Riemann Curvature Tensor - Coordinate Expression

Coordinate Expression

Converting to the tensor index notation, the Riemann curvature tensor is given by

where are the coordinate vector fields. The above expression can be written using Christoffel symbols:

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}

(see also the list of formulas in Riemannian geometry).

The Riemann curvature tensor is also the commutator of the covariant derivative of an arbitrary covector with itself:

since the connection is torsionless, which means that the torsion tensor vanishes.

This formula is often called the Ricci identity. This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor. In this way, the tensor character of the set of quantities is proved.

This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows

\begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; \gamma \delta} - T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; \delta \gamma} = \, & - R^{\alpha_1}{}_{\rho \gamma \delta} T^{\rho \alpha_2 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} - \cdots - R^{\alpha_r}{}_{\rho \gamma \delta} T^{\alpha_1 \cdots \alpha_{r-1} \rho}{}_{\beta_1 \cdots \beta_s} \\ & + \, R^\sigma{}_{\beta_1 \gamma \delta} T^{\alpha_1 \cdots \alpha_r}{}_{\sigma \beta_2 \cdots \beta_s} + \cdots + R^\sigma{}_{\beta_s \gamma \delta} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_{s-1} \sigma} \,. \end{align}

This formula also applies to tensor densities without alteration, because for the Levi-Civita (not generic) connection one gets:

\nabla_{\mu}(\sqrt{g}\,)\equiv (\sqrt{g}\,)_{;\mu}=0 \, , \quad {\mathrm{where}} \quad {g}=|{\mathrm{det}}(g_{\mu\nu})|\, .

It is sometimes convenient to also define the purely covariant version by

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