riemann-curvature-tensor-of-S5.egi

-- Parameters
def x := [| θ, φ, ψ, η, δ |]

def X := [| r * cos θ
          , r * sin θ * cos φ
          , r * sin θ * sin φ * cos ψ
          , r * sin θ * sin φ * sin ψ * cos η
          , r * sin θ * sin φ * sin ψ * sin η * cos δ
          , r * sin θ * sin φ * sin ψ * sin η * sin δ
          |]

-- Local basis
def e_i_j : Matrix MathExpr := ∂/∂ X_j x~i

-- Metric tensors
def g_i_j := generateTensor (\[a, b] -> V.* e_a e_b) [5, 5]
def g~i~j := M.inverse g_#_#

assertEqual "Metric tensor g_1_#"
  g_1_#
  [| r^2, 0, 0, 0, 0 |]_#
assertEqual "Metric tensor g_2_#"
  g_2_#
  [| 0, r^2 * (sin θ)^2, 0, 0, 0 |]_#

-- Christoffel symbols
def Γ_i_j_k := (1 / 2) * (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i)

def Γ~i_j_k := withSymbols [m]
  g~i~m . Γ_m_j_k

assertEqual "Christoffel symbols of the second kind Γ~1_#_#"
  Γ~1_#_#
  [| [| 0, 0, 0, 0, 0 |]
   , [| 0, -1 * sin θ * cos θ, 0, 0, 0 |]
   , [| 0, 0, -1 * sin θ * cos θ * (sin φ)^2, 0, 0 |]
   , [| 0, 0, 0, -1 * sin θ * cos θ * (sin φ)^2 * (sin ψ)^2, 0 |]
   , [| 0, 0, 0, 0, -1 * sin θ * cos θ * (sin φ)^2 * (sin ψ)^2 * (sin η)^2 |]
   |]_#_#
assertEqual "Christoffel symbols of the second kind Γ~2_#_#"
  Γ~2_#_#
  [| [| 0, (cos θ) / (sin θ), 0, 0, 0 |]
   , [| (cos θ) / (sin θ), 0, 0, 0, 0 |]
   , [| 0, 0, -1 * sin φ * cos φ, 0, 0 |]
   , [| 0, 0, 0, -1 * sin φ * cos φ * (sin ψ)^2, 0 |]
   , [| 0, 0, 0, 0, -1 * sin φ * cos φ * (sin ψ)^2 * (sin η)^2 |]
   |]_#_#

-- Riemann curvature
def 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_2"
  R~#_#_1_2
  [| [| 0, (sin θ)^2, 0, 0, 0 |], [| -1, 0, 0, 0, 0 |], [| 0, 0, 0, 0, 0 |], [| 0, 0, 0, 0, 0 |], [| 0, 0, 0, 0, 0 |] |]~#_#
assertEqual "Riemann curvature R~#_#_2_1"
  R~#_#_2_1
  [| [| 0, -1 * (sin θ)^2, 0, 0, 0 |], [| 1, 0, 0, 0, 0 |], [| 0, 0, 0, 0, 0 |], [| 0, 0, 0, 0, 0 |], [| 0, 0, 0, 0, 0 |] |]~#_#

-- Ricci curvature
def Ric_i_j := withSymbols [m]
  sum (contract R~m_i_m_j)

assertEqual "Ricci curvature Ric_1_#"
  Ric_1_#
  [| 4, 0, 0, 0, 0 |]_#
assertEqual "Ricci curvature Ric_2_#"
  Ric_2_#
  [| 0, 4 * (sin θ)^2, 0, 0, 0 |]_#

-- Scalar curvature
def scalarCurvature := withSymbols [i, j]
  g~i~j . Ric_i_j

assertEqual "scalar curvature"
  scalarCurvature
  (20 / r^2)