curvature-form.egi

x := [| θ, φ |]

g_i_j := [| [| r^2, 0 |], [| 0, r^2 * (sin θ)^2 |] |]_i_j
g~i~j := [| [| 1 / r^2, 0 |], [| 0, 1 / (r^2 * (sin θ)^2) |] |]~i~j

Γ_j_l_k := (1 / 2) * (∂/∂ g_j_l x~k + ∂/∂ g_j_k x~l - ∂/∂ g_k_l x~j)

Γ~i_k_l := withSymbols [j] g~i~j . Γ_j_l_k

R~i_j_k_l := withSymbols [m]
               ∂/∂ Γ~i_j_l x~k - ∂/∂ Γ~i_j_k x~l + Γ~m_j_l . Γ~i_m_k - Γ~m_j_k . Γ~i_m_l

assertEqual "Riemann curvature" R~#_#_1_1 [| [| 0, 0 |], [| 0, 0 |] |]~#_#
assertEqual "Riemann curvature" R~#_#_1_2 [| [| 0, (sin θ)^2 |], [| -1, 0 |] |]~#_#
assertEqual "Riemann curvature" R~#_#_2_1 [| [| 0, -1 * (sin θ)^2 |], [| 1, 0 |] |]~#_#
assertEqual "Riemann curvature" R~#_#_2_2 [| [| 0, 0 |], [| 0, 0 |] |]~#_#

ω := Γ~#_#_#

d %t := !(flip ∂/∂) x t

infixl expression 7 ∧

(∧) %x %y := x !. y

Ω := withSymbols [i, j]
       antisymmetrize (d ω~i_j + ω~i_k ∧ ω~k_j)

assertEqual "Curvature form" Ω~#_#_1_1 [| [| 0, 0 |], [| 0, 0 |] |]~#_#
assertEqual "Curvature form" Ω~#_#_1_2 [| [| 0, (sin θ)^2  / 2|], [| -1 / 2, 0 |] |]~#_#
assertEqual "Curvature form" Ω~#_#_2_1 [| [| 0, -1 * (sin θ)^2 / 2 |], [| 1 / 2, 0 |] |]~#_#
assertEqual "Curvature form" Ω~#_#_2_2 [| [| 0, 0 |], [| 0, 0 |] |]~#_#