In case there's any confusion, the following computations belong to a setting where

R^\mu_{\nu \rho \sigma} = \partial_\rho \Gamma^\mu_{\nu\sigma} - \partial_\sigma \Gamma^\mu_{\nu\rho}+ \Gamma^\tau_{\nu\sigma}\Gamma^\mu_{\tau\rho}-\Gamma^\tau_{\nu\rho}\Gamma^\mu_{\tau\sigma}

and of course

\Gamma^\mu_{\nu\rho} = \frac 1 2 g^{\mu\sigma}(g_{\sigma\nu,\rho}+g_{\sigma\rho,\nu}-g_{\nu\rho,\sigma})

Warped Procucts

Suppose you have a space with coordinates x^i, y^A of the form


so that schematically

 & h_{AB}\left(y\right)

This is a so-called warped product. A common question is "What is the form of the Ricci tensor?"

Let the dimension of the x subspace - that is to say, how many values i can range over - be denoted by D. Then one obtains the result

R_{AB} & = & \hat{R}_{AB}-D\frac{\nabla_{A}\nabla_{B}N}{N}\\
R_{ij} & = & \tilde{R}_{ij}-H_{ij}\left[N\Delta N+\left(D-1\right)h^{AB}\nabla_{A}N\cdot\nabla_{B}N\right]\\
R_{Ai} & = & 0

where \hat{R}_{AB} is the Ricci tensor of the metric h_{AB} and \tilde{R}_{ij} is the Ricci tensor of the metric H_{ij}. Note that \nabla_{A}=\nabla_{A}^{\left(y\right)} is the covariant derivative with respect to the h_{AB} metric.

Observe that the form of the modification to \tilde{R}_{ij} is a multiple of the metric tensor for the x subspace.

Conformal Rescalings

Another very common computation involves the rescaling

\tilde{g}_{\mu\nu}(x) = e^{2\phi(x)}g_{\mu\nu}(x)

A direct computation in local inertial coordinates for the new metric \tilde{g} gives the result

\tilde{R}_{\nu\sigma} & = & R_{\nu\sigma}+(2-D)\left[ \nabla_\nu\nabla_\sigma \phi - \nabla_\nu \phi \nabla_\sigma \phi \right] \\
 & & - g_{\nu\sigma} \left[ \nabla^\mu \nabla_\mu \phi + (D-2) \nabla_\mu \phi \nabla^\mu \phi \right] \\
\tilde{R} & = & R+2(1-D)\nabla^{2}\phi+\left(D-2\right)\left(1-D\right)\left(\nabla\phi\right)^{2}

where all covariant derivatives are measured in the unaltered metric g, and all indices are raised and lowered using this metric too.

Warped Products and Conformal Rescalings

A cheat sheet

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