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    "# Euler Form of the Two-Sphere\n",
    "\n",
    "The Euler class of an oriented rank-two tangent bundle is represented\n",
    "by a curvature two-form.  For the round sphere,\n",
    "\n",
    "$$\\int_{S^2} e(TS^2)=\\chi(S^2)=2.$$\n",
    "\n",
    "We compute the metric from the embedding, transform the connection to\n",
    "an orthonormal frame, and apply Cartan's curvature equation.\n"
   ]
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   "source": [
    "## Embedding and induced metric\n",
    "\n",
    "The radius-$r$ sphere is parametrized by $(\\theta,\\phi)$.  Dot products\n",
    "of its coordinate tangent vectors produce the metric rather than taking\n",
    "it as an input.\n"
   ]
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    "declare symbol r, θ, φ : MathValue\n",
    "\n",
    "def x : Vector MathValue := [| θ, φ |]\n",
    "def X : Vector MathValue :=\n",
    "  [| r * sin θ * cos φ\n",
    "   , r * sin θ * sin φ\n",
    "   , r * cos θ |]\n",
    "\n",
    "def e_i_j : Matrix MathValue := ∂/∂ X_j x~i\n",
    "def g_i_j : Matrix MathValue :=\n",
    "  generateTensor (\\[a, b] -> V.* e_a_# e_b_#) [2, 2]\n",
    "def g~i~j : Matrix MathValue := M.inverse g_#_#\n"
   ]
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       "$\\begin{pmatrix} r^{2} & 0 \\\\ 0 & \\sin(θ)^{2} r^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
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    "g_#_#\n"
   ]
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    "## Levi-Civita connection and orthonormal frame\n",
    "\n",
    "The diagonal vielbein rescales the coordinate basis by $r$ and\n",
    "$r\\sin\\theta$.  Under a frame change $A$, the connection transforms as\n",
    "\n",
    "$$\\omega=A^{-1}\\omega_0A+A^{-1}dA.$$\n"
   ]
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    "def Γ_i_j_k : Tensor MathValue :=\n",
    "  (1 / 2) *\n",
    "    (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i)\n",
    "\n",
    "def Γ~i_j_k : Tensor MathValue := withSymbols [m]\n",
    "  g~i~m . Γ_m_j_k\n",
    "\n",
    "def A : Matrix MathValue :=\n",
    "  [| [| 1 / r, 0 |], [| 0, 1 / (r * sin θ) |] |]\n",
    "\n",
    "def d (t : Tensor MathValue) : Tensor MathValue :=\n",
    "  !(flip ∂/∂) x t\n",
    "\n",
    "def ω0~i_j : Matrix MathValue := Γ~i_j_#\n",
    "def ω~i_j : Tensor MathValue := withSymbols [a, b]\n",
    "  (M.inverse A)~i_a . ω0~a_b . A~b_j\n",
    "  + (M.inverse A)~i_a . d A~a_j\n"
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   "source": [
    "## Curvature and Euler form\n",
    "\n",
    "Cartan's second equation gives $\\Omega$.  In two dimensions the\n",
    "Pfaffian reduces to the difference of the two off-diagonal curvature\n",
    "components.\n"
   ]
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   "source": [
    "def Ω~i_j : Tensor MathValue := withSymbols [k]\n",
    "  antisymmetrize (d ω~i_j + ω~i_k ∧ ω~k_j)\n",
    "\n",
    "def eulerForm : Tensor MathValue :=\n",
    "  (1 / (4 * π)) * withSymbols [t1, t2]\n",
    "    (Ω~1_2_t1_t2 - Ω~2_1_t1_t2)\n"
   ]
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       "$\\begin{pmatrix} 0 & \\frac{1}{4} \\sin(θ) π^{-1} \\\\ \\frac{-1}{4} \\sin(θ) π^{-1} & 0 \\\\ \\end{pmatrix}$"
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    "eulerForm\n"
   ]
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    "The upper tensor component is\n",
    "\n",
    "$$e_{12}=\\frac{\\sin\\theta}{4\\pi}.$$\n",
    "\n",
    "A full antisymmetric tensor stores both $e_{12}$ and $e_{21}$, so\n",
    "the corresponding oriented differential-form density is\n",
    "\n",
    "$$e(TS^2)=\\frac{\\sin\\theta}{2\\pi}\\,d\\theta\\wedge d\\phi.$$\n",
    "\n",
    "Therefore\n",
    "\n",
    "$$\\int_0^{2\\pi}\\!\\int_0^\\pi\n",
    "  \\frac{\\sin\\theta}{2\\pi}\\,d\\theta\\,d\\phi=2,$$\n",
    "\n",
    "recovering the Euler characteristic of the sphere.  The radius cancels,\n",
    "illustrating that the Euler number is topological rather than metric-size\n",
    "dependent.\n"
   ]
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