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

See also

References

  1. Kuiper, N. H. (1949). "On conformally flat spaces in the large". Annals of Mathematics 50: 916–924. doi:10.2307/1969587. Retrieved March 3, 2016.


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