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Conformally flat

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Conformally flat

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 flat Riemannian manifold is conformally equivalent to the round sphere.

See also

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