Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-z5d2w Total loading time: 0.455 Render date: 2021-11-27T12:42:35.178Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Introduction

Published online by Cambridge University Press:  04 August 2010

E. Bujalance
Affiliation:
Universidad National de Educación a Distancia, Madrid
A. F. Costa
Affiliation:
Universidad National de Educación a Distancia, Madrid
E. Martínez
Affiliation:
Universidad National de Educación a Distancia, Madrid
Get access

Summary

Riemann surfaces have played a central role in mathematics ever since their introduction by Riemann in his dissertation in 1851; for a biography of Riemann, see Riemann, topology and physics, Birkhäuser, 1987 by M. Monastyrsky. Following Riemann, we first consider a Riemann surface to be the natural maximal domain of some analytic function under analytic continuation, and this point of view enables one to put the theory of ‘manyvalued functions’ on a firm foundation. However, one soon realises that Riemann surfaces are the natural spaces on which one can study complex analysis and then an alternative definition presents itself, namely that a Riemann surface is a one dimensional complex manifold. This is the point of view developed by Weyl in his classic text (The concept of a Riemann surface, Addison-Wesley, 1964) and this idea leads eventually on to the general theory of manifolds. These two points of view raise interesting questions. If we start with a Riemann surface as an abstract manifold, how do we know that it supports analytic functions? On the other hand, if we develop Riemann surfaces from the point of view of analytic continuation, how do we know that in this way we get all complex manifolds of one (complex) dimension? Fortunately, it turns out that these two different views of a Riemann surface are indeed identical.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Send book to Kindle

To send this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Introduction
  • Edited by E. Bujalance, Universidad National de Educación a Distancia, Madrid, A. F. Costa, Universidad National de Educación a Distancia, Madrid, E. Martínez, Universidad National de Educación a Distancia, Madrid
  • Book: Topics on Riemann Surfaces and Fuchsian Groups
  • Online publication: 04 August 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511569272.002
Available formats
×

Send book to Dropbox

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox.

  • Introduction
  • Edited by E. Bujalance, Universidad National de Educación a Distancia, Madrid, A. F. Costa, Universidad National de Educación a Distancia, Madrid, E. Martínez, Universidad National de Educación a Distancia, Madrid
  • Book: Topics on Riemann Surfaces and Fuchsian Groups
  • Online publication: 04 August 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511569272.002
Available formats
×

Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

  • Introduction
  • Edited by E. Bujalance, Universidad National de Educación a Distancia, Madrid, A. F. Costa, Universidad National de Educación a Distancia, Madrid, E. Martínez, Universidad National de Educación a Distancia, Madrid
  • Book: Topics on Riemann Surfaces and Fuchsian Groups
  • Online publication: 04 August 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511569272.002
Available formats
×