A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.

In practice, the metric tensor, i.e., the geodesics the angles by moving from one to the other, as well as keeping the null geodesics unchanged, is known as the conformal factor is a point on the manifold.

and a smooth function is flat (i.e. the curvature.

Some authors use the definition of locally conformally flat and reserve the definition of conformally flat.

Examples

• Every manifold with constant sectional curvature is conformally flat.

Every 2-dimensional pseudo-Riemannian manifold is conformally flat.

• The line element of the two dimensional spherical coordinates, like the one used in the geographic coordinate system, and is not flat but with the stereographic projection is the distance from the origin of the flat space,.

• 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.

• The stereographic projection provides a coordinate system for the sphere in which conformal flatness is explicit, as the metric is proportional to the flat one.

  In general relativity, conformally flat manifolds are often used, for instance, in describing the Friedmann–Lemaître–Robertson–Walker metric. However, this is not always possible, as in the case of the Kerr metric, which does not admit any conformally flat slices.

• An example of conformally flat metric can be obtained by transforming the Kruskal–Szekeres coordinates for the Schwarzschild metric.

See also

• Weyl–Schouten theorem

• Conformal geometry

• Yamabe problem

References