Conformally flat manifold
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.
Examples
- 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 Euclidean Riemannian manifold is conformally equivalent to the round sphere.[1]
- In general relativity conformally flat manifolds can often be used, for example to describe Friedmann–Lemaître–Robertson–Walker metric.[2] However it was also shown that there are no conformally flat slices of the Kerr spacetime.[3]
gollark: Ah, so I can just do whatever, then.
gollark: .
gollark: Well this is useless
gollark: How does managing usernames work?
gollark: You do need to have admined me though!
References
- Kuiper, N. H. (1949). "On conformally flat spaces in the large". Annals of Mathematics. 50 (4): 916–924. doi:10.2307/1969587. JSTOR 1969587.
- Garecki, Janusz (2008). "On Energy of the Friedman Universes in Conformally Flat Coordinates". Acta Physica Polonica B. 39 (4): 781–797. arXiv:0708.2783. Bibcode:2008AcPPB..39..781G.
- Garat, Alcides; Price, Richard H. (2000-05-18). "Nonexistence of conformally flat slices of the Kerr spacetime". Physical Review D. 61 (12): 124011. arXiv:gr-qc/0002013. Bibcode:2000PhRvD..61l4011G. doi:10.1103/PhysRevD.61.124011. ISSN 0556-2821.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.