euler-form-of-T2.egi

-- Euler form of T2 (Torus)

declare symbol θ, φ, a, b

def x := [| θ, φ |]

def X := [| '(a * cos θ + b) * cos φ, '(a * cos θ + b) * sin φ, a * sin θ |]

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

-- Metric tensor
def g_i_j := generateTensor (\[a, b] -> V.* e_a e_b) [2, 2]
def g~i~j := M.inverse g_#_#

assertEqual "Metric tensor"
  g_#_#
  [| [| a^2, 0 |], [| 0, '(a * cos θ + b)^2 |] |]_#_#

-- 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

-- Connection 1-form
def ω0 := Γ~#_#_#

-- Vielbein
def A := [| [| 1 / a, 0 |], [| 0, 1 / '(a * cos θ + b) |] |]

-- Transformed connection
def d A := (flip ∂/∂) x~# A_#_#

def ω := withSymbols [i, j, k, l]
  (M.inverse A)~i_j . ω0~j_k . A~k_l + (M.inverse A)~i_j . d A~j_l

-- Curvature form
def wedge X Y := X !. Y

def Ω := withSymbols [i, j, k]
  dfNormalize (d ω~i_j + wedge ω~i_k ω~k_j)

-- Euler form
def eulerForm := (1 / (2 * π)) * (Ω~1_2 - Ω~2_1)

-- The Euler form integrates to the Euler characteristic χ = 0 for T²
assertEqual "Euler form of T2"
  eulerForm
  [| [| cos θ / ('(a * cos θ + b) * a * π), 0 |], [| 0, cos θ / ('(a * cos θ + b) * a * π) |] |]