Conformally flat manifold

The upper manifold is flat. The lower one is not, but it is conformal to the first one

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 ${\displaystyle g}$ of the manifold ${\displaystyle M}$ has to be conformal to the flat metric ${\displaystyle \eta }$, i.e., the geodesics maintain in all points of ${\displaystyle M}$ the angles by moving from one to the other, as well as keeping the null geodesics unchanged, ^[1] that means there exists a function ${\displaystyle \lambda (x)}$ such that ${\displaystyle g(x)=\lambda ^{2}(x)\,\eta }$, where ${\displaystyle \lambda (x)}$ is known as the conformal factor and ${\displaystyle x}$ is a point on the manifold.

More formally, let ${\displaystyle (M,g)}$ be a pseudo-Riemannian manifold. Then ${\displaystyle (M,g)}$ is conformally flat if for each point ${\displaystyle x}$ in ${\displaystyle M}$, there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ and a smooth function ${\displaystyle f}$ defined on ${\displaystyle U}$ such that ${\displaystyle (U,e^{2f}g)}$ is flat (i.e. the curvature of ${\displaystyle e^{2f}g}$ vanishes on ${\displaystyle U}$). The function ${\displaystyle f}$ need not be defined on all of ${\displaystyle M}$.

Some authors use the definition of locally conformally flat when referred to just some point ${\displaystyle x}$ on ${\displaystyle M}$ and reserve the definition of conformally flat for the case in which the relation is valid for all ${\displaystyle x}$ on ${\displaystyle M}$.

Examples

• Every manifold with constant sectional curvature is conformally flat.
• Every 2-dimensional pseudo-Riemannian manifold is conformally flat. ^[1]
  • The line element of the two dimensional spherical coordinates, like the one used in the geographic coordinate system,

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

  ${\displaystyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\,={\frac {4}{(1+r^{2})^{2}}}(dx^{2}+dy^{2})}$.

• 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

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

becomes

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

which is the flat metric times the conformal factor ${\displaystyle 1-{\frac {2GM}{r}}}$. ^[7]

See also

• Weyl–Schouten theorem
• conformal geometry
• Yamabe problem

References

1. Ray D'Inverno. "6.3 The Weil tensor". Introducing Einstein's Relativity. pp. 88–89.
2. Spherical coordinate system - Integration and differentiation in spherical coordinates
3. Stereographic projection - Properties. The Riemann's formula
4. Kuiper, N. H. (1949). "On conformally flat spaces in the large". Annals of Mathematics. 50 (4): 916–924. doi:10.2307/1969587. JSTOR   1969587.
5. Garecki, Janusz (2008). "On Energy of the Friedman Universes in Conformally Flat Coordinates". Acta Physica Polonica B. 39 (4): 781–797. arXiv: 0708.2783 . Bibcode:2008AcPPB..39..781G.
6. Garat, Alcides; Price, Richard H. (2000-05-18). "Nonexistence of conformally flat slices of the Kerr spacetime". Physical Review D. 61 (12): 124011. arXiv: gr-qc/0002013 . Bibcode:2000PhRvD..61l4011G. doi:10.1103/PhysRevD.61.124011. ISSN   0556-2821. S2CID   119452751.
7. Ray D'Inverno. "17.2 The Kruskal solution". Introducing Einstein's Relativity. pp. 230–231.