curvature-form.egi

declare symbol r, θ, φ: MathExpr

-- Parameters and metric tensor
def x : Vector MathExpr := [| θ, φ |]

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

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

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

-- Riemann curvature
def R~i_j_k_l : Tensor MathExpr := 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 |] |]~#_#

-- Exterior derivative
def d (t : Tensor MathExpr) : Tensor MathExpr := !(flip ∂/∂) x t

-- Connection form
def ω~i_j : Matrix MathExpr := Γ~i_j_#

-- Curvature form
def Ω~i_j : Tensor MathExpr := withSymbols [k]
  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 |] |]~#_#