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In the mathematical field of
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, the existence of
isothermal coordinates In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric l ...
for a ( pseudo-)
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
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 Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
and
Jan Arnoldus Schouten Jan Arnoldus Schouten (28 August 1883 – 20 January 1971) was a Dutch mathematician and Professor at the Delft University of Technology. He was an important contributor to the development of tensor calculus and Ricci calculus, and was one of the ...
) characterizes the existence of isothermal coordinates by certain equations to be satisfied by the
Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
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 In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
, the
Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
, and the
scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geome ...
, the
Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tida ...
of a pseudo-Riemannian metric of dimension is given by :W_=R_-\frac+\frac(g_g_-g_g_). The
Schouten tensor In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for by: :P=\frac \left(\mathrm -\frac g\right)\, \Leftrightarrow \mathrm=(n-2) P + J g \, , where Ric is the Ricci tensor (defined ...
is defined via the Ricci and scalar curvatures by :S_=\fracR_-\frac. As can be calculated by the
Bianchi identities In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie alg ...
, these satisfy the relation that :\nabla^jW_=\frac(\nabla_kS_-\nabla_lS_). The Weyl–Schouten theorem says that for any pseudo-Riemannian manifold of dimension : * If then the manifold is conformally flat if and only if its Weyl tensor is zero. * If 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 , 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 : :\nabla_i\omega_j=\frac\omega_i\omega_j-\fracg^\omega_p\omega_qg_-S_ Let denote the tensor on the right-hand side. The Frobenius theorem states that the above equation is locally solvable if and only if :\partial_k\Gamma_^p\omega_p+\Gamma_^pF_^+\fracF_^\omega_j+\frac\omega_iF_^-\frac\partial_kg^\omega_p\omega_qg_-\fracg^\omega_pF_^g_-\fracg^\omega_p\omega_q\partial_kg_-\partial_kS_ is symmetric in and for any 1-form . A direct cancellation of termsThis uses the identity W_=R_-\fracg_S_+\fracg_S_+\fracg_S_-\fracg_S_. shows that this is the case if and only if :^p\omega_p=\nabla_kS_-\nabla_iS_ for any 1-form . If 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 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 solving the given equation. From the symmetry of the right-hand side, it follows that must be a closed form. The Poincaré lemma then implies that there is a real-valued function with . Due to the formula for the
Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
under a conformal change of metric, the (locally defined) pseudo-Riemannian metric is Ricci-flat. If then every Ricci-flat metric is flat, and if then every Ricci-flat and Weyl-flat metric is flat.


See also

*
Yamabe problem The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds: By computing a formula for how the scalar curvatur ...


References

Notes. Sources. * * * * * {{DEFAULTSORT:Weyl-Schouten theorem Theorems in Riemannian geometry