Conformally flat manifold

A (<a href="/facts/Pseudo-Riemannian_manifold/VFXsVjqb">pseudo</a>-)<a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> is conformally flat if each point has a neighborhood that can be mapped to <a href="/facts/Flat_manifold/YBQfE8mn">flat space</a> by a <a href="/facts/Conformal_transformation/cEpQnMad">conformal transformation</a>.
In practice, the <a href="/facts/Metric_tensor/tbqek1gJ">metric</a> 
 
 
 
 g
 
 
 {\displaystyle g}
 
 of the manifold 
 
 
 
 M
 
 
 {\displaystyle M}
 
 has to be conformal to the flat metric 
 
 
 
 η
 
 
 {\displaystyle \eta }
 
, i.e., the <a href="/facts/Geodesic/fbWR6l80">geodesics</a> maintain in all points of 
 
 
 
 M
 
 
 {\displaystyle M}
 
 the angles by moving from one to the other, as well as keeping the null geodesics unchanged, that means there exists a function 
 
 
 
 λ
 (
 x
 )
 
 
 {\displaystyle \lambda (x)}
 
 such that 
 
 
 
 g
 (
 x
 )
 =
 
 λ
 
 2
 
 
 (
 x
 )
 
 η
 
 
 {\displaystyle g(x)=\lambda ^{2}(x)\,\eta }
 
, where 
 
 
 
 λ
 (
 x
 )
 
 
 {\displaystyle \lambda (x)}
 
 is known as the <a href="/facts/Conformal_geometry/tuU181cE">conformal factor</a> and 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is a point on the manifold.
More formally, let 
 
 
 
 (
 M
 ,
 g
 )
 
 
 {\displaystyle (M,g)}
 
 be a pseudo-Riemannian manifold. Then 
 
 
 
 (
 M
 ,
 g
 )
 
 
 {\displaystyle (M,g)}
 
 is conformally flat if for each point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 in 
 
 
 
 M
 
 
 {\displaystyle M}
 
, there exists a neighborhood 
 
 
 
 U
 
 
 {\displaystyle U}
 
 of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and a <a href="/facts/Smooth_function/mffL4ch2">smooth function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 defined on 
 
 
 
 U
 
 
 {\displaystyle U}
 
 such that 
 
 
 
 (
 U
 ,
 
 e
 
 2
 f
 
 
 g
 )
 
 
 {\displaystyle (U,e^{2f}g)}
 
 is <a href="/facts/Flat_manifold/YBQfE8mn">flat</a> (i.e. the <a href="/facts/Riemann_curvature_tensor/KyeHCWEh">curvature</a> of 
 
 
 
 
 e
 
 2
 f
 
 
 g
 
 
 {\displaystyle e^{2f}g}
 
 vanishes on 
 
 
 
 U
 
 
 {\displaystyle U}
 
). The function 
 
 
 
 f
 
 
 {\displaystyle f}
 
 need not be defined on all of 
 
 
 
 M
 
 
 {\displaystyle M}
 
.
Some authors use the definition of locally conformally flat when referred to just some point 
 
 
 
 x
 
 
 {\displaystyle x}
 
 on 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and reserve the definition of conformally flat for the case in which the relation is valid for all 
 
 
 
 x
 
 
 {\displaystyle x}
 
 on 
 
 
 
 M
 
 
 {\displaystyle M}
 
.

Conformally flat manifold open-in-new

Conformally flat manifold