Conformally flat manifold

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