World Library  
Flag as Inappropriate
Email this Article

Conformally flat

Article Id: WHEBN0015897964
Reproduction Date:

Title: Conformally flat  
Author: World Heritage Encyclopedia
Language: English
Subject: Curved space
Publisher: World Heritage Encyclopedia

Conformally flat

A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.

More formally, let (M, g) be a pseudo-Riemannian manifold. Then (M, g) is conformally flat if for each point x in M, there exists a neighborhood U of x and a smooth function f defined on U such that (U, e2fg) is flat (i.e. the curvature of e2fg vanishes on U). The function f need not be defined on all of M.

Some authors use locally conformally flat to describe the above notion and reserve conformally flat for the case in which the function f is defined on all of M.


  • Every manifold with constant sectional curvature is conformally flat.
  • Every 2-dimensional pseudo-Riemannian manifold is conformally flat.
  • A 3-dimensional pseudo-Riemannian manifold is conformally flat if and only if the Cotton tensor vanishes.
  • An n-dimensional pseudo-Riemannian manifold for n ≥ 4 is conformally flat if and only if the Weyl tensor vanishes.
  • Every compact, simply connected, conformally flat Riemannian manifold is conformally equivalent to the round sphere.

See also

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.