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The Riemann curvature tensor can be expressed as:

$$R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}$$

where the $$\Gamma^{k}{}_{ij}$$ are the Christoffel symbols. $$\begin{align} \Gamma^m{}_{ij}&= g^{mk}\Gamma_{kij}\\[0.2em] & =\frac12\, g^{mk} \left( \frac{\partial}{\partial x^j} g_{ki} +\frac{\partial}{\partial x^i} g_{kj} -\frac{\partial}{\partial x^k} g_{ij} \right)\\ & \equiv\frac12\, g^{mk} \left( g_{ki,j} + g_{kj,i} - g_{ij,k} \right) \,. \end{align}$$ with $g_{ij}$ metric tensor of the manifold.

My question is:

given a manifold with metric tensor $g_{\mu\nu}$ we can calculate the Riemann tensor. But, given a $R^\rho{}_{\sigma\mu\nu}$, does exist only a $g_{\mu\nu}$ having that Riemann curvature tensor or there ar many metric tensors with the given $R^\rho{}_{\sigma\mu\nu}$? Thanks in advance.

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    $\begingroup$ 1) See M. Berger's wonderful "A panoramic view of Riemannian geometry" for a detailed discussion of your question. The short answer is that many metrics can share a curvature tensor. 2) Please never use the Christoffel symbols, but strive to express all calculations in coordinate invariant notation (for your own sanity's sake). There's a lovely discussion of the evolution of notations and concepts in Riemannian geometry in Vol. 2 of Spivak's opus, which is put to great use in Gallot, Hulin & Lafontaine's "Riemannian geometry", which is very good for a beginner. $\endgroup$ Commented Apr 24, 2012 at 15:34
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    $\begingroup$ This isn't how the align environment works; you need to use & to mark the spots where the alignment should take place. $\endgroup$
    – joriki
    Commented Apr 24, 2012 at 17:10

1 Answer 1

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A simple counterexample is a flat Euclidean space of any dimension. The Riemann curvature is uniformly zero. A diffeomorphism of the space back to itself stretching/shrinking/shearing the space can be thought of as a change of metric on the original space, but the curvature remains zero. This argument should generalize to any manifold with constant fixed curvature (n-spheres & hyperbolic geometries), but when the curvature is not constant an automorphism may not preserve the curvature at some points.

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