I am doing some mathematical exercises with 3D anti-de sitter face using the metric ds2=-(1+r2)dt2+(1+r2)-1+r2dφ2 I found the three geodesics from the Christoffel symbols, and they seem to look correct to me. d2t/dλ2+2(r+1/r)*(dt/dλ)(dr/dλ)=0 d2r/dλ2+(r+r3)*(dt/dλ)2-r/(r2+1)(dr/dλ)2-(r+r3)(dφ/dλ)2=0 d2φ/dλ2+2/r*(dφ/dλ)(dr/dλ)=0 When I started calculating the Riemann and Ricci Tensor however things started to look hairy Rφrφr = -(1+r2)-1 Rtφtφ = -(r+1/r)(r+r3) Rtrtr = -2+1/r2-(r+1/r)2 I found the other components of the Riemann tensor to be 0 which may have been where I went wrong. This led me to a messy Ricci Tensor and Ricci Scalar Rrr= -2+1/r2-(r+1/r)2-(1+r2)-1 Rφφ = -(r+1/r)(r+r3) R=guvRuv = Rrr(1+r2)+1/r2Rφφ R=-r4-4r2-3+1/r2 This for some reason doesn't look right to me. It leads to a super complicated stress tensor as well. What did I do wrong here?