Riemann Tensor in Polar Coordinates

ds² = dr² + r²dθ². (r, θ) = (x¹, x²).

gij =

1 0 0 r2

.

g_22,1 = 2r and other g_ab,c’s are 0.
Γ_{a bc} = ½(∂_cg_ab + ∂_bg_ca − ∂_ag_bc)
Γ_{1 22} = −r and Γ_{2 12} = Γ_{2 21} = r; others are 0.
Γ^a_bc = g^ax Γ_{x bc}
Γ¹₂₂ = −r and Γ²₂₁ = Γ²₁₂ = 1/r; other Γ^a_bc = 0.
R^𝜌_σμν = ∂_μΓ^𝜌_νσ − ∂_νΓ^𝜌_μσ + Γ^𝜌_μλΓ^λ_νσ − Γ^𝜌_νλΓ^λ_μσ
We enumerate the non-zero terms above:
∂₁Γ¹₂₂ = −1 (+=> R¹₂₁₂; −=> R¹₂₂₁)
∂₁Γ²₁₂ = −r⁻² (+=> R²₂₁₁; −=> R²₂₁₁)
∂₁Γ²₂₁ = −r⁻² (+=> R²₁₁₂; −=> R²₁₂₁)
Γ^𝜌_μλΓ^λ_νσ:
Γ¹₁₁Γ¹₁₁ + Γ¹₁₂Γ²₁₁ = 0
Γ¹₁₁Γ¹₁₂ + Γ¹₁₂Γ²₁₂ = 0
Γ¹₁₁Γ¹₂₁ + Γ¹₁₂Γ²₂₁ = 0
Γ¹₁₁Γ¹₂₂ + Γ¹₁₂Γ²₂₂ = 0

Γ¹₂₁Γ¹₁₁ + Γ¹₂₂Γ²₁₁ = 0
Γ¹₂₁Γ¹₁₂ + Γ¹₂₂Γ²₁₂ = −1 (+=> R¹₂₂₁; −=> R¹₂₁₂)
Γ¹₂₁Γ¹₂₁ + Γ¹₂₂Γ²₂₁ = −1 (+=> R¹₁₂₂; −=> R¹₁₂₂)
Γ¹₂₁Γ¹₂₂ + Γ¹₂₂Γ²₂₂ = 0

Γ²₁₁Γ¹₁₁ + Γ²₁₂Γ²₁₁ = 0
Γ²₁₁Γ¹₁₂ + Γ²₁₂Γ²₁₂ = r⁻² (+=> R²₂₁₁; −=> R²₂₁₁)
Γ²₁₁Γ¹₂₁ + Γ²₁₂Γ²₂₁ = r⁻² (+=> R²₁₁₂; −=> R²₁₂₁)
Γ²₁₁Γ¹₂₂ + Γ²₁₂Γ²₂₂ = 0

Γ²₂₁Γ¹₁₁ + Γ²₂₂Γ²₁₁ = 0
Γ²₂₁Γ¹₁₂ + Γ²₂₂Γ²₁₂ = 0
Γ²₂₁Γ¹₂₁ + Γ²₂₂Γ²₂₁ = 0
Γ²₂₁Γ¹₂₂ + Γ²₂₂Γ²₂₂ = −1 (+=> R²₂₂₂; −=> R²₂₂₂)

In summary:
R¹₂₁₂ = −1 − −1 = 0
R¹₂₂₁ = − −1 + −1 = 0
R²₁₁₂ = −r⁻² + r⁻² = 0
R²₁₂₁ = − −r⁻² − r⁻² = 0

It wasn’t easy but they managed to all cancel out! Also R_𝜌σμν = 0. This gave me tools to find a bug in this, which I fixed.
Here is a version for the unit sphere.