# Double of a Riemann surface

A two-sheeted covering surface $ W $
of a finite Riemann surface $ R $.
Each interior point $ p \in R $
is brought into correspondence with a pair of points $ p $
and $ \widetilde{p} $
of the double $ W $;
in other words, two conjugate points $ p $
and $ \widetilde{p} $
are situated over $ p $.
Each point $ q $
of the boundary of $ R $
is brought into correspondence with a point $ q \in W $.
Moreover, two non-intersecting neighbourhoods of the points $ p , \widetilde{p} \in W $
are situated over each neighbourhood of an interior point $ p \in R $.
If $ z $
is a local uniformizing parameter in a neighbourhood of the interior point $ p \in R $,
it will also be a local uniformizing parameter in a $ W $-
neighbourhood of one out of the two conjugate points of $ W $
lying over $ p $,
say in a $ W $-
neighbourhood of the point $ p \in W $;
then, in a $ W $-
neighbourhood of the conjugate point $ \widetilde{p} $,
the complex conjugate $ \overline{z}\; $
of the variable $ z $
will be a local uniformizing parameter. If $ z $
is a local uniformizing parameter at a point $ q $
of the boundary of $ R $,
then the variable which is equal to $ z $
on one sheet of $ W $
and to $ \overline{z}\; $
on the other will be a local uniformizing parameter at the point $ q \in W $
lying over it.

In the case of a compact orientable Riemann surface $ R $, the double simply consists of two compact orientable Riemann surfaces, and its study is accordingly of no interest. In all other cases the double of the Riemann surface is a compact orientable Riemann surface. This fact permits one to simplify the study of certain problems in the theory of functions on $ R $ by reducing them to the study of functions on $ W $. The genus (cf. Genus of a surface) of $ W $ is $ g + m - 1 $, where $ g $ is the genus of $ R $ and $ m $ is the number of components of the boundary of $ R $, which are assumed to be non-degenerate. For instance, the double of a simply-connected plane domain is a sphere, while the double of an $ m $- connected plane domain is a sphere with $ m - 1 $ handles.

Analytic differentials on a Riemann surface (cf. Differential on a Riemann surface) $ R $ are analytic differentials on the double $ W $ characterized by the fact that they assume conjugate values at conjugate points of $ W $ and take real values at the points $ q \in W $ lying over points of the boundary of $ R $.

#### References

[1] | M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954) |

[2] | E. Picard, "Traité d'analyse" , 2 , Gauthier-Villars (1926) |

#### Comments

The process of constructing the double of a Riemann surface is called duplication. This process can be applied to any connected two-dimensional manifold $ M $ with boundary (cf. Boundary (of a manifold)) to yield a regular imbedding of $ M $ in a connected two-dimensional manifold (cf. [a1], § 13.H).

#### References

[a1] | L.V. Ahlfors, L. Sario, "Riemann surfaces" , Princeton Univ. Press (1974) |

[a2] | H.M. Farkas, I. Kra, "Riemann surfaces" , Springer (1980) |

[a3] | G. Springer, "Introduction to Riemann surfaces" , Addison-Wesley (1957) pp. Chapt.10 |

**How to Cite This Entry:**

Double of a Riemann surface.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Double_of_a_Riemann_surface&oldid=46772