{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "71a07d52",
   "metadata": {},
   "source": [
    "# Riemann Curvature Tensor of $S^5$\n",
    "\n",
    "A round 5-sphere of radius $r$ has constant sectional curvature\n",
    "$1/r^2$.  This notebook builds its metric, Levi-Civita connection, Riemann\n",
    "tensor, Ricci tensor, and scalar curvature in indexed Egison notation.\n",
    "\n",
    "We use the convention\n",
    "\n",
    "$$\n",
    "R^i{}_{jkl}\n",
    "  = \\partial_k\\Gamma^i{}_{jl}\n",
    "  - \\partial_l\\Gamma^i{}_{jk}\n",
    "  + \\Gamma^m{}_{jl}\\Gamma^i{}_{mk}\n",
    "  - \\Gamma^m{}_{jk}\\Gamma^i{}_{ml}.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "838f9451",
   "metadata": {},
   "source": [
    "## Hyperspherical chart\n",
    "\n",
    "The coordinates are $x=(\\theta, \\phi, \\psi, \\eta, \\delta)$.  The standard embedding\n",
    "$X:S^5\\hookrightarrow\\mathbb{R}^6$ is built by\n",
    "successively multiplying by sines: $X_1=r\\cos \\theta$,\n",
    "$X_2=r\\sin \\theta\\cos \\phi$, and so on.  It makes\n",
    "$X\\mathbin{\\cdot}X=r^2$ manifest.\n",
    "\n",
    "The chart excludes the usual coordinate poles; those singularities are\n",
    "features of hyperspherical coordinates, not of the round geometry.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "4220bd6c",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol r, θ, φ, ψ, η, δ: MathValue\n",
    "\n",
    "def x : Vector MathValue := [| θ, φ, ψ, η, δ |]\n",
    "\n",
    "def X : Vector MathValue := [| r * cos θ, r * sin θ * cos φ, r * sin θ * sin φ * cos ψ, r * sin θ * sin φ * sin ψ * cos η, r * sin θ * sin φ * sin ψ * sin η * cos δ, r * sin θ * sin φ * sin ψ * sin η * sin δ |]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "e4338c48",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} \\cos(θ) r \\\\ \\cos(φ) \\sin(θ) r \\\\ \\cos(ψ) \\sin(θ) \\sin(φ) r \\\\ \\cos(η) \\sin(θ) \\sin(φ) \\sin(ψ) r \\\\ \\cos(δ) \\sin(η) \\sin(θ) \\sin(φ) \\sin(ψ) r \\\\ \\sin(δ) \\sin(η) \\sin(θ) \\sin(φ) \\sin(ψ) r\\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "X\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd88c7c6",
   "metadata": {},
   "source": [
    "## Metric and inverse metric\n",
    "\n",
    "Differentiating the embedding gives an orthogonal coordinate basis.  Its\n",
    "line element is\n",
    "\n",
    "$$\n",
    "ds^2=r^2\\left(d\\theta^2 + \\sin^2 \\theta d\\phi^2 + \\sin^2 \\theta \\sin^2 \\phi d\\psi^2 + \\sin^2 \\theta \\sin^2 \\phi \\sin^2 \\psi d\\eta^2 + \\sin^2 \\theta \\sin^2 \\phi \\sin^2 \\psi \\sin^2 \\eta d\\delta^2\\right).\n",
    "$$\n",
    "\n",
    "Egison differentiates the embedding to obtain the coordinate tangent\n",
    "vectors and constructs every component as their dot product,\n",
    "$g_{ij}=\\partial_iX\\mathbin{\\cdot}\\partial_jX$.  The inverse metric is\n",
    "then computed from that induced metric.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "303f0391",
   "metadata": {},
   "outputs": [],
   "source": [
    "def e_i_j : Matrix MathValue := ∂/∂ X_j x~i\n",
    "\n",
    "def g_i_j : Matrix MathValue :=\n",
    "  generateTensor (\\[a, b] -> V.* e_a_# e_b_#) [5, 5]\n",
    "\n",
    "def g~i~j : Matrix MathValue := M.inverse g_#_#\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "7e231bdc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} r^{2} \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0\\\\ \\end{pmatrix}_{\\#}^{\\;}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g_1_#\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "65392fa4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ \\sin(η)^{2} \\sin(θ)^{2} \\sin(φ)^{2} \\sin(ψ)^{2} r^{2}\\\\ \\end{pmatrix}_{\\#}^{\\;}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g_5_#\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "1eb23aef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} r^{-2} \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0\\\\ \\end{pmatrix}_{\\;}^{\\#}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g~1~#\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5dc79c4e",
   "metadata": {},
   "source": [
    "## Levi-Civita connection\n",
    "\n",
    "The Christoffel symbols of the first kind and second kind are\n",
    "\n",
    "$$\n",
    "\\Gamma_{ijk}=\\frac12\n",
    "(\\partial_jg_{ik}+\\partial_kg_{ij}-\\partial_ig_{jk}),\n",
    "\\qquad\n",
    "\\Gamma^i{}_{jk}=g^{im}\\Gamma_{mjk}.\n",
    "$$\n",
    "\n",
    "Repeated symbolic indices are contracted by `.`.  The `withSymbols`\n",
    "block makes the dummy index local to the definition.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "7ec741ce",
   "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": 8,
   "id": "c2ac3a67",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$-\\cos(θ) \\sin(θ)$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Γ~1_2_2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "2d3b7434",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\cos(θ) \\sin(θ)^{-1}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Γ~2_1_2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "77decc44",
   "metadata": {},
   "source": [
    "## Riemann tensor\n",
    "\n",
    "The following definition is a direct transcription of the stated\n",
    "convention.  The two output cells sample the same coordinate\n",
    "two-plane with the first two tensor slots exchanged; their different\n",
    "coordinate factors are exactly what one expects in a non-orthonormal\n",
    "coordinate basis.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "36174543",
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "40c87ece",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\sin(θ)^{2}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "R~1_2_1_2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "e46443ed",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$-1$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "R~2_1_1_2\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ae1a8e44",
   "metadata": {},
   "source": [
    "## Ricci and scalar curvature\n",
    "\n",
    "Contracting the first and third Riemann indices gives\n",
    "\n",
    "$$\n",
    "\\operatorname{Ric}_{ij}=R^m{}_{imj},\n",
    "\\qquad\n",
    "\\mathcal{R}=g^{ij}\\operatorname{Ric}_{ij}.\n",
    "$$\n",
    "\n",
    "For a round $S^5$ the coordinate-free prediction is\n",
    "\n",
    "$$\n",
    "\\operatorname{Ric}=\\frac{4}{r^2}g,\n",
    "\\qquad\n",
    "\\mathcal{R}=\\frac{20}{r^2}.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "7e28b7c5",
   "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"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93c0b5c1",
   "metadata": {},
   "source": [
    "## Interpretation\n",
    "\n",
    "The sampled components are coordinate dependent, but their contractions\n",
    "recover the invariant statement: every tangent two-plane has sectional\n",
    "curvature $1/r^2$, the metric is Einstein, and the scalar curvature is\n",
    "$20/r^2$.  Factors such as $\\sin \\theta$ vanish\n",
    "at chart poles because the coordinate frame degenerates there; the\n",
    "curvature itself remains smooth.  The Ricci and scalar-curvature definitions are present, but their full contractions are intentionally not output cells: the selected Riemann components demonstrate the constant-curvature pattern without forcing a large symbolic expansion.\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Egison",
   "language": "egison",
   "name": "egison"
  },
  "language_info": {
   "codemirror_mode": "egison",
   "file_extension": ".egi",
   "mimetype": "text/x-egison",
   "name": "egison"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}