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   "source": [
    "# Curvature Two-Form on the Sphere\n",
    "\n",
    "The Riemann tensor can be computed directly from Christoffel symbols or\n",
    "assembled as the curvature of the connection one-form.  Cartan's second\n",
    "structure equation is\n",
    "\n",
    "$$\\Omega^i{}_j=d\\omega^i{}_j+\\omega^i{}_k\\wedge\\omega^k{}_j,$$\n",
    "\n",
    "with $\\Omega^i{}_j=\\tfrac12R^i{}_{jkl}\\,dx^k\\wedge dx^l$.\n"
   ]
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   "source": [
    "## Round metric on $S^2$\n",
    "\n",
    "For a sphere of radius $r$ with coordinates $(\\theta,\\phi)$,\n",
    "\n",
    "$$g_{ij}=\\operatorname{diag}(r^2,r^2\\sin^2\\theta).$$\n"
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    "declare symbol r, θ, φ : MathValue\n",
    "\n",
    "def x : Vector MathValue := [| θ, φ |]\n",
    "def g_i_j : Matrix MathValue :=\n",
    "  [| [| r ^ 2, 0 |], [| 0, r ^ 2 * (sin θ) ^ 2 |] |]_i_j\n",
    "def g~i~j : Matrix MathValue :=\n",
    "  [| [| 1 / r ^ 2, 0 |]\n",
    "   , [| 0, 1 / (r ^ 2 * (sin θ) ^ 2) |] |]~i~j\n"
   ]
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       "$\\begin{pmatrix} r^{2} & 0 \\\\ 0 & \\sin(θ)^{2} r^{2} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$"
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   "source": [
    "g_#_#\n"
   ]
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   "source": [
    "## Direct Riemann tensor\n",
    "\n",
    "The Levi-Civita connection follows from metric compatibility and zero\n",
    "torsion.  Egison's repeated symbolic indices perform the contraction\n",
    "over $m$.\n"
   ]
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   "source": [
    "def Γ_j_l_k : Tensor MathValue :=\n",
    "  (1 / 2) *\n",
    "    (∂/∂ g_j_l x~k + ∂/∂ g_j_k x~l - ∂/∂ g_k_l x~j)\n",
    "\n",
    "def Γ~i_k_l : Tensor MathValue :=\n",
    "  withSymbols [j] g~i~j . Γ_j_l_k\n",
    "\n",
    "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"
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       "$\\begin{pmatrix} 0 & \\sin(θ)^{2} \\\\ -1 & 0 \\\\ \\end{pmatrix}_{\\;\\#}^{\\#\\;}$"
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   "source": [
    "R~#_#_1_2\n"
   ]
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   "source": [
    "## Cartan's curvature form\n",
    "\n",
    "Regard the last Christoffel index as the one-form index.  The exterior\n",
    "derivative adds one form index, and `antisymmetrize` projects the result\n",
    "onto a genuine two-form.\n"
   ]
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  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "74ecd7e6",
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   "source": [
    "def d (t : Tensor MathValue) : Tensor MathValue :=\n",
    "  !(flip ∂/∂) x t\n",
    "\n",
    "def ω~i_j : Matrix MathValue := Γ~i_j_#\n",
    "\n",
    "def Ω~i_j : Tensor MathValue := withSymbols [k]\n",
    "  antisymmetrize (d ω~i_j + ω~i_k ∧ ω~k_j)\n"
   ]
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       "$\\begin{pmatrix} 0 & \\frac{1}{2} \\sin(θ)^{2} \\\\ \\frac{-1}{2} & 0 \\\\ \\end{pmatrix}_{\\;\\#}^{\\#\\;}$"
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   "source": [
    "Ω~#_#_1_2\n"
   ]
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   "source": [
    "The curvature-form component is one half of the corresponding direct\n",
    "Riemann component, exactly as\n",
    "$\\Omega^i{}_j=\\tfrac12R^i{}_{jkl}dx^k\\wedge dx^l$ requires.  The two\n",
    "computational paths therefore agree while exposing different geometry:\n",
    "the first uses coordinate indices, and the second treats curvature as\n",
    "the field strength of a connection.\n"
   ]
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