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## Main Question or Discussion Point

I am trying to prove a concept that was presented in my course but I am having a bit of an issue with understanding the final result. The concept was that in 3 dimension the vanishing of the Riemann tensor leaves only the trivial solution for the vacuum equations. I started by subbing Eq.(1) into Eq.(2) and solving for R (I took the trace in case you were wondering - my result was 6[itex]\Lambda[/itex]) and then subbed then this answer back in to solve for [itex]R_{\alpha\beta}[/itex] (answer was 2[itex]\Lambda[/itex][itex]g_{\alpha\beta}[/itex]). I then subbed both of these into (3) and solved. However, as you can see my answer is not zero. If anyone could point out a mistake or if I just haven't properly conceptualized the answer I would appreciate the help, thanks.

(1) [itex]G_{\alpha\beta}[/itex]=[itex]R_{\alpha\beta}[/itex]-[itex]\frac{1}{2}[/itex][itex]g_{\alpha\beta}[/itex]R

(2) [itex]G_{\alpha\beta}[/itex]+[itex]\Lambda[/itex][itex]g_{\alpha\beta}[/itex]=0

(3) [itex]R_{\alpha\beta\gamma\delta}[/itex] = [itex]g_{\alpha\gamma}[/itex][itex]R_{\beta\delta}[/itex]+[itex]g_{\beta\delta}[/itex][itex]R_{\alpha\gamma}[/itex]-[itex]g_{\alpha\delta}[/itex][itex]R_{\beta\gamma}[/itex]-[itex]g_{\beta\gamma}[/itex][itex]R_{\alpha\delta}[/itex]-[itex]\frac{R}{2}[/itex]([itex]g_{\alpha\gamma}[/itex][itex]g_{\beta\delta}[/itex]-[itex]g_{\alpha\delta}[/itex][itex]g_{\beta\gamma}[/itex])

= [itex]\Lambda[/itex]([itex]g_{\alpha\gamma}[/itex][itex]g_{\beta\delta}[/itex]-[itex]g_{\alpha\delta}[/itex][itex]g_{\beta\gamma}[/itex])

(1) [itex]G_{\alpha\beta}[/itex]=[itex]R_{\alpha\beta}[/itex]-[itex]\frac{1}{2}[/itex][itex]g_{\alpha\beta}[/itex]R

(2) [itex]G_{\alpha\beta}[/itex]+[itex]\Lambda[/itex][itex]g_{\alpha\beta}[/itex]=0

(3) [itex]R_{\alpha\beta\gamma\delta}[/itex] = [itex]g_{\alpha\gamma}[/itex][itex]R_{\beta\delta}[/itex]+[itex]g_{\beta\delta}[/itex][itex]R_{\alpha\gamma}[/itex]-[itex]g_{\alpha\delta}[/itex][itex]R_{\beta\gamma}[/itex]-[itex]g_{\beta\gamma}[/itex][itex]R_{\alpha\delta}[/itex]-[itex]\frac{R}{2}[/itex]([itex]g_{\alpha\gamma}[/itex][itex]g_{\beta\delta}[/itex]-[itex]g_{\alpha\delta}[/itex][itex]g_{\beta\gamma}[/itex])

= [itex]\Lambda[/itex]([itex]g_{\alpha\gamma}[/itex][itex]g_{\beta\delta}[/itex]-[itex]g_{\alpha\delta}[/itex][itex]g_{\beta\gamma}[/itex])