X is called symmetric if it admits an anti-conformal involution s : X → X which we call a symmetry of X . The quotient surface S = X /〈s〉 is a Klein surface. By a Klein surface we mean a surface with a dianalytic structure (see [1]). Here X is called the complex double of S . The algebraic genus of S is then defined to be the topological genus of X . It is known that the Klein surfaces of algebraic genus one are the Möbius band, the annulus and the Klein bottle. In this paper we study the
automorphisms of these surfaces. We do not claim originality of the work. However, it contains something demonstrative of the method, not readily available in the literature, which may be helpful to those who are not experts but wish to understand the subject.

Keywords

Klein yüzeyi, Möbius şeridi, silindir, Klein şişesi, otomorfizma

References

• N.L. Alling and N. Greenleaf, Foundation of the Theory of Klein Surfaces. (Lecture Notes in Math. Vol. 219, Springer-verlag, 1971).
• I. Bârz_, ‘The Dianalytic Morphisms of the Klein Bottles’, (In: Complex Analysis, Lecture Notes in Math. Vol. 1351 (1988), pp.38-51.)
• G.A. Jones and D. Singerman, Complex Functions: An Algebraic and Geometric Viewpoint (Cambridge University Press, 1987).