24Manifold_RiemannCurvature.ipynb

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   "source": [
    "# 24. Riemannian curvature tensor of type (0,4)\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)."
   ]
  },
  {
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    {
     "data": {
      "text/plain": [
       "'SageMath version 9.6, Release Date: 2022-05-15'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version()"
   ]
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   "source": [
    "Assume now that $M$ is a Riemannian manifold with metric $g$ and $\\nabla$ is the corresponding Levi-Civita connection. We define **Riemannian curvature tensor** of  type $(0,4)$ by\n",
    "\n",
    "\\begin{equation}\n",
    "R(X, Y, Z , W ) = g(R(X, Y )Z , W ),\\quad \\text{for } X,Y,Z,W\\in \\mathfrak{X}(M).\n",
    "\\tag{24.1}\n",
    "\\end{equation}\n",
    "\n",
    "Since $g$ is non-singular (cf. [notebook 22](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/22Manifold_Riemann.ipynb)) the curvature (0,4)-type tensor contains the same information as the corresponding (1,3)-type tensor from the [previous notebook](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/23Manifold_Curvature.ipynb). \n",
    "\n",
    "In a local coordinate system $(x^1,\\ldots,x^n)$ we have\n",
    "\n",
    "\\begin{equation}\n",
    "R_{ijkl}=g_{im}R^m_{jkl}.\n",
    "\\tag{24.1'}\n",
    "\\end{equation}"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Symmetries of Riemannian curvature tensor\n",
    "\n",
    "<br>\n",
    "\n",
    "If $X, Y, Z , W$ are vector fields in $M$ and $∇$ is the Levi–Civita connection, then\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{matrix}\n",
    "R(X, Y, Z , W ) + R(Y, Z , X, W ) + R(Z , X, Y, W ) = 0,\\\\\n",
    "R(X, Y, Z , W ) = −R(Y, X, Z , W ),\\\\\n",
    "R(X, Y, Z , W ) = −R(X, Y, W, Z ),\\\\\n",
    "R(X, Y, Z , W ) = R(Z , W, X, Y ).\n",
    "\\end{matrix}\n",
    "\\tag{24.2}\n",
    "\\end{equation}\n",
    "\n",
    "<br>\n",
    "\n",
    "Since the Levi-Civita connections are torsion-free, the first Bianchi identity (23.4) reduces to\n",
    "$R(X, Y)Z + R(Z, X)Y + R(Y, Z)X=0$ and implies the first relation.\n",
    "\n",
    "The antisymmetry $R(X, Y) = −R(Y, X)$ implies the second equality.\n",
    "\n",
    "The third equality is equivalent to $R(X, Y, Z , Z ) = 0.$ Indeed, if it holds then clearly $R(X, Y, Z , Z ) = 0.$ Conversely, if $R(X, Y, Z , Z ) = 0$, we have\n",
    "\n",
    "$$R(X, Y, Z + W, Z + W ) = 0\\\\\n",
    "⇔R(X, Y, Z , Z ) + R(X, Y, Z, W ) +R(X, Y, W, Z ) + R(X, Y, W, W )=0\\\\\n",
    "⇔ R(X, Y, Z , W ) + R(X, Y, W, Z ) = 0.$$"
   ]
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   "source": [
    "To prove $R(X, Y, Z , Z ) = 0$, let us note that\n",
    "since the Levi-Civita connection is compatible with the metric, we have\n",
    "\n",
    "$$ g(∇_X ∇_Y Z , Z ))=X ( g(∇_Y Z , Z ))-g( ∇_Y Z , ∇_X Z),\\\\\n",
    "g(∇_Y ∇_X Z , Z ))=Y ( g(∇_X Z , Z ))-g( ∇_X Z , ∇_Y Z).\n",
    "$$\n",
    "\n",
    "It is also clear, that for connections compatible with the metric, taking $V=[X,Y]$ we obtain \n",
    "\n",
    "$$V(g(Z,Z))=2g(\\nabla_V Z,Z) \\Longrightarrow \n",
    "[X,Y](g(Z , Z))=2 g(∇_{[X,Y ]} Z , Z). $$\n",
    "\n",
    "Therefore\n",
    "$$\n",
    "R(X, Y, Z , Z ) = \n",
    "g(∇_X ∇_Y Z , Z) − g(∇_Y ∇_X Z , Z) − g(∇_{[X,Y ]} Z , Z)\\\\\n",
    "= X(g( ∇_Y Z , Z)) − g(∇_Y Z , ∇_X Z)\\\\\n",
    "− Y(g( ∇_X Z , Z))+ g(∇_X Z , ∇_Y Z)\\\\\n",
    "− \\frac{1}{2}[X, Y ](g( Z , Z))\n",
    "$$\n",
    "But we have also\n",
    "$$\n",
    "g(\\nabla_YZ,Z)=Y(g(Z,Z))-g(Z,\\nabla_YZ),\\\\\n",
    "g(\\nabla_XZ,Z)=X(g(Z,Z))-g(Z,\\nabla_XZ),\n",
    "$$\n",
    "\n",
    "and consequently\n",
    "$$\n",
    "g(\\nabla_YZ,Z)=\\frac{1}{2}Y(g(Z,Z)),\\\\\n",
    "g(\\nabla_XZ,Z)=\\frac{1}{2}X(g(Z,Z)),\n",
    "$$\n",
    "\n",
    "therefore\n",
    "\n",
    "$$\n",
    "R(X, Y, Z , Z )\\\\\n",
    "=\\frac{1}{2}X(Y( g(Z , Z ))-\\frac{1}{2}Y(X( g(Z , Z ))-\\frac{1}{2}[X,Y]( g(Z , Z ))\\\\\n",
    "=\\frac{1}{2}[X,Y]( g(Z , Z ))-\\frac{1}{2}[X,Y]( g(Z , Z ))=0.\n",
    "$$"
   ]
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   "source": [
    "In the proof of the fourth relation we use the first one to obtain\n",
    "\n",
    "$$\n",
    "R(X, Y, Z , W )+R(Y, Z , X, W )+R(Z , X, Y, W )=0,\\\\\n",
    "R(Y, Z, W, X)  +R(Z , W, Y, X )+R(W, Y, Z , X )=0,\\\\\n",
    "R(Z , W, X, Y )+R(W, X, Z , Y )+R(X, Z , W, Y )=0,\\\\\n",
    "R(W, X, Y, Z )+R(X, Y, W, Z )+R(Y, W, X, Z )=0.\n",
    "$$\n",
    "\n",
    "Adding side by side and using the third equality we obtain\n",
    "\n",
    "$$\n",
    "R(Z , X, Y, W ) + R(W, Y, Z , X ) + R(X, Z , W, Y ) + R(Y, W, X, Z ) = 0.\n",
    "$$\n",
    "\n",
    "Using the second and third relation, we have\n",
    "\n",
    "$$2R(Z , X, Y, W ) − 2R(Y, W, Z , X ) = 0.$$"
   ]
  },
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   "source": [
    "\n",
    "<br>\n",
    "\n",
    "The component-wise version of (24.2) reads as follows:\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{matrix}\n",
    "R_{i jkl} + R_{ikl j} + R_{il jk} = 0,\\\\\n",
    "R_{ijkl} = −R_{jikl},\\\\\n",
    "R_{ijkl} = −R_{ijlk},\\\\\n",
    "R_{ijkl} = R_{klij}.\n",
    "\\end{matrix}\n",
    "\\tag{24.2'}\n",
    "\\end{equation}"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "<br>\n",
    "\n",
    "### The case of of 2-dimensional manifolds in $R^3$\n",
    "\n",
    "<br>\n",
    "\n",
    "If $n=2$, then the indices take values 1 or 2 (or 0,1 if Python is used). The antisymmetry in $i,j$ and $k,l$ means that for nonzero components of of the tensor $R_{ijkl}$, we have  $i\\not=j$ and $k\\not=l$. Furthermore changing the order of $i,j$ or $k,l$ we only change the sign of the corresponding components $R_{ijkl}$. Thus, for manifolds of dimension $n =\n",
    "2,$ there is only one independent component of the Riemannian curvature tensor, namely, $R_{1212}$. The remaining ones vanish or can be obtained by permutations of indices $(i,j)=(1,2), \\ (k,l)=(1,2)$ and an appropriate change of signs."
   ]
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    "Suppose we are given a surface in Euclidean 3-space \n",
    "\n",
    "$$ x=x(u,v),\\ \\ y=y(u,v),\\ \\ z=z(u,v),\\ \\ (u,v)\\in V\\subset R^2,$$\n",
    "\n",
    "and a non-singular\n",
    "point $P=(x_0 , y_0 , z_0 )=(x(u_0,v_0),y(u_0,v_0),z(u_0,v_0))$ on it. i.e.,  \n",
    "\n",
    "$$\n",
    "\\text{rank}\\left.\n",
    "\\left(\n",
    "\\begin{matrix}\n",
    "x_u & y_u & z_u\\\\\n",
    "x_v & y_v & z_v\n",
    "\\end{matrix}\n",
    "\\right)\n",
    "\\right|_{(u_0,v_0)}=2.\n",
    "$$\n",
    "\n",
    "We shall assume  that the $z$-axis is perpendicular to the tangent plane to the surface at the point $P,$ in which\n",
    "case the $x$-axis and $y$-axis will be parallel to it. The surface may then be given\n",
    "locally about (i.e. in a neighborhood of) the point $P$ by an equation\n",
    "of the form $z = f(x, y)$, where $z_0 = f(x_0 , y_0 )$ and\n",
    "\n",
    "$$\\frac{\\partial f}{\\partial x}\\Big|_{\\substack{x=x_0\\\\y=y_0}}\n",
    "=\\frac{\\partial f}{\\partial y}\\Big|_{\\substack{x=x_0\\\\y=y_0}}=0, \\quad \\text{i.e.}\\quad \n",
    "\\text{grad}\\, f\\,\\Big|_{\\substack{x=x_0\\\\y=y_0}}=0.\n",
    "$$\n",
    "\n",
    "Given a surface $z = f(x, y)$, and a point $P=(x_0 , y_0 , z_0 )$ on it\n",
    "at which grad $f = 0$, we can define the **Gaussian curvature $K$** of the surface at $(x_0,y_0,z_0)$ to be the determinant of the matrix \n",
    "\n",
    "$$K=\\det (f_{x^ix^j})\\Big|_{\\substack{x=x_0\\\\y=y_0}}=\n",
    "\\det \\left(\n",
    "\\begin{matrix}\n",
    "f_{xx} & f_{xy}\\\\\n",
    "f_{yx} & f_{yy}\n",
    "\\end{matrix}\n",
    "\\right)\\Big|_{\\substack{x=x_0\\\\y=y_0}}.\n",
    "$$\n",
    "\n",
    "The metric on such surface is  defined by\n",
    "\n",
    "$$ g_{11}=1+f^2_x,\\quad g_{12}=f_xf_y,\\quad g_{22}=1+f_y^2.$$\n",
    "\n",
    "By the assumption $\\text{grad}\\, f\\,\\Big|_{\\substack{x=x_0\\\\y=y_0}}=0$ at point $P$, the  derivatives of $g_{ij}$ with respect to $x$ and $y$ vanish, so all Levi-Civita connection coefficients $\\Gamma^m_{ij}=0$ at $P$. \n",
    "\n",
    "Recall the formulas defining the Riemannian curvature tensor at $P$ ( here, as usual  $(g_{ij})$ is the matrix of metric components and $(g^{ij})$ its inverse).\n",
    "\n",
    "$$\\displaystyle\\Gamma_{rs}^q=\\frac{1}{2}g^{tq}\\Big(\\frac{\\partial g_{rt}}{\\partial x^s}\n",
    "+\\frac{\\partial g_{ts}}{\\partial x^r}-\n",
    "\\frac{\\partial g_{sr}}{\\partial x^t}\\Big),\\\\\n",
    "R^m_{kij}=\\frac{\\partial \\Gamma^m_{kj}}{\\partial x^i}\n",
    "-\\frac{\\partial \\Gamma^m_{ki}}{\\partial x^j}\n",
    "+\\Gamma^m_{li}\\Gamma^l_{kj}\n",
    "-\\Gamma^m_{lj}\\Gamma^l_{ki},\\\\\n",
    "R_{ijkl}=g_{im}R^m_{jkl}\n",
    "=g_{im}\n",
    "\\Big(\\frac{\\partial \\Gamma^m_{jl}}{\\partial x^k}\n",
    "-\\frac{\\partial \\Gamma^m_{jk}}{\\partial x^l}\\Big),\\\\\n",
    "$$\n",
    "(in the last formula we used the fact that $\\Gamma^m_{ki}=0$ at $P=(x_0,y_0,z_0)$).\n",
    "To obtain the derivatives of Christoffel symbols we need their values in a neighborhood of $P$\n",
    "\n",
    "$$\\Gamma^m_{jl}=\\frac{1}{2}g^{pm}\\Big(\\frac{\\partial g_{jp}}{\\partial x^l}\n",
    "+\\frac{\\partial g_{pl}}{\\partial x^j}-\n",
    "\\frac{\\partial g_{lj}}{\\partial x^p}\\Big),\n",
    "$$\n",
    "so using the fact that derivatives of $g_{ij}$ vanish at $P$ we obtain\n",
    "\n",
    "$$\\frac{\\partial \\Gamma^m_{jl}}{\\partial x^k}\n",
    "=\\frac{1}{2}g^{pm}\\big(\\frac{\\partial^2 g_{jp}}{\\partial x^k\\partial x^l}\n",
    "+\\frac{\\partial^2 g_{pl}}{\\partial x^k\\partial x^j}-\n",
    "\\frac{\\partial^2 g_{lj}}{\\partial x^k\\partial x^p}\\Big).\n",
    "$$\n",
    "Similarly\n",
    "\n",
    "$$\\Gamma^m_{jk}=\\frac{1}{2}g^{pm}\\Big(\\frac{\\partial g_{jp}}{\\partial x^k}\n",
    "+\\frac{\\partial g_{pk}}{\\partial x^j}-\n",
    "\\frac{\\partial g_{kj}}{\\partial x^p}\\Big),\n",
    "$$\n",
    "so\n",
    "$$\n",
    "\\frac{\\partial \\Gamma^m_{jk}}{\\partial x^l}\n",
    "=\\frac{1}{2}g^{pm}\\big(\\frac{\\partial^2 g_{jp}}{\\partial x^l\\partial x^k}\n",
    "+\\frac{\\partial^2 g_{pk}}{\\partial x^l\\partial x^j}-\n",
    "\\frac{\\partial^2 g_{kj}}{\\partial x^l\\partial x^p}\\Big).\n",
    "$$"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since $g_{im}g^{pm}=\\delta_i^p$, then $R_{ijkl}$ is a difference of two expressions:\n",
    "\n",
    "$$\\frac{1}{2}\\delta_i^p\\big(\\frac{\\partial^2 g_{jp}}{\\partial x^k\\partial x^l}\n",
    "+\\frac{\\partial^2 g_{pl}}{\\partial x^k\\partial x^j}-\n",
    "\\frac{\\partial^2 g_{lj}}{\\partial x^k\\partial x^p}\\big)\n",
    "=\n",
    "\\frac{1}{2}\\big(\\frac{\\partial^2 g_{ji}}{\\partial x^k\\partial x^l}\n",
    "+\\frac{\\partial^2 g_{il}}{\\partial x^k\\partial x^j}-\n",
    "\\frac{\\partial^2 g_{lj}}{\\partial x^k\\partial x^i}\\big)\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\\frac{1}{2}\\delta_i^p\\big(\\frac{\\partial^2 g_{jp}}{\\partial x^l\\partial x^k}\n",
    "+\\frac{\\partial^2 g_{pk}}{\\partial x^l\\partial x^j}-\n",
    "\\frac{\\partial^2 g_{kj}}{\\partial x^l\\partial x^p}\\big)\n",
    "=\n",
    "\\frac{1}{2}\\big(\\frac{\\partial^2 g_{ji}}{\\partial x^l\\partial x^k}\n",
    "+\\frac{\\partial^2 g_{ik}}{\\partial x^l\\partial x^j}-\n",
    "\\frac{\\partial^2 g_{kj}}{\\partial x^l\\partial x^i}\\big).\n",
    "$$\n",
    "\n",
    "Thus\n",
    "\n",
    "$$R_{ijkl}=\\frac{1}{2}\\Big(\n",
    "\\frac{\\partial^2 g_{il}}{\\partial x^k\\partial x^j}+\n",
    "\\frac{\\partial^2 g_{kj}}{\\partial x^l\\partial x^i}-\n",
    "\\frac{\\partial^2 g_{lj}}{\\partial x^k\\partial x^i}-\n",
    "\\frac{\\partial^2 g_{ik}}{\\partial x^l\\partial x^j}\\Big).\n",
    "$$\n",
    "\n",
    "For $(i,j,k,l)=(1,2,1,2)$ we obtain\n",
    "$$R_{1212}=\\frac{1}{2}\\Big(\n",
    "\\frac{\\partial^2 g_{12}}{\\partial x^1\\partial x^2}+\n",
    "\\frac{\\partial^2 g_{12}}{\\partial x^2\\partial x^1}-\n",
    "\\frac{\\partial^2 g_{22}}{\\partial x^1\\partial x^1}-\n",
    "\\frac{\\partial^2 g_{11}}{\\partial x^2\\partial x^2}\\Big)\\\\\n",
    "=\\frac{1}{2}\\Big(\\frac{\\partial^2 g_{12}}{\\partial x\\partial y}+\n",
    "\\frac{\\partial^2 g_{12}}{\\partial y\\partial x}-\n",
    "\\frac{\\partial^2 g_{22}}{\\partial x^2}-\n",
    "\\frac{\\partial^2 g_{11}}{\\partial y^2}\\Big)\n",
    ".\n",
    "$$\n",
    "\n",
    "Since at $P$ \n",
    "$$(g_{11})_{yy}=2(f_{xy})^2,\\quad (g_{22})_{xx}=2(f_{xy})^2,\\quad (g_{12})_{xy}=f_{xx}f_{yy}+(f_{xy})^2,\n",
    "$$\n",
    "we have at $P$\n",
    "$$R_{1212}=\\frac{1}{2}\\big(2(f_{xx}f_{yy}+(f_{xy})^2)\n",
    "-2(f_{xy})^2\n",
    "-2(f_{xy})^2\\big)=f_{xx}f_{yy}-(f_{xy})^2=\\det\n",
    "\\left(\n",
    "\\begin{matrix}\n",
    "f_{xx} & f_{xy}\\\\\n",
    "f_{yx} & f_{yy}\n",
    "\\end{matrix}\n",
    "\\right)=K.\n",
    "$$\n"
   ]
  },
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   "source": [
    "<br>\n",
    "\n",
    "### Ricci tensor and scalar curvature\n",
    "\n",
    "<br>\n",
    "\n",
    "**Ricci tensor** can be  defined by\n",
    "\n",
    "\\begin{equation}\n",
    "Ric(u,v)=R(e^i,u,e_i,v),\n",
    "\\tag{24.3}\n",
    "\\end{equation}\n",
    "\n",
    "for any vector fields $u,v$, where $e_i$ is any vector frame and $e^i$ the corresponding coframe.\n",
    "\n",
    "In a Riemannian manifold with the metric $g$ the components of Ricci tensor are defined by\n",
    "\n",
    "$$Ric_{ij}=R^k_{ikj}=g^{kl}R_{likj}=g^{lk}R_{kjli},$$\n",
    "\n",
    "and the **scalar curvature** by\n",
    "\n",
    "\\begin{equation}\n",
    "R=g^{ij}Ric_{ij}.\n",
    "\\tag{24.4}\n",
    "\\end{equation}\n",
    "\n",
    "In the case of two-dimensional manifold we have $R=2K$. In fact\n",
    "\n",
    "$$R=g^{ij}Ric_{ij}=g^{ij}g^{lk}R_{kjli}.$$ \n",
    "\n",
    "From the symmetry properties of $R_{ijkl}$ it follows that in the obtained sum of $2^4$ elements only 4 are non-zero:\n",
    "\n",
    "$$R=g^{22}g^{11}R_{1212}+g^{12}g^{21}R_{1221}+\n",
    "g^{21}g^{12}R_{2112}+g^{11}g^{22}R_{2121}\\\\\n",
    "=g^{22}g^{11}R_{1212}-g^{12}g^{21}R_{1212}-g^{21}g^{12}R_{1212}+g^{11}g^{22}R_{1212}\\\\\n",
    "=2(g^{11}g^{22}-(g^{12})^2)R_{1212}=2\\det (g^{ij})R_{1212}=\\frac{2}{\\det (g_{ij})}R_{1212}.\n",
    "$$\n",
    "\n",
    "**Remark** in the last calculations  the condition $\\text{grad}\\, f\\,\\Big|_{\\substack{x=x_0\\\\y=y_0}}=0,$ was not used.\n",
    "\n",
    "\n",
    "If as previously, the coordinates at $P$ are such that $\\text{grad}\\, f\\,\\Big|_{\\substack{x=x_0\\\\y=y_0}}=0,$ we have $ \\det (g_{ij})\\Big|_{\\substack{x=x_0\\\\y=y_0}}=1$ and $R=2K$. Since $R$ and $K$ are scalars, and consequently, are coordinate-independent, we conclude that \n",
    "\n",
    "\\begin{equation}\n",
    "R = 2K\n",
    "\\tag{24.5}\n",
    "\\end{equation}\n",
    "\n",
    "at every (non-singular) point of the surface."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Remark.** For  non-singular  surfaces represented locally in the form $z=f(x,y)$ (without the condition $\\text{grad}\\, f\\,\\Big|_{\\substack{x=x_0\\\\y=y_0}}=0$) the Gauss curvature is defined by\n",
    "$$K=\\frac{f_{xx}f_{yy}-(f_{xy})^2}{(1+(f_x)^2+(f_y)^2)^2}.$$\n",
    "\n",
    "**Remark.**\n",
    "The calculations above give us the formula\n",
    "$$R_{1212}=\\frac{R}{2}\\det(g_{ij}).$$\n",
    "\n",
    "It can be generalized to all components of the tensor $R_{ijkl}$ as follows:\n",
    "\n",
    "$$R_{ijkl}=\\frac{R}{2}(g_{ik}g_{jl}-g_{il}g_{jk}).$$\n",
    "\n",
    "In fact, an explicit calculations using the symmetry properties of $R_{ijkl}$ show that\n",
    "\n",
    "\\begin{eqnarray}\n",
    " (i,j,k,l)=(1,2,1,2)\\quad \\Rightarrow & \\quad\n",
    "\\frac{R}{2}(g_{11}g_{22}-g_{12}g_{21})=\\frac{R}{2}\\det(g_{ij})=R_{1212},\\\\\n",
    "(i,j,k,l)=(2,1,1,2)\\quad \\Rightarrow & \\quad\n",
    "\\frac{R}{2}(g_{21}g_{12}-g_{22}g_{11})=-\\frac{R}{2}\\det(g_{ij})=R_{2112},\\\\\n",
    "(i,j,k,l)=(1,2,2,1)\\quad \\Rightarrow & \\quad\n",
    "\\frac{R}{2}(g_{12}g_{21}-g_{11}g_{22})=-\\frac{R}{2}\\det(g_{ij})=R_{1221},\\\\\n",
    "(i,j,k,l)=(2,1,2,1)\\quad \\Rightarrow & \\quad\n",
    "\\frac{R}{2}(g_{22}g_{11}-g_{21}g_{12})=\\frac{R}{2}\\det(g_{ij})=R_{2121},\\\\\n",
    "\\end{eqnarray}\n",
    "\n",
    "and for the remaining sequences of indices, both sides are zero.\n",
    "\n"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "<br>\n",
    "\n",
    "**Example 24.1**\n",
    "\n",
    "In $R^2$ with $g=dx\\otimes dx+dy\\otimes dy$, we have $R^i_{mkl}=0$ and $K=\\frac{R}{2}=0.$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field Riem(g) of type (1,3) on the Euclidean plane E^2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Riem(g) = 0"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.<x,y>=EuclideanSpace()      # Euclidean space E^2\n",
    "print(E.metric().riemann())   # curvature tensor of (1,3)-type\n",
    "E.metric().riemann().disp()   # show the curvature tensor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The method `ricci` gives the Ricci tensor: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Field of symmetric bilinear forms Ric(g) on the Euclidean plane E^2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Ric(g) = 0"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(E.metric().ricci())    # Ricci curvature tensor\n",
    "E.metric().ricci().disp()    # show the result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Ricci scalar curvature can be obtained using `ricci_scalar` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.metric().ricci_scalar().expr()  # Ricci scalar curvature"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 24.2**\n",
    "\n",
    "The same can be done without `EuclideanSpace` command."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Tensor field Riem(g) of type (1,3) on the 2-dimensional differentiable manifold R^2\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\mathrm{Riem}\\left(g\\right) = 0\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\mathrm{Riem}\\left(g\\right) = 0$"
      ],
      "text/plain": [
       "Riem(g) = 0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "M=Manifold(2,name='R^2',start_index=1) # manifold R2\n",
    "c_xy.<x,y>=M.chart()        # chart on R^2\n",
    "g = M.metric('g');          # metric on R^2\n",
    "g[:]=[[1,0],[0,1]]          # nonzero components\n",
    "Riem=g.riemann()            # curvature (1,3)-type tensor \n",
    "print(Riem)                 # show information on curv. tensor\n",
    "Riem.disp()                 # show the curvature tensor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Field of symmetric bilinear forms Ric(g) on the 2-dimensional differentiable manifold R^2\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\mathrm{Ric}\\left(g\\right) = 0\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\mathrm{Ric}\\left(g\\right) = 0$"
      ],
      "text/plain": [
       "Ric(g) = 0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(g.ricci())            # Ricci curvature tensor information\n",
    "g.ricci().disp()            # show  Ricci tensor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Ricci scalar curvature: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 0\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.ricci_scalar().expr()     # Ricci scalar curvature "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 24.3**\n",
    "\n",
    "For the metric on the sphere $S^2\\subset R^3$, we have \n",
    "$𝑔=d𝜃⊗d𝜃+sin(𝜃)^2d𝜙⊗d𝜙$, $K=R/2=1.$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle g = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle g = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi$"
      ],
      "text/plain": [
       "g = dtheta⊗dtheta + sin(theta)^2 dphi⊗dphi"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "M = Manifold(3, 'R^3')              # R^3 ambient space for S^2\n",
    "c_xyz.<x,y,z> = M.chart()           # chart on R^3\n",
    "N = Manifold(2, 'N',start_index=1)  # sphere S^2\n",
    "c_sph.<theta,phi>=N.chart()         # chart on S^2\n",
    "psi = N.diff_map(M, (sin(theta)*cos(phi), \n",
    "                     sin(theta)*sin(phi),\n",
    "                     cos(theta)),\n",
    "   name='psi',latex_name=r'\\psi')   # embedding S^2 -> R^3\n",
    "g0=M.metric('g0')                   # metric g0 on R^3\n",
    "g0[0,0],g0[1,1],g0[2,2]=1,1,1       # components of g0\n",
    "g=N.metric('g')                     # metric on S^2\n",
    "g.set( psi.pullback(g0) )           # g is the pullback of g0\n",
    "g.disp()                            # show g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 1}\\,2\\,1\\,2}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\sin\\left(\\theta\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 1}\\,2\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\sin\\left(\\theta\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 2}\\,1\\,1\\,2}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -1 \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 2}\\,1\\,2\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & 1 \\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 1}\\,2\\,1\\,2}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\sin\\left(\\theta\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 1}\\,2\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\sin\\left(\\theta\\right)^{2} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 2}\\,1\\,1\\,2}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -1 \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 2}\\,1\\,2\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & 1 \\end{array}$"
      ],
      "text/plain": [
       "Riem(g)^1_212 = sin(theta)^2 \n",
       "Riem(g)^1_221 = -sin(theta)^2 \n",
       "Riem(g)^2_112 = -1 \n",
       "Riem(g)^2_121 = 1 "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Riem=g.riemann()                    # curvature (1,3)-tensor\n",
    "Riem.display_comp(coordinate_labels=False)  # show components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Field of symmetric bilinear forms Ric(g) on the 2-dimensional differentiable manifold N\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\mathrm{Ric}\\left(g\\right) = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\mathrm{Ric}\\left(g\\right) = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi$"
      ],
      "text/plain": [
       "Ric(g) = dtheta⊗dtheta + sin(theta)^2 dphi⊗dphi"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(g.ricci())                   # Ricci curvature tensor\n",
    "g.ricci().disp()                   # show Ricci tensor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us show how to apply (24.3).\n",
    "\n",
    "First we have to define the frame and coframe fields:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "fr=c_sph.frame()[:]                # frame\n",
    "cfr=c_sph.coframe()[:]             # coframe"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "next we define the matrix of coefficient of the Ricci tensor field according to (24.3). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\left[1, 0\\right], \\left[0, \\sin\\left(\\theta\\right)^{2}\\right]\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\left[1, 0\\right], \\left[0, \\sin\\left(\\theta\\right)^{2}\\right]\\right]$"
      ],
      "text/plain": [
       "[[1, 0], [0, sin(theta)^2]]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ric_coeff=[[sum([Riem(cfr[i],fr[k],fr[i],fr[l]).expr() \n",
    "                 for i in range(2)])\n",
    " for k in range(2)] for l in range(2)]  # cf. (24.3)\n",
    "Ric_coeff[:]                       # Ricci ten. components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we define the Ricci tensor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle Ric = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle Ric = \\mathrm{d} \\theta\\otimes \\mathrm{d} \\theta + \\sin\\left(\\theta\\right)^{2} \\mathrm{d} \\phi\\otimes \\mathrm{d} \\phi$"
      ],
      "text/plain": [
       "Ric = dtheta⊗dtheta + sin(theta)^2 dphi⊗dphi"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ric=N.tensor_field(0,2,'Ric')     # Ricci curvature tensor\n",
    "Ric[:]=Ric_coeff                  # define all components\n",
    "Ric.disp()                        # show Ricci tensor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ricci scalar curvature:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 2\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.ricci_scalar().expr()           # Ricci scalar curvature"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Formula (24.4) can be used as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 2\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 2$"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.inverse().contract(Ric).trace().expr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "According to (24.5), the Gaussian curvature is  1."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 24.4**\n",
    "\n",
    "Consider the Poincare half-plane $y>0$ with the metric defined by\n",
    "$g_{11}=\\frac{1}{y^2},\\ \\ g_{22}=\\frac{1}{y^2}, \\ $ and  remaining components equal to zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = Manifold(2, 'M', start_index=1)  # Poincare half-plane\n",
    "X.<x,y> = M.chart(coord_restrictions=lambda x,y: y>0)\n",
    "g = M.metric('g')                    # metric on M\n",
    "g[1,1], g[2,2] = 1/y^2, 1/y^2        # nonzero components of g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute (1,3) type curvature tensor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 1}\\,2\\,1\\,2}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{1}{y^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 1}\\,2\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{1}{y^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 2}\\,1\\,1\\,2}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{1}{y^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 2}\\,1\\,2\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{1}{y^{2}} \\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{lcl} \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 1}\\,2\\,1\\,2}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{1}{y^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 1}\\,2\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{1}{y^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 2}\\,1\\,1\\,2}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{1}{y^{2}} \\\\ \\mathrm{Riem}\\left(g\\right)_{\\phantom{\\, 2}\\,1\\,2\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{1}{y^{2}} \\end{array}$"
      ],
      "text/plain": [
       "Riem(g)^1_212 = -1/y^2 \n",
       "Riem(g)^1_221 = y^(-2) \n",
       "Riem(g)^2_112 = y^(-2) \n",
       "Riem(g)^2_121 = -1/y^2 "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Riem=g.riemann()                     # (1,3) type curvature tensor\n",
    "Riem.display_comp(coordinate_labels=False)  # show nonzero comp."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and next the Riemannian curvature tensor of type (0,4), using (24.1):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{lcl} R_{\\,1\\,2\\,1\\,2}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{1}{y^{4}} \\\\ R_{\\,1\\,2\\,2\\,1}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{1}{y^{4}} \\\\ R_{\\,2\\,1\\,1\\,2}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{1}{y^{4}} \\\\ R_{\\,2\\,1\\,2\\,1}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{1}{y^{4}} \\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{lcl} R_{\\,1\\,2\\,1\\,2}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{1}{y^{4}} \\\\ R_{\\,1\\,2\\,2\\,1}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{1}{y^{4}} \\\\ R_{\\,2\\,1\\,1\\,2}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{1}{y^{4}} \\\\ R_{\\,2\\,1\\,2\\,1}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{1}{y^{4}} \\end{array}$"
      ],
      "text/plain": [
       "R_1212 = -1/y^4 \n",
       "R_1221 = y^(-4) \n",
       "R_2112 = y^(-4) \n",
       "R_2121 = -1/y^4 "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R=M.tensor_field(0,4,'R')         # R is (0,4)-type tensor field\n",
    "R[:]=g.contract(Riem)[:]                # apply (24.1)\n",
    "R.display_comp(coordinate_labels=False) # show (0,4) type ten. R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now compute the Ricci curvature tensor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\mathrm{Ric}\\left(g\\right) = -\\frac{1}{y^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x -\\frac{1}{y^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\mathrm{Ric}\\left(g\\right) = -\\frac{1}{y^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x -\\frac{1}{y^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y$"
      ],
      "text/plain": [
       "Ric(g) = -1/y^2 dx⊗dx - 1/y^2 dy⊗dy"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.ricci().disp()                      # Ricci curv. tensor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and Ricci  curvature scalar:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle -2\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle -2$"
      ],
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.ricci_scalar().expr()               # Ricci scalar curv."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "<br>\n",
    "\n",
    "**Example 24.5**\n",
    "\n",
    "Poincare disk is the set $\\ \\ \\{(x,y)\\in R^2:x^2+y^2<1\\}\\ \\ $ with the metric defined by\n",
    "$g_{11}=\\frac{4}{(1-x^2-y^2)^2},\\ \\ g_{22}=\\frac{4}{(1-x^2-y^2)^2}\\ \\ $ and remaining components equal to zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = Manifold(2, 'M', start_index=1)  # Poincare disc\n",
    "                # Cartesian coordinates on the Poincaré disk:\n",
    "X.<x,y> = M.chart('x:(-1,1) y:(-1,1)', \n",
    "    coord_restrictions=lambda x,y: x^2+y^2<1)\n",
    "g = M.metric('g')                    #  metric on M\n",
    "                                     # components of g\n",
    "g[1,1], g[2,2] = 4/(1-x^2-y^2)^2,4/(1-x^2-y^2)^2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute (1,3)-type curvature tensor. Since `g.riemann()` gives an unsimplified result, we simplify the components one by one."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{lcl} R_{\\phantom{\\, 1}\\,2\\,1\\,2}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\\\ R_{\\phantom{\\, 1}\\,2\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\\\ R_{\\phantom{\\, 2}\\,1\\,1\\,2}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\\\ R_{\\phantom{\\, 2}\\,1\\,2\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{lcl} R_{\\phantom{\\, 1}\\,2\\,1\\,2}^{\\,1\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\\\ R_{\\phantom{\\, 1}\\,2\\,2\\,1}^{\\,1\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\\\ R_{\\phantom{\\, 2}\\,1\\,1\\,2}^{\\,2\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\\\ R_{\\phantom{\\, 2}\\,1\\,2\\,1}^{\\,2\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\end{array}$"
      ],
      "text/plain": [
       "R^1_212 = -4/(x^2 + y^2 - 1)^2 \n",
       "R^1_221 = 4/(x^2 + y^2 - 1)^2 \n",
       "R^2_112 = 4/(x^2 + y^2 - 1)^2 \n",
       "R^2_121 = -4/(x^2 + y^2 - 1)^2 "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Riem = g.riemann();           # curvature (1,3) type tensor\n",
    "                              # factor components\n",
    "Riem2=[[[[Riem[a,b,c,d].factor() for a in [1,2]] \n",
    "         for b in [1,2]] for c in [1,2]] for d in[1,2]];\n",
    "R=M.tensor_field(1,3,'R')     #  tensor field of type (1,3)\n",
    "R[X.frame(),:]=Riem2          # simplified components\n",
    "R.display_comp(coordinate_labels=False)  # show R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ricci curvature tensor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\mathrm{Ric}\\left(g\\right) = -\\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x -\\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\mathrm{Ric}\\left(g\\right) = -\\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x -\\frac{4}{{\\left(x^{2} + y^{2} - 1\\right)}^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y$"
      ],
      "text/plain": [
       "Ric(g) = -4/(x^2 + y^2 - 1)^2 dx⊗dx - 4/(x^2 + y^2 - 1)^2 dy⊗dy"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ric=g.ricci()                # Ricci curvature tensor\n",
    "Ric.apply_map(factor)        # factor components\n",
    "Ric.disp()                   # show Ricci curv.tensor  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ricci curvature scalar:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle -2\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle -2$"
      ],
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.ricci_scalar().expr()     # Ricci scalar curvature "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 24.6**\n",
    "\n",
    "Consider $R^2$ with the  matrix of metric components:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left(\\begin{array}{rr}\n",
       "4 \\, x^{2} + 1 & 4 \\, x y \\\\\n",
       "4 \\, x y & 4 \\, y^{2} + 1\n",
       "\\end{array}\\right)\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left(\\begin{array}{rr}\n",
       "4 \\, x^{2} + 1 & 4 \\, x y \\\\\n",
       "4 \\, x y & 4 \\, y^{2} + 1\n",
       "\\end{array}\\right)$"
      ],
      "text/plain": [
       "[4*x^2 + 1     4*x*y]\n",
       "[    4*x*y 4*y^2 + 1]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix([[1+4*x^2,4*x*y],[4*x*y,1+4*y^2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "N=Manifold(2,name='R^2',start_index=1) # Manifold R^2\n",
    "X.<x,y>=N.chart()                      # chart on R^2\n",
    "g = N.metric('g');                     # metric g on R^2\n",
    "g[:]=[[1+4*x^2,4*x*y],[4*x*y,1+4*y^2]] # components of g\n",
    "Riem = g.riemann();                    # (1,3) type curv. tens.\n",
    "R=N.tensor_field(0,4,'R')              # (0,4) type curv. tens."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute Riemannian (0,4)-type tensor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{lcl} R_{\\,1\\,2\\,1\\,2}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{4}{4 \\, x^{2} + 4 \\, y^{2} + 1} \\\\ R_{\\,1\\,2\\,2\\,1}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{4}{4 \\, x^{2} + 4 \\, y^{2} + 1} \\\\ R_{\\,2\\,1\\,1\\,2}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{4}{4 \\, x^{2} + 4 \\, y^{2} + 1} \\\\ R_{\\,2\\,1\\,2\\,1}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{4}{4 \\, x^{2} + 4 \\, y^{2} + 1} \\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{lcl} R_{\\,1\\,2\\,1\\,2}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{4}{4 \\, x^{2} + 4 \\, y^{2} + 1} \\\\ R_{\\,1\\,2\\,2\\,1}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{4}{4 \\, x^{2} + 4 \\, y^{2} + 1} \\\\ R_{\\,2\\,1\\,1\\,2}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{4}{4 \\, x^{2} + 4 \\, y^{2} + 1} \\\\ R_{\\,2\\,1\\,2\\,1}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{4}{4 \\, x^{2} + 4 \\, y^{2} + 1} \\end{array}$"
      ],
      "text/plain": [
       "R_1212 = 4/(4*x^2 + 4*y^2 + 1) \n",
       "R_1221 = -4/(4*x^2 + 4*y^2 + 1) \n",
       "R_2112 = -4/(4*x^2 + 4*y^2 + 1) \n",
       "R_2121 = 4/(4*x^2 + 4*y^2 + 1) "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R[:]=g.contract(Riem)[:]                   # use (24.1)\n",
    "R.display_comp(coordinate_labels=False)    # show (0,4) t. R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ricci curvature tensor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\mathrm{Ric}\\left(g\\right) = \\frac{4 \\, {\\left(4 \\, x^{2} + 1\\right)}}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x + \\frac{16 \\, x y}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}} \\mathrm{d} x\\otimes \\mathrm{d} y + \\frac{16 \\, x y}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}} \\mathrm{d} y\\otimes \\mathrm{d} x + \\frac{4 \\, {\\left(4 \\, y^{2} + 1\\right)}}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\mathrm{Ric}\\left(g\\right) = \\frac{4 \\, {\\left(4 \\, x^{2} + 1\\right)}}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x + \\frac{16 \\, x y}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}} \\mathrm{d} x\\otimes \\mathrm{d} y + \\frac{16 \\, x y}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}} \\mathrm{d} y\\otimes \\mathrm{d} x + \\frac{4 \\, {\\left(4 \\, y^{2} + 1\\right)}}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y$"
      ],
      "text/plain": [
       "Ric(g) = 4*(4*x^2 + 1)/(4*x^2 + 4*y^2 + 1)^2 dx⊗dx + 16*x*y/(4*x^2 + 4*y^2 + 1)^2 dx⊗dy + 16*x*y/(4*x^2 + 4*y^2 + 1)^2 dy⊗dx + 4*(4*y^2 + 1)/(4*x^2 + 4*y^2 + 1)^2 dy⊗dy"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ric=g.ricci()                          # Ricci curvature tensor\n",
    "Ric.apply_map(factor)                  # factor components\n",
    "Ric.disp()                             # show the result"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and Ricci  scalar curvature:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\frac{4}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\frac{4}{{\\left(4 \\, x^{2} + 4 \\, y^{2} + 1\\right)}^{2}}$"
      ],
      "text/plain": [
       "4/(4*x^2 + 4*y^2 + 1)^2"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(g.ricci_scalar().expr()/2)      # Ricci curvature scalar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 24.7**\n",
    "\n",
    "Consider the embedding  of the hyperboloid into $R^3$ defined by \n",
    "$\\ \\ \\psi(u,v)= (u, v,u^2-v^2)$. The metric on the hyperboloid can be defined as  the pullback $\\psi^*g_0\\ \\ $ under $\\psi$ where $g_0$ is the standard metric on $R^3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle g = \\left( 4 \\, u^{2} + 1 \\right) \\mathrm{d} u\\otimes \\mathrm{d} u -4 \\, u v \\mathrm{d} u\\otimes \\mathrm{d} v -4 \\, u v \\mathrm{d} v\\otimes \\mathrm{d} u + \\left( 4 \\, v^{2} + 1 \\right) \\mathrm{d} v\\otimes \\mathrm{d} v\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle g = \\left( 4 \\, u^{2} + 1 \\right) \\mathrm{d} u\\otimes \\mathrm{d} u -4 \\, u v \\mathrm{d} u\\otimes \\mathrm{d} v -4 \\, u v \\mathrm{d} v\\otimes \\mathrm{d} u + \\left( 4 \\, v^{2} + 1 \\right) \\mathrm{d} v\\otimes \\mathrm{d} v$"
      ],
      "text/plain": [
       "g = (4*u^2 + 1) du⊗du - 4*u*v du⊗dv - 4*u*v dv⊗du + (4*v^2 + 1) dv⊗dv"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "M = Manifold(3, 'R3')                  # manifold R^3\n",
    "c_xyz.<x,y,z> = M.chart()              # Cart. coord. on R^3\n",
    "N=Manifold(2,name='R2',start_index=1)  # hyperboloid N\n",
    "c_uv.<u,v>=N.chart()                   # coordinates on N\n",
    "g0 = M.metric('g0');                   # standard metr, of R^3\n",
    "g0[:]=[[1,0,0],[0,1,0],[0,0,1]];       # components of g0\n",
    "psi = N.diff_map(M, (u, v,u^2-v^2) ,name='psi')  # embedding\n",
    "g=N.metric('g')                        # metric on N\n",
    "g.set(psi.pullback(g0))                # g is pullback of g0\n",
    "g.disp()                               # show g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the Riemannian curvature tensor of type (0,4)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\begin{array}{lcl} R_{\\,1\\,2\\,1\\,2}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{4}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\\\ R_{\\,1\\,2\\,2\\,1}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{4}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\\\ R_{\\,2\\,1\\,1\\,2}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{4}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\\\ R_{\\,2\\,1\\,2\\,1}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{4}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\end{array}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\begin{array}{lcl} R_{\\,1\\,2\\,1\\,2}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}} & = & -\\frac{4}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\\\ R_{\\,1\\,2\\,2\\,1}^{\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 2}\\phantom{\\, 1}} & = & \\frac{4}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\\\ R_{\\,2\\,1\\,1\\,2}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 1}\\phantom{\\, 2}} & = & \\frac{4}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\\\ R_{\\,2\\,1\\,2\\,1}^{\\phantom{\\, 2}\\phantom{\\, 1}\\phantom{\\, 2}\\phantom{\\, 1}} & = & -\\frac{4}{4 \\, u^{2} + 4 \\, v^{2} + 1} \\end{array}$"
      ],
      "text/plain": [
       "R_1212 = -4/(4*u^2 + 4*v^2 + 1) \n",
       "R_1221 = 4/(4*u^2 + 4*v^2 + 1) \n",
       "R_2112 = 4/(4*u^2 + 4*v^2 + 1) \n",
       "R_2121 = -4/(4*u^2 + 4*v^2 + 1) "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Riem = g.riemann();                    # curv. (1,3) tensor\n",
    "R=N.tensor_field(0,4,'R')              # tensor field (0,4)-type\n",
    "R[:]=g.contract(Riem)[:]               # use (24.1)\n",
    "R.display_comp(coordinate_labels=False) # show (0,4) type ten. R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ricci tensor:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\mathrm{Ric}\\left(g\\right) = -\\frac{4 \\, {\\left(4 \\, u^{2} + 1\\right)}}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\otimes \\mathrm{d} u + \\frac{16 \\, u v}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\otimes \\mathrm{d} v + \\frac{16 \\, u v}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} v\\otimes \\mathrm{d} u -\\frac{4 \\, {\\left(4 \\, v^{2} + 1\\right)}}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} v\\otimes \\mathrm{d} v\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\mathrm{Ric}\\left(g\\right) = -\\frac{4 \\, {\\left(4 \\, u^{2} + 1\\right)}}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\otimes \\mathrm{d} u + \\frac{16 \\, u v}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} u\\otimes \\mathrm{d} v + \\frac{16 \\, u v}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} v\\otimes \\mathrm{d} u -\\frac{4 \\, {\\left(4 \\, v^{2} + 1\\right)}}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}} \\mathrm{d} v\\otimes \\mathrm{d} v$"
      ],
      "text/plain": [
       "Ric(g) = -4*(4*u^2 + 1)/(4*u^2 + 4*v^2 + 1)^2 du⊗du + 16*u*v/(4*u^2 + 4*v^2 + 1)^2 du⊗dv + 16*u*v/(4*u^2 + 4*v^2 + 1)^2 dv⊗du - 4*(4*v^2 + 1)/(4*u^2 + 4*v^2 + 1)^2 dv⊗dv"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ric=g.ricci()                         # Ricci curvature tensor\n",
    "Ric.apply_map(factor)                 # factor components\n",
    "Ric.disp()                            # show the Ricci tensor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ricci  scalar curvature:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle -\\frac{8}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle -\\frac{8}{{\\left(4 \\, u^{2} + 4 \\, v^{2} + 1\\right)}^{2}}$"
      ],
      "text/plain": [
       "-8/(4*u^2 + 4*v^2 + 1)^2"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(g.ricci_scalar().expr())      # Ricci scalar curvature "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What's next?\n",
    "\n",
    "Take a look at the notebook [Torsion and curvature forms](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/25Manifold_Torsion_Curvature_Forms.ipynb)."
   ]
  }
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