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    "# Euler Form of the Two-Torus\n",
    "\n",
    "A torus has regions of positive and negative Gaussian curvature, but\n",
    "their total cancels:\n",
    "\n",
    "$$\\int_{T^2}e(TT^2)=\\chi(T^2)=0.$$\n",
    "\n",
    "This notebook computes the local Euler form from an embedded torus and\n",
    "makes that cancellation explicit.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb3dad5f",
   "metadata": {},
   "source": [
    "## Embedded torus and metric\n",
    "\n",
    "Let $a$ be the tube radius and $b$ the distance from the tube center to\n",
    "the symmetry axis.  The opaque quotes around $a\\cos\\theta+b$ preserve a\n",
    "compact symbolic atom during intermediate matrix computations.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "441f6f87",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol θ, φ, a, b : MathValue\n",
    "\n",
    "def x : Vector MathValue := [| θ, φ |]\n",
    "def X : Vector MathValue :=\n",
    "  [| `(a * cos θ + b) * cos φ\n",
    "   , `(a * cos θ + b) * sin φ\n",
    "   , a * sin θ |]\n",
    "\n",
    "def e_i_j : Matrix MathValue := ∂/∂ X_j x~i\n",
    "def g_i_j : Matrix MathValue :=\n",
    "  generateTensor (\\[u, v] -> V.* e_u_# e_v_#) [2, 2]\n",
    "def g~i~j : Matrix MathValue := M.inverse g_#_#\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "008a9076",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} a^{2} & 0 \\\\ 0 & (\\cos(θ) a + b)^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g_#_#\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0a8ae65f",
   "metadata": {},
   "source": [
    "## Connection in an orthonormal frame\n",
    "\n",
    "The vielbein removes the coordinate scale factors.  As on the sphere,\n",
    "the inhomogeneous $A^{-1}dA$ term is essential when changing frames.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1d78635e",
   "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 / a, 0 |]\n",
    "   , [| 0, 1 / `(a * cos θ + b) |] |]\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 [u, v]\n",
    "  (M.inverse A)~i_u . ω0~u_v . A~v_j\n",
    "  + (M.inverse A)~i_u . d A~u_j\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e214987",
   "metadata": {},
   "source": [
    "The nonzero connection coefficient changes sign across the torus and\n",
    "drives the sign-changing curvature.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4b78f197",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\sin(θ)$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ω~1_2_2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f92fa3c1",
   "metadata": {},
   "source": [
    "## Curvature and Euler form\n",
    "\n",
    "We explicitly antisymmetrize the two form indices after applying\n",
    "Cartan's equation, then take the rank-two Pfaffian.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "d808dc93",
   "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": 6,
   "id": "88b639ca",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & \\frac{1}{4} \\cos(θ) π^{-1} \\\\ \\frac{-1}{4} \\cos(θ) π^{-1} & 0 \\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "eulerForm\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1005146",
   "metadata": {},
   "source": [
    "The upper tensor component is $\\cos\\theta/(4\\pi)$; including its\n",
    "antisymmetric partner gives the oriented density\n",
    "$\\cos\\theta\\,d\\theta\\wedge d\\phi/(2\\pi)$.  Its integral over one full\n",
    "meridian vanishes:\n",
    "\n",
    "$$\\int_0^{2\\pi}\\cos\\theta\\,d\\theta=0.$$\n",
    "\n",
    "Hence the positive outer curvature and negative inner curvature cancel,\n",
    "giving $\\chi(T^2)=0$ independently of $a$ and $b$.\n"
   ]
  }
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