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    "# 2次元球面のオイラー形式\n",
    "\n",
    "向き付けられた階数2の接束のオイラー類は、曲率2形式によって表されます。\n",
    "丸い球面に対しては\n",
    "\n",
    "$$\\int_{S^2} e(TS^2)=\\chi(S^2)=2.$$\n",
    "\n",
    "となります。埋め込みから計量を計算し、接続を正規直交枠へ変換してから、\n",
    "カルタンの曲率方程式を適用します。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb1398a8",
   "metadata": {},
   "source": [
    "## 埋め込みと誘導計量\n",
    "\n",
    "半径 $r$ の球面を $(\\theta,\\phi)$ でパラメーター表示します。計量を\n",
    "入力として与えるのではなく、座標接ベクトル同士の内積から導きます。\n"
   ]
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   "source": [
    "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "cf446863",
   "metadata": {},
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       "$\\begin{pmatrix} r^{2} & 0 \\\\ 0 & \\sin(θ)^{2} r^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
      ]
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   "source": [
    "g_#_#\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47571e36",
   "metadata": {},
   "source": [
    "## レヴィ・チヴィタ接続と正規直交枠\n",
    "\n",
    "対角な多脚場（vielbein）は、座標基底を $r$ と $r\\sin\\theta$ で\n",
    "スケール変換します。\n",
    "枠の変換 $A$ のもとで、接続は\n",
    "\n",
    "$$\\omega=A^{-1}\\omega_0A+A^{-1}dA.$$\n"
   ]
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   "cell_type": "code",
   "execution_count": 3,
   "id": "f85393f5",
   "metadata": {},
   "outputs": [],
   "source": [
    "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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   "cell_type": "markdown",
   "id": "9dcaffe7",
   "metadata": {},
   "source": [
    "## 曲率とオイラー形式\n",
    "\n",
    "カルタンの第2方程式から $\\Omega$ が得られます。2次元では、\n",
    "パフィアン（Pfaffian）は2つの非対角な曲率成分の差に帰着します。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "5da2150b",
   "metadata": {},
   "outputs": [],
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "33080a0d",
   "metadata": {},
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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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   "source": [
    "eulerForm\n"
   ]
  },
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   "id": "74af171b",
   "metadata": {},
   "source": [
    "上三角側のテンソル成分は\n",
    "\n",
    "$$e_{12}=\\frac{\\sin\\theta}{4\\pi}.$$\n",
    "\n",
    "です。完全な反対称テンソルは $e_{12}$ と $e_{21}$ の両方を保持するため、\n",
    "対応する向き付き微分形式の密度は\n",
    "\n",
    "$$e(TS^2)=\\frac{\\sin\\theta}{2\\pi}\\,d\\theta\\wedge d\\phi.$$\n",
    "\n",
    "となります。したがって\n",
    "\n",
    "$$\\int_0^{2\\pi}\\!\\int_0^\\pi\n",
    "  \\frac{\\sin\\theta}{2\\pi}\\,d\\theta\\,d\\phi=2,$$\n",
    "\n",
    "となり、球面のオイラー標数が得られます。半径は相殺されるため、オイラー数が\n",
    "計量の大きさに依存するのではなく、位相的な量であることが分かります。\n"
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