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 flat Riemannian manifold is conformally equivalent to the round sphere.
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