euler-form-of-T2.egi

-- Euler form of T2 (Torus)

declare symbol θ, φ, a, b : MathValue

def x : Vector MathValue := [| θ, φ |]

-- Use backtick `(...)` for opaque-atom quoting (so `(a*cos θ + b)^k stays
-- unexpanded). Apostrophe `'(...)` is the rule-suppression quote — it only
-- turns off `declare rule` rewriting during construction, it does not make
-- an opaque atom — so it would let intermediate computations explode the
-- polynomial.
def X : Vector MathValue := [| `(a * cos θ + b) * cos φ, `(a * cos θ + b) * sin φ, a * sin θ |]

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

-- Metric tensor
def g_i_j : Matrix MathValue := generateTensor (\[x, y] -> V.* e_x_# e_y_#) [2, 2]
def g~i~j : Matrix MathValue := M.inverse g_#_#

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

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

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

-- Vielbein (orthonormal frame)
def A : Matrix MathValue := [| [| 1 / a, 0 |], [| 0, 1 / `(a * cos θ + b) |] |]

-- Exterior derivative (paper canonical: !(flip ∂/∂) x t with disjoint completion)
def d (t : Tensor MathValue) : Tensor MathValue := !(flip ∂/∂) x t

-- Connection 1-form in coordinate basis (paper canonical: ω~i_j := Γ~i_j_#)
def ω0~i_j : Matrix MathValue := Γ~i_j_#

-- Connection 1-form in orthonormal basis (Cartan transformation):
--   ω = A⁻¹ ω₀ A + A⁻¹ dA
def ω~i_j : Tensor MathValue := withSymbols [u, v]
  (M.inverse A)~i_u . ω0~u_v . A~v_j + (M.inverse A)~i_u . d A~u_j

-- Component-wise checks (whole-vector compare currently mismatches on
-- index metadata even when values agree, so use element access).
assertEqual "ω~1_2_1" ω~1_2_1 0
assertEqual "ω~1_2_2" ω~1_2_2 ('sin θ)
assertEqual "ω~2_1_1" ω~2_1_1 0
assertEqual "ω~2_1_2" ω~2_1_2 (- ('sin θ))

-- Curvature 2-form (Cartan structure equation)
def Ω'~i_j : Tensor MathValue := withSymbols [k]
  d ω~i_j + ω~i_k ∧ ω~k_j
def Ω~i_j_t1_t2 : Tensor MathValue := Ω'~i_j_t1_t2 - Ω'~i_j_t2_t1

-- Euler form: e(T²) = (1/(4π)) (Ω₁² - Ω²₁)
def eulerForm : Tensor MathValue :=
  (1 / (4 * π)) * withSymbols [t1, t2] (Ω~1_2_t1_t2 - Ω~2_1_t1_t2)

-- The Euler form integrates to χ(T²) = 0.
-- In the orthonormal vielbein frame, the (a*(b+a*cos θ)) Jacobian factors
-- cancel out and the curvature 2-form reduces to cos θ /(2π) dθ∧dφ.
assertEqual "Euler form of T2"
  eulerForm
  [| [| 0, cos θ / (2 * π) |]
   , [| - cos θ / (2 * π), 0 |] |]