Specific proof of the Riemann tensor for FRW metric

[Chromatic_Universe]

Homework Statement

Prove R_ijkl= k/R² * (g_ik g_jl-g_il g_jk) where g_ik is the 3 metric for FRW universe and K =0,+1,-1, and i,j=1,2,3, that is, spatial coordinates.
.

Homework Equations

The Christoffel symbol definition:
Γ^μ_νρ = ½g^μσ(∂_ρg_νσ+∂_νg_ρσ-∂_σg_νρ)
and the Riemann tensor definition:
R^μ_νσρ = ∂_σΓ^μ_ρν-∂_ρΓ^μ_σν+Γ^μ_σλΓ^λ_ρν-Γ^μ_ρλΓ^λ_σν
and the FRLW metric, in the section:
Reduced-circumference polar coordinates (under general metric section)

The Attempt at a Solution

I cannot come to the general expression for the Christoffel symbols using g_ij. But the expression can be derived using Killing vectors for maximally symmetric space. For the FRW universe(homogeneous and isotropic), the same holds true, but I am finding it difficult to get to this expression without using Killing vectors, only using the definition of Christoffel symbols and Riemann tensors.

 

[stevendaryl]

Homework Statement

Prove R_ijkl= k/R² * (g_ik g_jl-g_il g_jk) where g_ik is the 3 metric for FRW universe and K =0,+1,-1
.

Homework Equations

The Christoffel symbol definition and the Riemann tensor definition

The Attempt at a Solution

I cannot come to the general expression for the Christoffel symbols using g_ij.

Well, to get started, can you write down the metric, and the definition of the Christoffel symbols and Riemann tensors? You have to show some work.

 

[Chromatic_Universe]

Well, to get started, can you write down the metric, and the definition of the Christoffel symbols and Riemann tensors? You have to show some work.

Edited the question! Thanks!