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  {
   "cell_type": "markdown",
   "id": "71d40661",
   "metadata": {},
   "source": [
    "# 埋め込みトーラス$T^2$のリーマン曲率テンソル\n",
    "\n",
    "管半径$a$、大半径$b$の環状トーラスを次のように埋め込みます。\n",
    "\n",
    "$$\n",
    "X(\\theta,\\phi)=\\bigl((b+a\\cos\\theta)\\cos\\phi,\n",
    "(b+a\\cos\\theta)\\sin\\phi,a\\sin\\theta\\bigr),\\qquad b>a>0.\n",
    "$$\n",
    "\n",
    "内在的に平坦な商空間としてのトーラスとは異なり、この埋め込み\n",
    "トーラスの曲率は場所によって符号が変わります。ここでは接続、\n",
    "リーマンテンソル、リッチテンソル、スカラー曲率を計算します。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8b3592df",
   "metadata": {},
   "source": [
    "## 座標と計量\n",
    "\n",
    "座標ベクトルは互いに直交し、その長さの2乗は$a^2$と\n",
    "$(b+a\\cos\\theta)^2$です。したがって、\n",
    "\n",
    "$$\n",
    "g=\\begin{pmatrix}a^2&0\\\\0&(b+a\\cos\\theta)^2\\end{pmatrix}.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "95c32f2b",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol a, b, θ, φ: 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",
    "\n",
    "def e_i_j : Matrix MathValue := ∂/∂ X_j x~i\n",
    "\n",
    "def g_i_j : Matrix MathValue :=\n",
    "  generateTensor (\\[u, v] -> V.* e_u_# e_v_#) [2, 2]\n",
    "\n",
    "def g~i~j : Matrix MathValue := M.inverse g_#_#\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "4b8cc8f5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} \\cos(φ) (\\cos(θ) a + b) \\\\ \\sin(φ) (\\cos(θ) a + b) \\\\ \\sin(θ) a\\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f275c364",
   "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": "code",
   "execution_count": 4,
   "id": "ba4e727c",
   "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": "62ffcc8b",
   "metadata": {},
   "source": [
    "## レヴィ・チヴィタ接続\n",
    "\n",
    "次の式を用い、第1添字を$g^{ij}$で上げます。\n",
    "$\\Gamma_{ijk}=\\tfrac12(\\partial_jg_{ik}+\\partial_kg_{ij}-\\partial_ig_{jk})$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "4228de9b",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Γ_i_j_k : Tensor MathValue :=\n",
    "  (1 / 2) * (∂/∂ 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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "f4c35e9a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & 0 \\\\ 0 & \\sin(θ) (\\cos(θ) a + b) a^{-1} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Γ~1_#_#\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ce671ad7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & -\\sin(θ) (\\cos(θ) a + b)^{-1} a \\\\ -\\sin(θ) (\\cos(θ) a + b)^{-1} a & 0 \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Γ~2_#_#\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bfea5de7",
   "metadata": {},
   "source": [
    "## リーマンテンソル\n",
    "\n",
    "末尾の2つの添字は反対称です。第1添字を下げると、$R_{ijkl}$で\n",
    "よく知られた添字対の対称性も明らかになります。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "0a547618",
   "metadata": {},
   "outputs": [],
   "source": [
    "def R~i_j_k_l : Tensor MathValue := withSymbols [m]\n",
    "  ∂/∂ Γ~i_j_l x~k - ∂/∂ Γ~i_j_k x~l\n",
    "    + Γ~m_j_l . Γ~i_m_k - Γ~m_j_k . Γ~i_m_l\n",
    "\n",
    "def R_i_j_k_l : Tensor MathValue := withSymbols [m]\n",
    "  g_i_m . R~m_j_k_l\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "16dc0741",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & \\cos(θ) (\\cos(θ) a + b) a^{-1} \\\\ -\\cos(θ) (\\cos(θ) a + b)^{-1} a & 0 \\\\ \\end{pmatrix}_{\\;\\#}^{\\#\\;}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "R~#_#_1_2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "7cfb8edc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\cos(θ) (\\cos(θ) a + b) a$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "R_1_2_1_2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "afa384c7",
   "metadata": {},
   "source": [
    "## リッチ曲率・スカラー曲率・ガウス曲率\n",
    "\n",
    "2次元では、スカラー曲率はガウス曲率の2倍、すなわち\n",
    "$\\mathcal R=2K$です。このトーラスでは、\n",
    "\n",
    "$$\n",
    "K(\\theta)=\\frac{\\cos\\theta}{a(b+a\\cos\\theta)}.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c62da1ed",
   "metadata": {},
   "outputs": [],
   "source": [
    "def Ric_i_j : Matrix MathValue := withSymbols [m]\n",
    "  sum (contract R~m_i_m_j)\n",
    "\n",
    "def scalarCurvature : MathValue := withSymbols [i, j]\n",
    "  g~i~j . Ric_i_j\n",
    "\n",
    "def gaussianCurvature : MathValue := scalarCurvature / 2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "846b78da",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} \\cos(θ) (\\cos(θ) a + b)^{-1} a & 0 \\\\ 0 & \\cos(θ) (\\cos(θ) a + b) a^{-1} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Ric_#_#\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "a128cfed",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$2 \\cos(θ) (\\cos(θ) a + b)^{-1} a^{-1}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "scalarCurvature\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "5779b1bc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\cos(θ) (\\cos(θ) a + b)^{-1} a^{-1}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gaussianCurvature\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b9496b2",
   "metadata": {},
   "source": [
    "## 幾何学的な意味\n",
    "\n",
    "トーラスの外側（$\\cos\\theta>0$）では曲率が正、内側\n",
    "（$\\cos\\theta<0$）では負となり、上端と下端の円周上では曲率が0で\n",
    "放物的になります。\n",
    "ガウス–ボンネ積分では正負の寄与が打ち消し合い、\n",
    "$\\chi(T^2)=0$と整合します。\n"
   ]
  }
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