How Do You Calculate the Ricci Tensor for the AdS Metric in 4 Dimensions?

[HamOnRye]

Consider the AdS metric in D+1 dimensions
$$ds^{2}=\frac{L^{2}}{z^{2}}\left(dz^{2}+\eta_{\mu\nu}dx^{\mu}dx^{\nu}\right)$$
I wanted to calculate the Ricci tensor for this metric for D=3. ([\eta_{\mu\nu} is the Minkowski metric in D dimensions)
I have found the following Christoffel symbols
$$\Gamma^{t}_{tz}=\frac{L^{2}}{z^{3}}, \quad \Gamma^{x}_{xz}=\Gamma^{y}_{yz}=\Gamma^{z}_{zz}=-\frac{L^{2}}{z^{3}}$$
From this point I wanted to determine the Riemann tensor in order to finally determine the Ricci tensor.
What I've got the following contributing Riemann tensors
$$R^{x}_{zxz}, \quad R^{y}_{zyz},\quad R^{t}_{ztz}$$
I also noticed that if I have a z-coordinate in the upper index for the Riemann tensor it will be zero no matter what I choose for the lower indices.
My problem is as follows, based on symmetry, the above Riemann tensors should also be zero but I can't see how. Did I make a mistake with my Christoffel symbols or anywhere else?
Any help is appreciated!

Tim

 

[egourgoulhon]

It seems that you made an error in the computation of the Christoffel symbols. They should be either zero or equal to +/- 1/z. See this CoCalc worksheet for the computation, as well as the expression of the Riemann and Ricci tensors.