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 tensor ${\displaystyle g}$ of the manifold ${\displaystyle M}$ has to be conformal to the flat metric tensor ${\displaystyle \eta }$, 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 ${\displaystyle M}$.

• 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 ${\displaystyle \textstyle g_{ik}=\left[{\begin{smallmatrix}1&0\\0&\sin ^{2}\theta \end{smallmatrix}}\right]}$ and is not flat but with the stereographic projection can be mapped to a flat space using the conformal factor ${\displaystyle \textstyle {\frac {2}{1+r^{2}}}}$, where ${\displaystyle r}$ is the distance from the origin of the flat space,^[3] obtaining ${\displaystyle \textstyle ds^{2}=d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\,={\frac {4}{(1+r^{2})^{2}}}(dx^{2}+dy^{2})}$.
• 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 are often used, for instance, in describing the Friedmann–Lemaître–Robertson–Walker metric.^[5] However, this is not always possible, as in the case of the Kerr metric, which does not admit any conformally flat slices.^{[6][further explanation needed]}

An example of conformally flat metric can be obtained by transforming the Kruskal–Szekeres coordinates for the Schwarzschild metric, which have line element ${\displaystyle \textstyle ds^{2}=\left(1-{2GM}/{r}\right)dv\,du}$ with metric tensor ${\displaystyle \textstyle g_{ik}=\left[{\begin{smallmatrix}0&1-{2GM}/{r}\\1-{2GM}/{r}&0\end{smallmatrix}}\right]}$ and so is not flat. But with the transformations ${\displaystyle t=(v+u)/2}$ and ${\displaystyle x=(v-u)/2}$ becomes ${\displaystyle \textstyle ds^{2}=\left(1-{2GM}/{r}\right)(dt^{2}-dx^{2})}$ with metric tensor ${\displaystyle \textstyle g_{ik}=\left[{\begin{smallmatrix}1-{2GM}/{r}&0\\0&-1+{2GM}/{r}\end{smallmatrix}}\right]}$, which is the flat metric times the conformal factor ${\displaystyle \textstyle 1-{2GM}/{r}}$.^[7]

• Weyl–Schouten theorem
• Conformal geometry
• Yamabe problem

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