25Manifold_Torsion_Curvature_Forms.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 25. Torsion and curvature forms\n",
    "\n",
    "This notebook is part of the [Introduction to manifolds in SageMath](https://sagemanifolds.obspm.fr/intro_to_manifolds.html) by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'SageMath version 9.6, Release Date: 2022-05-15'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To describe connections one can use  bases not necessarily\n",
    "induced by coordinate systems. We can use sets $\\{e_1 , . . . , e_n \\}$ of smooth vector\n",
    "fields defined on some open subset $U$ of the manifold $M$ such that, at each point $p ∈ U $, the tangent vectors $e_i$  at $p$ form a basis of $T_p M$. Let the set of 1-forms $\\{e^1 , . . . , e^n \\}$ be its dual basis (i.e.  $e^i (e_j ) = δ_{ij} $). If there exists a coordinate system $(x_1 , . . . , x_n )$ such\n",
    "that $e_i = \\frac{∂}{∂ x^i}$ or, equivalently, $e^i = dx^i$ , we will say that the basis ${e_1 , . . . , e_n }$ is\n",
    "**holonomic**. In the present notebook the set $\\{e_1,\\ldots e_n\\}$ will denote **not necessarily holonomic basis**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Connection forms\n",
    "\n",
    "<br>\n",
    "\n",
    "If $∇$ is a connection on a manifold $ M$, the **connection forms**, $\\omega^i_j$ , with\n",
    "respect to the basis $\\{e_1 , . . . , e_n \\}$, are the 1-forms defined by\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\omega_j^i(X)=e^i(\\nabla_Xe_j),\\quad \\text{for} X\\in \\mathfrak{X}(M),\n",
    "\\tag{25.1}\n",
    "\\end{eqnarray}\n",
    "or equivalently\n",
    "\\begin{eqnarray}\n",
    "\\nabla_Xe_j=\\omega_j^i(X)e_i,\\quad \\text{for} X\\in \\mathfrak{X}(M).\n",
    "\\tag{25.1'}\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 25.1**\n",
    "\n",
    "On the two-dimensional manifold we have four connection forms $\\omega^i_j$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\omega^1_{\\ \\, 1}, \\omega^1_{\\ \\, 2}, \\omega^2_{\\ \\, 1}, \\omega^2_{\\ \\, 2}\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\omega^1_{\\ \\, 1}, \\omega^1_{\\ \\, 2}, \\omega^2_{\\ \\, 1}, \\omega^2_{\\ \\, 2}\\right]$"
      ],
      "text/plain": [
       "[1-form nabla connection 1-form (1,1) on the 2-dimensional differentiable manifold M,\n",
       " 1-form nabla connection 1-form (1,2) on the 2-dimensional differentiable manifold M,\n",
       " 1-form nabla connection 1-form (2,1) on the 2-dimensional differentiable manifold M,\n",
       " 1-form nabla connection 1-form (2,2) on the 2-dimensional differentiable manifold M]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "M = Manifold(2, 'M', start_index=1)     # 2-dim. manifold M\n",
    "c_xy.<x,y> = M.chart()                  # chart on M\n",
    "nab = M.affine_connection('nabla', r'\\nabla')  # affine connection\n",
    "nab[1,1,1] = 0                                 # all con. coeff. zero\n",
    "[nab.connection_form(i,j) for i in [1,2] for j in [1,2]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "If we define\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\Gamma ^i_{jk}=\\omega^i_j(e_k),\n",
    "\\tag{25.2}\n",
    "\\end{eqnarray}\n",
    "\n",
    "then for $X\\in \\mathfrak{X}(M)$, we have $X=e^k(X)e_k$ and\n",
    "\n",
    "$$\n",
    "\\omega^i_j(X)=\\omega^i_j(e^k(X)e_k)=e^k(X)\\omega^i_j(e_k)=\\Gamma^i_{jk}e^k(X),$$\n",
    "\n",
    "i.e.,\n",
    "\\begin{eqnarray}\n",
    "\\omega_j^i=\\Gamma^i_{jk}e^k,\n",
    "\\tag{25.3}\n",
    "\\end{eqnarray}\n",
    "and according to (25.1')\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\nabla_{e_i}e_j=\\omega_j^k(e_i)e_k=\\Gamma^k_{ji}e_k,\n",
    "\\tag{25.4}\n",
    "\\end{eqnarray}\n",
    "\n",
    "which is analogous to (21.5) but  holds also for non-holonomic bases."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Torsion forms\n",
    "\n",
    "<br>\n",
    "\n",
    "If $T (X, Y) = ∇ X Y − ∇ Y X − [X, Y],\\ \\ X, Y ∈ \\mathfrak{X}(M)\\ \\ $ is the torsion, then the **torsion 2-forms** $\\theta^i$ with respect to $\\{e_1,\\ldots,e_n\\}$ are defined by\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\theta^i(X,Y)=e^i(T(X,Y)),\\quad \\text{for}\\ X,Y\\in \\mathfrak{X}(M),\n",
    "\\tag{25.5}\n",
    "\\end{eqnarray}\n",
    "or equivalently\n",
    "\n",
    "\\begin{eqnarray}\n",
    "T(X,Y)=\\theta^i(X,Y)e_i,\\quad \\text{for}\\ X,Y\\in \\mathfrak{X}(M).\n",
    "\\tag{25.6}\n",
    "\\end{eqnarray}\n",
    "\n",
    "Since $T$ is antisymmetric, each $\\theta^i$ is 2-form.\n",
    "\n",
    "<br>\n",
    "\n",
    "**Example 25.2**\n",
    "\n",
    "On the two-dimensional manifold we have two torsion 2-forms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\theta^1, \\theta^2\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\theta^1, \\theta^2\\right]$"
      ],
      "text/plain": [
       "[2-form torsion (1) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) on the 2-dimensional differentiable manifold M,\n",
       " 2-form torsion (2) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) on the 2-dimensional differentiable manifold M]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "M = Manifold(2, 'M', start_index=1)    # 2-dim. manifold M\n",
    "c_xy.<x,y> = M.chart()                 # chart on M\n",
    "nab = M.affine_connection('nabla', r'\\nabla') # affine connection\n",
    "nab[1,1,1] = 0                         # all con. coeff. zero\n",
    "[nab.torsion_form(i) for i in [1,2]]   # torsion 2-forms on M"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### First Cartan structure equation\n",
    "\n",
    "<br>\n",
    "\n",
    "Assume we have  vector fields $\\ X\\ $ and $\\ Y=Y^ie_i=e^i(Y)e_i.\\ $ From (21.14) it follows \n",
    "\n",
    "\\begin{equation}\n",
    "\\nabla_Xt(Y)=X(t(Y))-t(\\nabla_XY),\n",
    "\\tag{*}\n",
    "\\end{equation}\n",
    "\n",
    "for 1-forms $t$ on $M$, so\n",
    "\n",
    "$$\\nabla_Xe^i(Y)=X(e^i(Y))-e^i(\\nabla_XY)\n",
    "=X(Y^i)-e^i(\\nabla_X(Y^je_j))\\\\\n",
    "=X(Y^i)-e^i(Y^j\\nabla_Xe_j+(XY^j)e_j)\\\\\n",
    "=X(Y^i)-Y^je^i(\\nabla_Xe_j)-e^i((XY^j)e_j)\\\\\n",
    "=X(Y^i)-Y^je^i(\\nabla_Xe_j)-X(Y^i)\\\\\n",
    "=-Y^j\\omega^i_j(X)=-\\omega^i_j(X)e^j(Y),\n",
    "$$\n",
    "so\n",
    "\\begin{eqnarray}\n",
    "\\nabla_Xe^i=-\\omega^i_j(X)e^j.\n",
    "\\tag{25.7}\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using (*) once again we obtain\n",
    "\n",
    "$$\n",
    "e^i(\\nabla_XY)=X(e^i(Y))-(\\nabla_Xe^i)(Y),\\\\\n",
    "e^i(\\nabla_YX)=Y(e^i(X))-(\\nabla_Ye^i)(X).\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From (16.6) we know that\n",
    "\n",
    "$$\n",
    "(d\\omega)(X,Y)=X\\omega(Y)-Y\\omega(X)-\\omega([X,Y]),\n",
    "$$\n",
    "\n",
    "and from the definition of the exterior product\n",
    "\n",
    "$$\n",
    "t\\wedge s=t\\otimes s-s\\otimes t.\n",
    "$$\n",
    "\n",
    "From those facts \n",
    "and (25.7) we  obtain\n",
    "\n",
    "$$\\theta^i(X,Y)=e^i(T(X,Y))=e^i(\\nabla_XY-\\nabla_YX-[X,Y])\\\\\n",
    "=X(e^i(Y))-(\\nabla_Xe^i)(Y)-Y(e^i(X))+(\\nabla_Ye^i)(X)-e^i([X,Y])\\\\\n",
    "=X(e^i(Y))-Y(e^i(X)-e^i([X,Y])-(\\nabla_Xe^i)(Y)+(\\nabla_Ye^i)(X)\\\\\n",
    "=de^i(X,Y)+\\omega^i_j(X)e^j(Y)-\\omega^i_j(Y)e^j(X)\\\\\n",
    "=(de^i+\\omega^i_j\\wedge e^j)(X,Y).\n",
    "$$\n",
    "\n",
    "We have proved **the first Cartan structural equation**\n",
    "\n",
    "\\begin{eqnarray}\n",
    "\\theta^i=de^i+\\omega^i_j\\wedge  e^j.\n",
    "\\tag{25.8}\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Curvature forms\n",
    "\n",
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $\\ \\ R(X, Y)= ∇_X ∇_Y  − ∇_Y ∇_X − ∇_{[X,Y]},\\ \\\n",
    " X, Y ∈ \\mathfrak{X}(M)\\ \\ $ is the curvature, then the  **curvature forms** $\\Omega^i_j(X,Y)$ with respect to the basis $\\{e_1 , e_2 , . . . , e_n \\}$, are defined by\n",
    " \n",
    "\\begin{equation}\n",
    "\\Omega^i_j(X,Y)=e^i(R(X,Y)e_j),\\quad X, Y ∈ \\mathfrak{X}(M),\n",
    "\\tag{25.9}\n",
    "\\end{equation}\n",
    "\n",
    "or equivalently\n",
    "\n",
    "\\begin{equation}\n",
    "R(X,Y)e_j=\\Omega^i_j(X,Y)e_i,\\quad X, Y ∈ \\mathfrak{X}(M).\n",
    "\\tag{25.9'}\n",
    "\\end{equation}\n",
    "\n",
    "<br>\n",
    "\n",
    "**Example 25.3**\n",
    "\n",
    "On the two-dimensional manifold we have four curvature forms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\Omega^1_{\\ \\, 1}, \\Omega^1_{\\ \\, 2}, \\Omega^2_{\\ \\, 1}, \\Omega^2_{\\ \\, 2}\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\Omega^1_{\\ \\, 1}, \\Omega^1_{\\ \\, 2}, \\Omega^2_{\\ \\, 1}, \\Omega^2_{\\ \\, 2}\\right]$"
      ],
      "text/plain": [
       "[2-form curvature (1,1) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) on the 2-dimensional differentiable manifold M,\n",
       " 2-form curvature (1,2) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) on the 2-dimensional differentiable manifold M,\n",
       " 2-form curvature (2,1) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) on the 2-dimensional differentiable manifold M,\n",
       " 2-form curvature (2,2) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) on the 2-dimensional differentiable manifold M]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "M = Manifold(2,'M',start_index=1)   # manifold M\n",
    "c_xy.<x,y> = M.chart()              # chart on M\n",
    "nab = M.affine_connection('nabla', r'\\nabla') # connection\n",
    "nab[1,1,1] = 0                      # all coefficients zero\n",
    "[nab.curvature_form(i,j) for i in [1,2] for j in [1,2]]\n",
    "                                    # curvature forms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Second Cartan structural equations \n",
    "\n",
    "<br>\n",
    "\n",
    "If we use the formulas\n",
    "$$ R(X, Y)= ∇_X ∇_Y  − ∇_Y ∇_X − ∇_{[X,Y]},\\\\\n",
    "\\nabla_Xe_j=\\omega^i_j(X)e_i,\\quad \\omega^i_j(X)=e^i(\\nabla_Xe_j),\\\\\n",
    "\\nabla_X(\\omega^k_j(Y)e_k)=X(\\omega^k_j(Y))e_k+\\omega^k_j(Y)\\nabla_Xe_k,\\\\\n",
    "(d\\omega)(X,Y)=X\\omega(Y)-Y\\omega(X)-\\omega([X,Y]),\\\\\n",
    "t\\wedge s=t\\otimes s-s\\otimes t,\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "then we can check that\n",
    "$$\\Omega^i_j(X,Y)=e^i(∇_X ∇_Y e_j − ∇_Y ∇_X e_j− ∇_{[X,Y]}e_j)\\\\\n",
    "=e^i(\\nabla_X(\\omega^k_j(Y)e_k)-\\nabla_Y(\\omega^k_j(X)e_k))-\\omega^i_j([X,Y])\\\\\n",
    "=e^i(X(\\omega^k_j(Y))e_k+\\omega^k_j(Y)\\nabla_Xe_k-Y(\\omega^k_j(X))e_k-\\omega^k_j(X)\\nabla_Ye_k)-\\omega^i_j([X,Y])\\\\\n",
    "=(X(\\omega^k_j(Y))e^i(e_k)+\\omega_j^k(Y)e^i(\\nabla_Xe_k)\\\\\n",
    "-Y(\\omega^k_j(X))e^i(e_k)-\\omega^k_j(X)e^i(\\nabla_Ye_k))-\n",
    "\\omega^i_j([X,Y])\\\\\n",
    "=X(\\omega^i_j(Y))+\\omega_j^k(Y)\\omega^i_k(X)-Y(\\omega^i_j(X))-\\omega^k_j(X)\\omega^i_k(Y)-\\omega^i_j([X,Y])\\\\\n",
    "=X(\\omega^i_j(Y))-Y(\\omega^i_j(X))-\\omega^i_j([X,Y])+\\omega_k^i(X)\\omega^k_j(Y)-\\omega^i_k(Y)\\omega_j^k(X)\\\\\n",
    "=d\\omega^i_j(X,Y)+(\\omega^i_k\\wedge \\omega^k_j)(X,Y).\n",
    "$$\n",
    "\n",
    "We have proved **the second Cartan structural equation**\n",
    "\n",
    "\\begin{equation}\n",
    "\\Omega^i_j=d\\omega^i_j+\\omega^i_k\\wedge \\omega^k_j.\n",
    "\\tag{25.10}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Observation:** $\\hskip0.5cm$  If $A_{kl}$ is an antisymmetric matrix, then for 1-forms $e^i$ on $M$ and $X,Y\\in\\mathfrak{X}(M)$\n",
    "$$(A_{kl}e^k\\wedge e^l)(X,Y)=A_{kl}(e^k\\otimes e^l-e^l\\otimes e^k)(X,Y)\\\\\n",
    "=A_{kl}(e^k(X) e^l(Y)-e^l(X) e^k(Y)\n",
    "=A_{kl}X^kY^l-A_{kl}X^lY^k\\\\\n",
    "=A_{kl}X^kY^l+A_{lk}X^lY^k\n",
    "=2A_{kl}X^kY^l.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "If the components of the torsion and the curvature tensors  with respect to the basis $\\{e_1 , . . . , e_n \\}$ are defined by \n",
    "$$T (e_i , e_j ) = T^k_{i j} e_k,\\quad  R(e_i , e_j )e_k = R^l_{ki j}e_l,\n",
    "$$\n",
    "then\n",
    "\\begin{equation}\n",
    "\\begin{matrix}\n",
    "\\theta^i=\\frac{1}{2}T^i_{jk}e^j\\wedge e^k,\\\\\n",
    "\\Omega^i_j=\\frac{1}{2}R^i_{jkl}e^k\\wedge e^l.\n",
    "\\end{matrix}\n",
    "\\tag{25.11}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In fact\n",
    "$$\\theta^i(X,Y)=e^i(T(X,Y))=e^i(T(X^ke_k,Y^le_l))\\\\\n",
    "=e^i(X^kY^lT^m(e_k,e_l)e_m)=e^i(X^kY^lT^m_{kl}e_m)\\\\\n",
    "=X^kY^lT^m_{kl}e^l(e_m)=X^kY^lT^m_{kl}\\delta^i_m=X^kY^lT^i_{kl}\\\\\n",
    "=\\frac{1}{2}(T^i_{kl}e^k\\wedge e^l)(X,Y),\n",
    "$$\n",
    "and\n",
    "$$\n",
    "\\Omega^i_j(X,Y)=e^i(R(X,Y)e_j)=e^i(R(X^ke_k,Y^le_l)e_j)\\\\\n",
    "=X^kY^lR^m_{jkl}e^i(e_m)=X^kY^lR^m_{jkl}\\delta^i_m\\\\\n",
    "=X^kY^lR^i_{jkl}=\\frac{1}{2}(R^i_{jkl}e^k\\wedge e^l)(X,Y).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "<br>\n",
    "\n",
    "**Example 25.4**  \n",
    "\n",
    "Consider the half-plane $y>0$ with the connection coefficients with respect to the bases $\\ \\ e_1=\\frac{\\partial}{\\partial x},\\ \\ e_2=\\frac{\\partial}{\\partial y},\\ \\  e^1=dx,\\ \\  e^2=dy\\ \\ $ equal to\n",
    "$$\\Gamma^1_{12}=\\Gamma^1_{21}=\\Gamma^2_{22}=-\\frac{1}{y}, \\quad\\Gamma^2_{11}=\\frac{1}{y},$$\n",
    "and all other coefficients equal to zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{lcl} \\Gamma_{\\phantom{\\, 1}\\,1\\,2}^{\\,1\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{1}{y} \\\\ \\Gamma_{\\phantom{\\, 1}\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{1}{y} \\\\ \\Gamma_{\\phantom{\\, 2}\\,1\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}} & = & \\frac{1}{y} \\\\ \\Gamma_{\\phantom{\\, 2}\\,2\\,2}^{\\,2\\phantom{\\, 2}\\phantom{\\, 2}} & = & -\\frac{1}{y} \\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{lcl} \\Gamma_{\\phantom{\\, 1}\\,1\\,2}^{\\,1\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{1}{y} \\\\ \\Gamma_{\\phantom{\\, 1}\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{1}{y} \\\\ \\Gamma_{\\phantom{\\, 2}\\,1\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}} & = & \\frac{1}{y} \\\\ \\Gamma_{\\phantom{\\, 2}\\,2\\,2}^{\\,2\\phantom{\\, 2}\\phantom{\\, 2}} & = & -\\frac{1}{y} \\end{array}$"
      ],
      "text/plain": [
       "Gam^1_12 = -1/y \n",
       "Gam^1_21 = -1/y \n",
       "Gam^2_11 = 1/y \n",
       "Gam^2_22 = -1/y "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "M = Manifold(2, 'M', start_index=1)             # manifold M\n",
    "c_xy.<x,y> = M.chart()                          # chart on M\n",
    "nab = M.affine_connection('nabla', r'\\nabla')   # connection on M\n",
    "                                                # con. coefficients\n",
    "nab[1,1,2], nab[1,2,1], nab[2,2,2], nab[2,1,1] = -1/y, -1/y, -1/y, 1/y\n",
    "nab.display(coordinate_labels=False)            # show conn. coeff.\n",
    "                                                # (only nonzero)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Connection forms $\\omega^i_j$ can be computed using the formula\n",
    "(25.3), $\\ \\ \\omega_j^i=\\Gamma^i_{jk}e^k$:\n",
    "$$\\omega^1_1=\\Gamma^1_{11}e^1+\\Gamma^1_{12}e^2=-\\frac{1}{y}dy,\\\\\n",
    "\\omega^1_2=\\Gamma^1_{21}e^1+\\Gamma^1_{22}e^2=-\\frac{1}{y}dx,\\\\\n",
    "\\omega^2_1=\\Gamma^2_{11}e^1+\\Gamma^2_{12}e^2=\\frac{1}{y}dx,\\\\\n",
    "\\omega^2_2=\\Gamma^2_{21}e^1+\\Gamma^2_{22}e^2=-\\frac{1}{y}dy.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\omega^1_{\\ \\, 1} = -\\frac{1}{y} \\mathrm{d} y, \\omega^1_{\\ \\, 2} = -\\frac{1}{y} \\mathrm{d} x, \\omega^2_{\\ \\, 1} = \\frac{1}{y} \\mathrm{d} x, \\omega^2_{\\ \\, 2} = -\\frac{1}{y} \\mathrm{d} y\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\omega^1_{\\ \\, 1} = -\\frac{1}{y} \\mathrm{d} y, \\omega^1_{\\ \\, 2} = -\\frac{1}{y} \\mathrm{d} x, \\omega^2_{\\ \\, 1} = \\frac{1}{y} \\mathrm{d} x, \\omega^2_{\\ \\, 2} = -\\frac{1}{y} \\mathrm{d} y\\right]$"
      ],
      "text/plain": [
       "[nabla connection 1-form (1,1) = -1/y dy,\n",
       " nabla connection 1-form (1,2) = -1/y dx,\n",
       " nabla connection 1-form (2,1) = 1/y dx,\n",
       " nabla connection 1-form (2,2) = -1/y dy]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[nab.connection_form(i,j).display()             # connection forms\n",
    " for i in [1,2] for j in [1,2]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "Torsion forms can be computed from the first Cartan structure equation (25.8), $\\ \\ \\theta^i=de^i+\\omega^i_j\\wedge  e^j$:\n",
    "\n",
    "$$\\theta^1=de^1+\\omega^1_1\\wedge e^1+\\omega^1_2\\wedge e^2\n",
    "=d(dx)-\\frac{1}{y}dy\\wedge dx-\\frac{1}{y}dx\\wedge dy=0,\\\\\n",
    "\\theta^2=de^2+\\omega^2_1\\wedge e^1+\\omega^2_2\\wedge e^2\n",
    "=d(dy)+\\frac{1}{y}dx\\wedge dx-\\frac{1}{y}dy\\wedge dy=0.\\\\\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\theta^1 = 0, \\theta^2 = 0\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\theta^1 = 0, \\theta^2 = 0\\right]$"
      ],
      "text/plain": [
       "[torsion (1) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = 0,\n",
       " torsion (2) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = 0]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[nab.torsion_form(i).disp() for i in [1,2]]    # torsion forms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second Cartan structure equation (25.10):\n",
    "$\\ \\Omega^i_j=d\\omega^i_j+\\omega^i_k\\wedge \\omega^k_j$ \n",
    "can be used in calculation of curvature forms:\n",
    "\n",
    "$$\n",
    "\\Omega_1^1=d\\omega^1_1+\\omega^1_1\\wedge \\omega^1_1+\\omega^1_2\\wedge \\omega^2_1\\\\=d\\Big(-\\frac{1}{y}dy\\Big)-\\Big(\\frac{1}{y}dx\\Big)\\wedge \\Big(\\frac{1}{y}dx\\Big)\\\\\n",
    "=\\frac{1}{y^2}dy\\wedge dy=0,$$\n",
    "<br>\n",
    "\n",
    "\n",
    "$$\n",
    "\\Omega^1_2=d\\omega^1_2+\\omega^1_1\\wedge \\omega^1_2+\\omega^1_2\\wedge \\omega^2_2\\\\\n",
    "=d\\Big(-\\frac{1}{y}dx\\Big)\n",
    "+\\Big(-\\frac{1}{y}dy\\Big)\\wedge \\Big(-\\frac{1}{y}dx\\Big)\n",
    "+\\Big(-\\frac{1}{y}dx\\Big)\\wedge \\Big(-\\frac{1}{y}dy\\Big)\\\\\n",
    "=\\frac{1}{y^2}dy\\wedge dx+\\frac{1}{y^2}dy\\wedge dx+\\frac{1}{y^2}dx\\wedge dy=-\\frac{1}{y^2}dx\\wedge dy,$$\n",
    "<br>\n",
    "\n",
    "$$\n",
    "\\Omega^2_1=d\\omega^2_1+\\omega^2_1\\wedge \\omega^1_1+\\omega^2_2\\wedge \\omega^2_1\\\\\n",
    "=d\\Big(-\\frac{1}{y}dx\\Big)\n",
    "+\\Big(\\frac{1}{y}dx\\Big)\\wedge \\Big(-\\frac{1}{y}dy\\Big)\n",
    "+\\Big(-\\frac{1}{y}dy\\Big)\\wedge \\Big(\\frac{1}{y}dx\\Big)\\\\\n",
    "=\\frac{1}{y^2}dy\\wedge dx-\\frac{1}{y^2}dx\\wedge dy-\\frac{1}{y^2}dy\\wedge dx=\\frac{1}{y^2}dx\\wedge dy,\n",
    "$$\n",
    "<br>\n",
    "\n",
    "$$\n",
    "\\Omega^2_2=d\\omega^2_2+\\omega^2_1\\wedge \\omega^1_2+\\omega^2_2\\wedge \\omega^2_2\\\\\n",
    "=d\\Big(-\\frac{1}{y}dy\\Big)+\n",
    "\\Big(\\frac{1}{y}dx\\Big)\\wedge \\Big(-\\frac{1}{y}dx\\Big)\\\\\n",
    "=\\frac{1}{y^2}dy\\wedge dy-\\frac{1}{y^2}dx\\wedge dx=0.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\Omega^1_{\\ \\, 1} = 0, \\Omega^1_{\\ \\, 2} = -\\frac{1}{y^{2}} \\mathrm{d} x\\wedge \\mathrm{d} y, \\Omega^2_{\\ \\, 1} = \\frac{1}{y^{2}} \\mathrm{d} x\\wedge \\mathrm{d} y, \\Omega^2_{\\ \\, 2} = 0\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\Omega^1_{\\ \\, 1} = 0, \\Omega^1_{\\ \\, 2} = -\\frac{1}{y^{2}} \\mathrm{d} x\\wedge \\mathrm{d} y, \\Omega^2_{\\ \\, 1} = \\frac{1}{y^{2}} \\mathrm{d} x\\wedge \\mathrm{d} y, \\Omega^2_{\\ \\, 2} = 0\\right]$"
      ],
      "text/plain": [
       "[curvature (1,1) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = 0,\n",
       " curvature (1,2) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = -1/y^2 dx∧dy,\n",
       " curvature (2,1) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = y^(-2) dx∧dy,\n",
       " curvature (2,2) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = 0]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[nab.curvature_form(i,j).display()            # curvature forms\n",
    "   for i in [1,2] for j in [1,2]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "Since by (25.11) $\\quad \\Omega^i_j=\\frac{1}{2}R^i_{jkl}e^k\\wedge e^l$, and $R^i_{jkl}$ is antisymmetric in $k,l$, then\n",
    "$$\\Omega_2^1=\\frac{1}{2}R^1_{2kl}e^k\\wedge e^l=\n",
    "\\frac{1}{2}(R^1_{212}dx\\wedge dy+R^1_{221}dy\\wedge dx)\\\\\n",
    "=\\frac{1}{2}(R^1_{212}dx\\wedge dy+R^1_{212}dx\\wedge dy)=R^1_{212}dx\\wedge dy=-\\frac{1}{y^2}dx\\wedge dy,\n",
    "$$\n",
    "\n",
    "Using an analogous calculation for $\\Omega^2_1$, we see that the different from zero components \n",
    "of the curvature tensor are equal to:\n",
    "\n",
    "$R^1_{212} =-\\frac{1}{y^2},\\ \\ \n",
    "R^1_{221} =\\frac{1}{y^2},\\ \\ R^2_{112}=\\frac{1}{y^2},\\ \\ R^2_{121}=-\\frac{1}{y^2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{lcl} R_{\\phantom{\\, 1}\\,2\\,1\\,2}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{1}{y^{2}} \\\\ R_{\\phantom{\\, 1}\\,2\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{1}{y^{2}} \\\\ R_{\\phantom{\\, 2}\\,1\\,1\\,2}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{1}{y^{2}} \\\\ R_{\\phantom{\\, 2}\\,1\\,2\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{1}{y^{2}} \\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{lcl} R_{\\phantom{\\, 1}\\,2\\,1\\,2}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{1}{y^{2}} \\\\ R_{\\phantom{\\, 1}\\,2\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{1}{y^{2}} \\\\ R_{\\phantom{\\, 2}\\,1\\,1\\,2}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{1}{y^{2}} \\\\ R_{\\phantom{\\, 2}\\,1\\,2\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{1}{y^{2}} \\end{array}$"
      ],
      "text/plain": [
       "R^1_212 = -1/y^2 \n",
       "R^1_221 = y^(-2) \n",
       "R^2_112 = y^(-2) \n",
       "R^2_121 = -1/y^2 "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = M.tensor_field(1,3,'R')              # tensor field of (1,3)-type\n",
    "R[:] = nab.riemann()[:]                  # R <- curv. (1,3) tensor f.\n",
    "R.display_comp(coordinate_labels=False)      # show R\n",
    "#nab.riemann().display_comp(coordinate_labels=False)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 25.5** \n",
    "\n",
    "Consider the plane $R^2$ with Christoffel symbols\n",
    "$$Γ^1_{11}= Γ^1_{22}=\\frac{4u}{1+u^2+4v^2},\\\\\n",
    "Γ^2_{11}= Γ^2_{22}=\\frac{4v}{1+u^2+4v^2},\n",
    "$$\n",
    "and the remaining symbols equal to 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "%display latex\n",
    "N = Manifold(2,name='R2',start_index=1)     # manifold N\n",
    "c_uv.<u,v> = N.chart()                      # chart on N\n",
    "nab = N.affine_connection('nab1')           # connection coefficients\n",
    "nab[:] = [[[4*u/(4*u^2 + 4*v^2 + 1), 0],    # define all coefficients\n",
    "           [0, 4*u/(4*u^2 + 4*v^2 + 1)]], \n",
    "          [[4*v/(4*u^2 + 4*v^2 + 1), 0], \n",
    "           [0, 4*v/(4*u^2 + 4*v^2 + 1)]]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute connection forms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\omega^1_{\\ \\, 1} = \\left( \\frac{4 \\, u}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\right) \\mathrm{d} u, \\omega^1_{\\ \\, 2} = \\left( \\frac{4 \\, u}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\right) \\mathrm{d} v, \\omega^2_{\\ \\, 1} = \\left( \\frac{4 \\, v}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\right) \\mathrm{d} u, \\omega^2_{\\ \\, 2} = \\left( \\frac{4 \\, v}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\right) \\mathrm{d} v\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\omega^1_{\\ \\, 1} = \\left( \\frac{4 \\, u}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\right) \\mathrm{d} u, \\omega^1_{\\ \\, 2} = \\left( \\frac{4 \\, u}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\right) \\mathrm{d} v, \\omega^2_{\\ \\, 1} = \\left( \\frac{4 \\, v}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\right) \\mathrm{d} u, \\omega^2_{\\ \\, 2} = \\left( \\frac{4 \\, v}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\right) \\mathrm{d} v\\right]$"
      ],
      "text/plain": [
       "[nab1 connection 1-form (1,1) = 4*u/(4*u^2 + 4*v^2 + 1) du,\n",
       " nab1 connection 1-form (1,2) = 4*u/(4*u^2 + 4*v^2 + 1) dv,\n",
       " nab1 connection 1-form (2,1) = 4*v/(4*u^2 + 4*v^2 + 1) du,\n",
       " nab1 connection 1-form (2,2) = 4*v/(4*u^2 + 4*v^2 + 1) dv]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[nab.connection_form(i,j).display()      # connection forms\n",
    "   for i in [1,2] for j in [1,2]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and  torsion forms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\theta^1 = 0, \\theta^2 = 0\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\theta^1 = 0, \\theta^2 = 0\\right]$"
      ],
      "text/plain": [
       "[torsion (1) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) = 0,\n",
       " torsion (2) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) = 0]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[nab.torsion_form(i).disp()             # torsion forms\n",
    "     for i in [1,2]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let us try to compute curvature forms."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\Omega^1_{\\ \\, 1}, \\Omega^1_{\\ \\, 2}, \\Omega^2_{\\ \\, 1}, \\Omega^2_{\\ \\, 2}\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\Omega^1_{\\ \\, 1}, \\Omega^1_{\\ \\, 2}, \\Omega^2_{\\ \\, 1}, \\Omega^2_{\\ \\, 2}\\right]$"
      ],
      "text/plain": [
       "[2-form curvature (1,1) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) on the 2-dimensional differentiable manifold R2,\n",
       " 2-form curvature (1,2) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) on the 2-dimensional differentiable manifold R2,\n",
       " 2-form curvature (2,1) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) on the 2-dimensional differentiable manifold R2,\n",
       " 2-form curvature (2,2) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) on the 2-dimensional differentiable manifold R2]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[nab.curvature_form(i,j)               # curvature forms -symbols\n",
    " for i in [1,2] for j in [1,2]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we show the results, we have to do some simplifications."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in [1,2]:                       # factor curvature forms\n",
    "    for j in [1,2]:\n",
    "        nab.curvature_form(i,j).comp()[:].apply_map(factor)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\Omega^1_{\\ \\, 1} = \\frac{16 \\, u v}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\wedge \\mathrm{d} v\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\Omega^1_{\\ \\, 1} = \\frac{16 \\, u v}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\wedge \\mathrm{d} v$"
      ],
      "text/plain": [
       "curvature (1,1) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) = 16*u*v/(4*u^2 + 4*v^2 + 1)^2 du∧dv"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\Omega^1_{\\ \\, 2} = \\frac{4 \\, {\\left(4 \\, v^{2} + 1\\right)}}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\wedge \\mathrm{d} v\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\Omega^1_{\\ \\, 2} = \\frac{4 \\, {\\left(4 \\, v^{2} + 1\\right)}}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\wedge \\mathrm{d} v$"
      ],
      "text/plain": [
       "curvature (1,2) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) = 4*(4*v^2 + 1)/(4*u^2 + 4*v^2 + 1)^2 du∧dv"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\Omega^2_{\\ \\, 1} = -\\frac{4 \\, {\\left(4 \\, u^{2} + 1\\right)}}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\wedge \\mathrm{d} v\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\Omega^2_{\\ \\, 1} = -\\frac{4 \\, {\\left(4 \\, u^{2} + 1\\right)}}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\wedge \\mathrm{d} v$"
      ],
      "text/plain": [
       "curvature (2,1) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) = -4*(4*u^2 + 1)/(4*u^2 + 4*v^2 + 1)^2 du∧dv"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\Omega^2_{\\ \\, 2} = -\\frac{16 \\, u v}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\wedge \\mathrm{d} v\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\Omega^2_{\\ \\, 2} = -\\frac{16 \\, u v}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\wedge \\mathrm{d} v$"
      ],
      "text/plain": [
       "curvature (2,2) of connection nab1 w.r.t. Coordinate frame (R2, (∂/∂u,∂/∂v)) = -16*u*v/(4*u^2 + 4*v^2 + 1)^2 du∧dv"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in [1,2]:                       # show curvature forms\n",
    "    for j in [1,2]:\n",
    "        show(nab.curvature_form(i,j).disp())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### <br>\n",
    "\n",
    "**Example 25.6**\n",
    "\n",
    "In  previous examples, the torsion forms were equal to zero.\n",
    "\n",
    "Let us consider a 2-dimensional manifold with \"random\" connection coefficients."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = Manifold(2, 'M', start_index=1)               # manifold M\n",
    "c_xy.<x,y> = M.chart()                            # chart on M\n",
    "nab = M.affine_connection('nabla', r'\\nabla')     # connection on M\n",
    "nab[1,1,1], nab[1,1,2] = x*y, x^2\n",
    "nab[1,2,1], nab[1,2,2] = -x^3, y^2\n",
    "nab[2,1,1], nab[2,1,2] = y^2, x*y\n",
    "nab[2,2,1], nab[2,2,2] = x^2+y^2+y^2, y^2-x^2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\left[\\left[x y, x^{2}\\right], \\left[-x^{3}, y^{2}\\right]\\right], \\left[\\left[y^{2}, x y\\right], \\left[x^{2} + 2 \\, y^{2}, -x^{2} + y^{2}\\right]\\right]\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\left[\\left[x y, x^{2}\\right], \\left[-x^{3}, y^{2}\\right]\\right], \\left[\\left[y^{2}, x y\\right], \\left[x^{2} + 2 \\, y^{2}, -x^{2} + y^{2}\\right]\\right]\\right]$"
      ],
      "text/plain": [
       "[[[x*y, x^2], [-x^3, y^2]], [[y^2, x*y], [x^2 + 2*y^2, -x^2 + y^2]]]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex                                   # show connection\n",
    "nab[:]                                           # coefficients"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the connection forms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\left[\\omega^1_{\\ \\, 1} = x y \\mathrm{d} x + x^{2} \\mathrm{d} y, \\omega^1_{\\ \\, 2} = -x^{3} \\mathrm{d} x + y^{2} \\mathrm{d} y\\right], \\left[\\omega^2_{\\ \\, 1} = y^{2} \\mathrm{d} x + x y \\mathrm{d} y, \\omega^2_{\\ \\, 2} = \\left( x^{2} + 2 \\, y^{2} \\right) \\mathrm{d} x + \\left( -x^{2} + y^{2} \\right) \\mathrm{d} y\\right]\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\left[\\omega^1_{\\ \\, 1} = x y \\mathrm{d} x + x^{2} \\mathrm{d} y, \\omega^1_{\\ \\, 2} = -x^{3} \\mathrm{d} x + y^{2} \\mathrm{d} y\\right], \\left[\\omega^2_{\\ \\, 1} = y^{2} \\mathrm{d} x + x y \\mathrm{d} y, \\omega^2_{\\ \\, 2} = \\left( x^{2} + 2 \\, y^{2} \\right) \\mathrm{d} x + \\left( -x^{2} + y^{2} \\right) \\mathrm{d} y\\right]\\right]$"
      ],
      "text/plain": [
       "[[nabla connection 1-form (1,1) = x*y dx + x^2 dy,\n",
       "  nabla connection 1-form (1,2) = -x^3 dx + y^2 dy],\n",
       " [nabla connection 1-form (2,1) = y^2 dx + x*y dy,\n",
       "  nabla connection 1-form (2,2) = (x^2 + 2*y^2) dx + (-x^2 + y^2) dy]]"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[[nab.connection_form(i,j).display()             # connection forms\n",
    "  for j in [1,2]] for i in [1,2]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall that the torsion was defined in (21.13) as \n",
    "\n",
    "$$T (X, Y) = ∇_X Y − ∇_Y X − [X, Y],\\quad \\text{for}\\quad X,Y\\in \\mathfrak{X}(M).$$\n",
    "\n",
    "In SageMath the `torsion` method returns the tensor of type (1,2) defined by\n",
    "\n",
    "$$\\tilde{T}(\\omega,X,Y)=\\omega(T(X,Y)),\\quad \\text{for}\\quad X,Y\\in \\mathfrak{X}(M), \\omega\\in \\Omega^1(M).\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field of type (1,2) on the 2-dimensional differentiable manifold M\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left( -x^{3} - x^{2} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( x^{3} + x^{2} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x + \\left( x^{2} - x y + 2 \\, y^{2} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( -x^{2} + x y - 2 \\, y^{2} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left( -x^{3} - x^{2} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( x^{3} + x^{2} \\right) \\frac{\\partial}{\\partial x }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x + \\left( x^{2} - x y + 2 \\, y^{2} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} x\\otimes \\mathrm{d} y + \\left( -x^{2} + x y - 2 \\, y^{2} \\right) \\frac{\\partial}{\\partial y }\\otimes \\mathrm{d} y\\otimes \\mathrm{d} x$"
      ],
      "text/plain": [
       "(-x^3 - x^2) ∂/∂x⊗dx⊗dy + (x^3 + x^2) ∂/∂x⊗dy⊗dx + (x^2 - x*y + 2*y^2) ∂/∂y⊗dx⊗dy + (-x^2 + x*y - 2*y^2) ∂/∂y⊗dy⊗dx"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(nab.torsion())                          # torsion (1,2) tensor\n",
    "nab.torsion().disp()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The torsion forms in our example are equal to:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\theta^1 = \\left( -x^{3} - x^{2} \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\theta^1 = \\left( -x^{3} - x^{2} \\right) \\mathrm{d} x\\wedge \\mathrm{d} y$"
      ],
      "text/plain": [
       "torsion (1) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = (-x^3 - x^2) dx∧dy"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\theta^2 = \\left( x^{2} - x y + 2 \\, y^{2} \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\theta^2 = \\left( x^{2} - x y + 2 \\, y^{2} \\right) \\mathrm{d} x\\wedge \\mathrm{d} y$"
      ],
      "text/plain": [
       "torsion (2) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = (x^2 - x*y + 2*y^2) dx∧dy"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in [1,2]:                             # torsion forms\n",
    "    show(nab.torsion_form(i).disp())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the curvature forms:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\Omega^1_{\\ \\, 1} = \\left( -x^{4} y - y^{4} + x \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\Omega^1_{\\ \\, 1} = \\left( -x^{4} y - y^{4} + x \\right) \\mathrm{d} x\\wedge \\mathrm{d} y$"
      ],
      "text/plain": [
       "curvature (1,1) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = (-x^4*y - y^4 + x) dx∧dy"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\Omega^1_{\\ \\, 2} = \\left( 2 \\, x^{5} + x y^{3} - 2 \\, y^{4} - {\\left(x^{3} + x^{2}\\right)} y^{2} \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\Omega^1_{\\ \\, 2} = \\left( 2 \\, x^{5} + x y^{3} - 2 \\, y^{4} - {\\left(x^{3} + x^{2}\\right)} y^{2} \\right) \\mathrm{d} x\\wedge \\mathrm{d} y$"
      ],
      "text/plain": [
       "curvature (1,2) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = (2*x^5 + x*y^3 - 2*y^4 - (x^3 + x^2)*y^2) dx∧dy"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\Omega^2_{\\ \\, 1} = \\left( x^{2} y^{2} + 2 \\, x y^{3} - y^{4} + {\\left(x^{3} - 1\\right)} y \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\Omega^2_{\\ \\, 1} = \\left( x^{2} y^{2} + 2 \\, x y^{3} - y^{4} + {\\left(x^{3} - 1\\right)} y \\right) \\mathrm{d} x\\wedge \\mathrm{d} y$"
      ],
      "text/plain": [
       "curvature (2,1) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = (x^2*y^2 + 2*x*y^3 - y^4 + (x^3 - 1)*y) dx∧dy"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\Omega^2_{\\ \\, 2} = \\left( y^{4} + {\\left(x^{4} - 4\\right)} y - 2 \\, x \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\Omega^2_{\\ \\, 2} = \\left( y^{4} + {\\left(x^{4} - 4\\right)} y - 2 \\, x \\right) \\mathrm{d} x\\wedge \\mathrm{d} y$"
      ],
      "text/plain": [
       "curvature (2,2) of connection nabla w.r.t. Coordinate frame (M, (∂/∂x,∂/∂y)) = (y^4 + (x^4 - 4)*y - 2*x) dx∧dy"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in [1,2]:                            # curvature forms\n",
    "    for j in [1,2]:\n",
    "        show(nab.curvature_form(i,j).disp())"
   ]
  }
 ],
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