Riemann Curvature Tensor of `S^5`

Dec 4 2016
The following code calculates the Riemann curvature tensor of `S^5`.
-- 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 := ∂/∂ X_j x~i

-- Metric tensors
def g{_i_j} := generateTensor (\[x, y] -> V.* e_x_# e_y_#) [5, 5]
def g{~i~j} := M.inverse g_#_#

-- 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

-- 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

--R~#_#_#_#

--def R{[_a_b][_c_d]} := withSymbols [i] g_a_i . R~i_b_c_d

-- Ricci curvature
def Ric[_a_b] := withSymbols [m, n]
  sum (contract (R~m_a_m_b))

Ric_#_# -- 7.422 sec

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

-- Conformal curvature tensor
def C_i_k_l_m := R_i_k_l_m +
                 (Ric_i_m . g_k_l - Ric_i_l . g_k_m + Ric_k_l . g_i_m - Ric_k_m . g_i_l) +
                 (scalarCurvature / 2) * (g_i_l . g_k_m - g_i_m . g_k_l)
--C_#_#_#_#

-- Wodzicki-Chern-Simons class
def S :=
 let (es, os) := evenAndOddPermutations 5 in
   sum (map (\σ -> R~u_1_s_(σ 1) . R~s_t_(σ 3)_(σ 2) . R~t_u_(σ 5)_(σ 4)) es) -
   sum (map (\σ -> R~u_1_s_(σ 1) . R~s_t_(σ 3)_(σ 2) . R~t_u_(σ 5)_(σ 4)) os)

--S -- 0 -- 16.957 sec
Links

Back to the Table of Contents