Weyl–Schouten theorem
Encyclopedia
The Weyl–Schouten theorem in mathematics says that a Riemannian manifold
of dimension n with n ≥ 3 is conformally flat
if and only if the Schouten tensor is a Codazzi tensor for n = 3, or the Weyl tensor
vanishes for n > 3.
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
of dimension n with n ≥ 3 is conformally flat
Conformally flat
A 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 be a pseudo-Riemannian manifold...
if and only if the Schouten tensor is a Codazzi tensor for n = 3, or the Weyl tensor
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...
vanishes for n > 3.