Conformally flat manifold explained

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

g of the manifold M has to be conformal to the flat metric η , i.e., the geodesics maintain in all points of M the angles by moving from one to the other, as well as keeping the null geodesics unchanged,^[1] that means there exists a function λ (x) such that g(x) = λ ^{2(x)} η , where λ (x) is known as the conformal factor and x is a point on the 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,e ^{2f} g) is flat (i.e. the curvature of e ^{2f} g vanishes on U ). The function f need not be defined on all of M .

Some authors use the definition of locally conformally flat when referred to just some point

x on M and reserve the definition of conformally flat for the case in which the relation is valid for all x on M .

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,

ds ^{2} = d\theta ^{2} + \sin ^{2} \theta d\phi ^{2} ,^[2] has metric tensor g _{ik} = \begin{bmatrix} 1 & 0 \ 0 & sin ^{2} \theta \end{bmatrix}  and is not flat but with the stereographic projection can be mapped to a flat space using the conformal factor 2 \over ( 1+ r ^{2)} , where r is the distance from the origin of the flat space,^[3] obtainingds ^{2} = d\theta ^{2} + \sin ^{2} \theta d\phi ^{2} = (4)/(( 1+ r ^{2)} ^{2}) (dx ^{2} + dy ^{2)} .

• 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.^[4]

• 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 can often be used, for example to describe Friedmann–Lemaître–Robertson–Walker metric.^[5] However it was also shown that there are no conformally flat slices of the Kerr spacetime.^[6]

For example, the Kruskal-Szekeres coordinates have line element

ds ^{2} = \left( 1- (2 GM)/(r) \right) dv du with metric tensor g _{ik} = \begin{bmatrix} 0 & 1- (2 GM)/(\ 1- r) (2 GM)/(r) & 0 \end{bmatrix} and so is not flat. But with the transformations t = (v + u)/ 2 and x = (v - u)/ 2

becomes

ds ^{2} = \left( 1- (2 GM)/(r) \right) (dt ^{2} - dx ^{2)} with metric tensor g _{ik} = \begin{bmatrix} 1- (2 GM)/(r) & 0 \ 0 & -1+ (2 GM)/(r) \end{bmatrix} ,

which is the flat metric times the conformal factor

1- (2 GM)/(r) .^[7]

See also

• Weyl–Schouten theorem
• conformal geometry
• Yamabe problem

Notes and References

1. Book: Ray D'Inverno. Introducing Einstein's Relativity. 88–89. 6.13 The Weyl tensor.
2. [Spherical coordinate system#Integration and differentiation in spherical coordinates|Spherical coordinate system - Integration and differentiation in spherical coordinates]
3. [Stereographic projection#Properties|Stereographic projection - Properties]
4. Kuiper. N. H.. On conformally flat spaces in the large. Annals of Mathematics. 1949. 50. 4. 916–924. 10.2307/1969587. 1969587. Kuiper.
5. Garecki. Janusz. 2008. On Energy of the Friedman Universes in Conformally Flat Coordinates. Acta Physica Polonica B. 39. 4. 781–797. 0708.2783. 2008AcPPB..39..781G.
6. Garat. Alcides. Price. Richard H.. 2000-05-18. Nonexistence of conformally flat slices of the Kerr spacetime. Physical Review D. en. 61. 12. 124011. 10.1103/PhysRevD.61.124011. gr-qc/0002013. 2000PhRvD..61l4011G. 119452751. 0556-2821.
7. Book: Ray D'Inverno. Introducing Einstein's Relativity. 230–231. 17.2 The Kruskal solution.