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Can ${g}^{\mu\sigma}{\Gamma}^{\rho}_{\sigma\nu}$ be written as ${\Gamma}^{\mu\rho}_{\nu}$? If so how come this symbol never appears in any GR book?

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Well, you can of course simply define a new symbol and use that notation, but no one does that because raising and lowering indices is an operation that has a well-defined, coordinate-free meaning on tensors (it has to do with something called the tangent-cotangent isomorphism), but the connection coefficients are not the components of a tensor.

Addendum. Since there has been a downvote, I'd like to quote Carroll's Spactime and Geometry to corroborate my assertion that physicists tend to avoid raising and lowering indices on symbols such as the Christoffel symbols because they are not components of a tensor;

...the connection coefficients are not the components of a tensor...This is why we are not so careful about index placement on the connection coefficients; they are not a tensor, and therefore you should not try to raise and lower their indices.

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  • $\begingroup$ Of course some authors do use $\Gamma_{\rho\mu\nu}$ for $g_{\rho\lambda} \Gamma^\lambda_{\mu\nu}$, but I find this somewhat confusing. $\endgroup$
    – user10851
    Jan 27 '14 at 2:48
  • $\begingroup$ @ChrisWhite Agreed. $\endgroup$ Jan 27 '14 at 2:48
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    $\begingroup$ @ChrisWhite: but that at least is less violent to the notation, because of the factor of $g^{rho\lambda}$ up front in the expression of $\Gamma^{\lambda}{}_{\mu\nu}$ in terms of the metric, so it's pretty safely coordinate independent. $\endgroup$ Jan 27 '14 at 3:26
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The real problem with your components ${\Gamma}^{\mu\rho}_{\nu} = {g}^{\mu\sigma}{\Gamma}^{\rho}_{\sigma\nu}$ and similarly with $\Gamma_{\rho\mu\nu}$ – the reason why the symbols are not used – is the symmetry, more precisely the absence of it.

While $\Gamma^{\alpha}_{\beta\gamma}$ is $\beta\gamma$ symmetric so it doesn't matter in which order the lower indices are written, the first two non-tensors don't enjoy the symmetry in the upper indices or the symmetry in the lower indices so one could easily misinterpret the symbol $\Gamma^{\mu\rho}_\nu$ as $\Gamma^{\rho\mu}_\nu$ which is actually something else. There is no way to uniquely determine whether the first upper index or the second upper index originated from a lower index that was raised. But the answer to this problematic question actually affects the values of the components.

Josh is right that the symbols are not tensors but if there were not the problem above, it would be very natural to define the new symbol according to your rule even for non-tensors.

To support my point by an additional argument, note that the Riemann tensor *is*a tensor but we must be very careful to write various raised components as $R^{\alpha}{}_{\beta\gamma\delta}$, for example, so that we may distinguish the first, second, third, and fourth index. Writing $R^\alpha_{\beta\gamma\delta}$ would be utterly ambiguous. But for $\Gamma^\alpha_{\beta\gamma}$, there is no unambiguous solution because in the standard notation I just wrote, the index $\alpha$ is directly above $\beta$ index so both of them are on the first spot and they consequently can't be raised or lowered.

To offer the "opposite" argument that the non-tensor character isn't the actual reason why the symbols are not used, note that $\partial^\alpha g_{\mu\nu}$ is used with the raised $\alpha$ index although the partial derivative isn't a tensor, either.

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