In paper by Barnich and Brandt Covariant theory of asymptotic symmetries,
conservation laws and central charges they defined the Riemann tensor like this:
$$R_{\rho\mu\nu}^{\quad \ \ \lambda}~=~\partial_\rho \Gamma_{\mu\nu}^{\ \ \ \ \ \lambda}+\Gamma_{\rho\sigma}^{\ \ \ \ \ \lambda}\Gamma_{\mu\nu}^{\ \ \ \ \ \sigma}-(\rho\leftrightarrow\mu).$$
Now I have taken the 'normal' definition of Riemann tensor and raised the first index, and lowered the last one, and if I do the same with original (lower the first one and raise the last one) I get the same expression, which means that this is ok.
But, why such definition? And does that mean that the Christoffel symbols have different definition compared to usual? I mean raised first and lowered last index.
Best Answer
Comments to the question (v1):
Beware that different authors have different conventions for the horizontal order of indices for the Christoffel symbols$^1$ $\Gamma^{\lambda}{}_{\mu\nu}$ and the Riemann curvature tensor $R^{\sigma}{}_{\mu\nu\lambda}$. Some may e.g. write $\Gamma_{\mu\nu}{}^{\lambda}$ and $R_{\mu\nu\lambda}{}^{\sigma}$ instead.
It may be useful to write objects such as $\Gamma^{\lambda}{}_{\mu\nu}$ and $R^{\sigma}{}_{\mu\nu\lambda}$ with covariant and contravariant indices not merely on top of each other a la $\Gamma^{\lambda}_{\mu\nu}$ and $R^{\sigma}_{\mu\nu\lambda}$ but horizontally displaced to keep track of the horizontal position of the indices. Then the indices can be raised or lowered by a metric $g_{\mu\nu}$ with no ambiguity in notation of which index was raised or lowered.
In case of supermanifolds, the indices may correspond to Grassmann-odd coordinates, and it may become a tedious exercise in book-keeping to assign consistent transformation laws with correct sign factors for tensor components under coordinate transformations of supercharts. Some choices of horizontal orders of indices may be more natural in the sense of minimizing the appearance of sign factors. We mention this 3rd point because both authors Barnich and Brandt are experts on BRST formalism, where Grassmann-odd variables play an essential role.
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$^1$ It is covenient to call $\Gamma^{\lambda}{}_{\mu\nu}$ Christoffel symbols even if the tangent-space connection $\nabla$ is not torsionfree nor compatible with a metric.