Conformally flat manifold
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 Euclidean Riemannian manifold is conformally equivalent to the round sphere.
- In general relativity conformally flat manifolds can often be used, for example to describe Friedmann–Lemaître–Robertson–Walker metric. However it was also shown that there are no conformally flat slices of the Kerr spacetime.
- 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.
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