Weyl–Schouten theorem

In the mathematical field of differential geometry, the existence of isothermal coordinates for a (pseudo-)Riemannian metric is often of interest. In the case of a metric on a two-dimensional space, the existence of isothermal coordinates is unconditional. For higher-dimensional spaces, the Weyl–Schouten theorem (named after Hermann Weyl and Jan Arnoldus Schouten) characterizes the existence of isothermal coordinates by certain equations to be satisfied by the Riemann curvature tensor of the metric.

Existence of isothermal coordinates is also called conformal flatness, although some authors refer to it instead as local conformal flatness; for those authors, conformal flatness refers to a more restrictive condition.

Theorem

In terms of the Riemann curvature tensor, the Ricci tensor, and the scalar curvature, the Weyl tensor of a pseudo-Riemannian metric $g$ of dimension $n$ is given by^[1]

W_{ijkl}=R_{ijkl}-{\frac {R_{ik}g_{jl}-R_{il}g_{jk}+R_{jl}g_{ik}-R_{jk}g_{il}}{n-2}}+{\frac {R}{(n-1)(n-2)}}(g_{jl}g_{ik}-g_{jk}g_{ik}).

The Schouten tensor is defined via the Ricci and scalar curvatures by^[1]

S_{ij}={\frac {2}{n-2}}R_{ij}-{\frac {Rg_{ij}}{(n-2)(n-1)}}.

As can be calculated by the Bianchi identities, these satisfy the relation that^[2]

\nabla ^{j}W_{ijkl}={\frac {n-3}{2}}(\nabla _{k}S_{il}-\nabla _{l}S_{ik}).

The Weyl–Schouten theorem says that for any pseudo-Riemannian manifold of dimension $n$ :^[3]

If $n \geq 4$ then the manifold is conformally flat if and only if its Weyl tensor is zero.
If $n = 3$ then the manifold is conformally flat if and only if its Schouten tensor is a Codazzi tensor.

As known prior to the work of Weyl and Schouten, in the case $n = 2$ , every manifold is conformally flat. In all cases, the theorem and its proof are entirely local, so the topology of the manifold is irrelevant.

There are varying conventions for the meaning of conformal flatness; the meaning as taken here is sometimes instead called local conformal flatness.

Sketch of proof

The only if direction is a direct computation based on how the Weyl and Schouten tensors are modified by a conformal change of metric. The if direction requires more work.

Consider the following equation for a 1-form $ω$ :

\nabla _{i}\omega _{j}={\frac {1}{2}}\omega _{i}\omega _{j}-{\frac {1}{4}}g^{pq}\omega _{p}\omega _{q}g_{ij}-S_{ij}

Let $F ω, g$ denote the tensor on the right-hand side. The Frobenius theorem^[4] states that the above equation is locally solvable if and only if

\partial _{k}\Gamma _{ij}^{p}\omega _{p}+\Gamma _{ij}^{p}F_{kp}^{\omega ,g}+{\frac {1}{2}}F_{ki}^{\omega ,g}\omega _{j}+{\frac {1}{2}}\omega _{i}F_{kj}^{\omega ,g}-{\frac {1}{4}}\partial _{k}g^{pq}\omega _{p}\omega _{q}g_{ij}-{\frac {1}{2}}g^{pq}\omega _{p}F_{kq}^{\omega ,g}g_{ij}-{\frac {1}{4}}g^{pq}\omega _{p}\omega _{q}\partial _{k}g_{ij}-\partial _{k}S_{ij}

is symmetric in $i$ and $k$ for any 1-form $ω$ . A direct cancellation of terms^[5] shows that this is the case if and only if

{W_{kij}}^{p}\omega _{p}=\nabla _{k}S_{ij}-\nabla _{i}S_{jk}

for any 1-form $ω$ . If $n = 3$ then the left-hand side is zero since the Weyl tensor of any three-dimensional metric is zero; the right-hand side is zero whenever the Schouten tensor is a Codazzi tensor. If $n \geq 4$ then the left-hand side is zero whenever the Weyl tensor is zero; the right-hand side is also then zero due to the identity given above which relates the Weyl tensor to the Schouten tensor.

As such, under the given curvature and dimension conditions, there always exists a locally defined 1-form $ω$ solving the given equation. From the symmetry of the right-hand side, it follows that $ω$ must be a closed form. The Poincaré lemma then implies that there is a real-valued function $u$ with $ω = du$ . Due to the formula for the Ricci curvature under a conformal change of metric, the (locally defined) pseudo-Riemannian metric $e u g$ is Ricci-flat. If $n = 3$ then every Ricci-flat metric is flat, and if $n \geq 4$ then every Ricci-flat and Weyl-flat metric is flat.^[3]

References

Notes.

^ ^a ^b Aubin 1998, Definition 4.23.
^ Aubin 1998, p. 118; Eisenhart 1926, p. 91.
^ ^a ^b Aubin 1998, Theorem 4.24; Eisenhart 1926, Section 28.
^ For the direct version being used, see Abraham, Marsden & Ratiu 1988, Example 6.4.25D; Lee 2013, Proposition 19.29; Warner 1983, Remarks 1.61.
^ This uses the identity $W_{kijp}=R_{kijp}-{\frac {1}{2}}g_{ip}S_{jk}+{\frac {1}{2}}g_{ij}S_{kp}+{\frac {1}{2}}g_{kp}S_{ij}-{\frac {1}{2}}g_{jk}S_{ip}.$

Sources.

Abraham, R.; Marsden, J. E.; Ratiu, T. (1988). Manifolds, tensor analysis, and applications. Applied Mathematical Sciences. Vol. 75 (Second edition of 1983 original ed.). New York: Springer-Verlag. doi:10.1007/978-1-4612-1029-0. ISBN 0-387-96790-7. MR 0960687. Zbl 0875.58002.
Aubin, Thierry (1998). Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics. Berlin: Springer-Verlag. doi:10.1007/978-3-662-13006-3. ISBN 3-540-60752-8. MR 1636569. Zbl 0896.53003.
Eisenhart, Luther Pfahler (1926). Riemannian geometry. Reprinted in 1997. Princeton: Princeton University Press. doi:10.1515/9781400884216. ISBN 0-691-02353-0. JFM 52.0721.01.
Lee, John M. (2013). Introduction to smooth manifolds. Graduate Texts in Mathematics. Vol. 218 (Second edition of 2003 original ed.). New York: Springer. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9981-8. MR 2954043. Zbl 1258.53002.
Warner, Frank W. (1983). Foundations of differentiable manifolds and Lie groups. Graduate Texts in Mathematics. Vol. 94 (Corrected reprint of the 1971 original ed.). New York–Berlin: Springer-Verlag. doi:10.1007/978-1-4757-1799-0. ISBN 0-387-90894-3. MR 0722297. Zbl 0516.58001.

[FOOTNOTEAubin1998Definition_4.23-1] Aubin 1998, Definition 4.23.

[FOOTNOTEAubin1998118Eisenhart192691-2] Aubin 1998, p. 118; Eisenhart 1926, p. 91.

[FOOTNOTEAubin1998Theorem_4.24Eisenhart1926Section_28-3] Aubin 1998, Theorem 4.24; Eisenhart 1926, Section 28.

[4] For the direct version being used, see Abraham, Marsden & Ratiu 1988, Example 6.4.25D; Lee 2013, Proposition 19.29; Warner 1983, Remarks 1.61.

[5] This uses the identity $W_{kijp}=R_{kijp}-{\frac {1}{2}}g_{ip}S_{jk}+{\frac {1}{2}}g_{ij}S_{kp}+{\frac {1}{2}}g_{kp}S_{ij}-{\frac {1}{2}}g_{jk}S_{ip}.$

[1]

[2]

[3]

[4]

[5]

Theorem

Sketch of proof

See also

References