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, e^2fg) is flat (i.e. the curvature of e^2fg 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 flat Riemannian manifold is conformally equivalent to the round sphere.^[1]

See also

• Weyl–Schouten theorem
• conformal geometry

References

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