{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Hyperbolic plane $\\mathbb{H}^2$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This Jupyter notebook illustrates some differential geometry capabilities of SageMath on the example of the hyperbolic plane. The corresponding tools have been developed within\n", "the [SageManifolds](https://sagemanifolds.obspm.fr) project." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A version of SageMath at least equal to 9.4 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.4.rc2, Release Date: 2021-08-12'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "version()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we set up the notebook to display mathematical objects using LaTeX formatting:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%display latex" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We also tell Maxima, which is used by SageMath for simplifications of symbolic expressions, that all computations involve real variables:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|real|\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|real|$$" ], "text/plain": [ "'real'" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "maxima_calculus.eval(\"domain: real;\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We declare $\\mathbb{H}^2$ as a 2-dimensional differentiable manifold:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2-dimensional differentiable manifold H2\n" ] }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbb{H}^2\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbb{H}^2$$" ], "text/plain": [ "2-dimensional differentiable manifold H2" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H2 = Manifold(2, 'H2', latex_name=r'\\mathbb{H}^2', start_index=1)\n", "print(H2)\n", "H2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We shall introduce charts on $\\mathbb{H}^2$ that are related to various models of the hyperbolic plane as submanifolds of $\\mathbb{R}^3$. Therefore, we start by declaring $\\mathbb{R}^3$ as a 3-dimensional manifold equiped with a global chart: the chart of Cartesian coordinates $(X,Y,Z)$:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{R}^3,(X, Y, Z)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{R}^3,(X, Y, Z)\\right)$$" ], "text/plain": [ "Chart (R3, (X, Y, Z))" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R3 = Manifold(3, 'R3', latex_name=r'\\mathbb{R}^3', start_index=1)\n", "X3. = R3.chart()\n", "X3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Hyperboloid model\n", "\n", "The first chart we introduce is related to the **hyperboloid model of $\\mathbb{H}^2$**, namely to the representation of $\\mathbb{H}^2$ as the upper sheet ($Z>0$) of the hyperboloid of two sheets defined in $\\mathbb{R}^3$ by the equation $X^2 + Y^2 - Z^2 = -1$:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(X, Y)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(X, Y)\\right)$$" ], "text/plain": [ "Chart (H2, (X, Y))" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_hyp. = H2.chart()\n", "X_hyp" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The corresponding embedding of $\\mathbb{H}^2$ in $\\mathbb{R}^3$ is" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_1:& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ & \\left(X, Y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(X, Y, \\sqrt{X^{2} + Y^{2} + 1}\\right) \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_1:& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ & \\left(X, Y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(X, Y, \\sqrt{X^{2} + Y^{2} + 1}\\right) \\end{array}$$" ], "text/plain": [ "Phi_1: H2 → R3\n", " (X, Y) ↦ (X, Y, Z) = (X, Y, sqrt(X^2 + Y^2 + 1))" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi1 = H2.diff_map(R3, [X, Y, sqrt(1+X^2+Y^2)], name='Phi_1', latex_name=r'\\Phi_1')\n", "Phi1.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "By plotting the chart $\\left(\\mathbb{H}^2,(X,Y)\\right)$ in terms of the Cartesian coordinates of $\\mathbb{R}^3$, we get a graphical view of $\\Phi_1(\\mathbb{H}^2)$:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(X_hyp.plot(X3, mapping=Phi1, number_values=15, color='blue'), \n", " aspect_ratio=1, figsize=7)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "A second chart is obtained from the polar coordinates $(r,\\varphi)$ associated with $(X,Y)$. Contrary to $(X,Y)$, the polar chart is not defined on the whole $\\mathbb{H}^2$, but on the complement $U$ of the segment $\\{Y=0, x\\geq 0\\}$: " ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Open subset U of the 2-dimensional differentiable manifold H2\n" ] } ], "source": [ "U = H2.open_subset('U', coord_def={X_hyp: (Y!=0, X<0)})\n", "print(U)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that (y!=0, x<0) stands for $y\\not=0$ OR $x<0$; the condition $y\\not=0$ AND $x<0$ would have been written [y!=0, x<0] instead." ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right)$$" ], "text/plain": [ "Chart (U, (r, ph))" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_pol. = U.chart(r'r:(0,+oo) ph:(0,2*pi):\\varphi')\n", "X_pol" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\varphi} :\\ \\left( 0 , 2 \\, \\pi \\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\varphi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$" ], "text/plain": [ "r: (0, +oo); ph: (0, 2*pi)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_pol.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We specify the transition map between the charts $\\left(U,(r,\\varphi)\\right)$ and $\\left(\\mathbb{H}^2,(X,Y)\\right)$ as $X=r\\cos\\varphi$, $Y=r\\sin\\varphi$:" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right) \\rightarrow \\left(U,(X, Y)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right) \\rightarrow \\left(U,(X, Y)\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (U, (r, ph)) to Chart (U, (X, Y))" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_hyp = X_pol.transition_map(X_hyp, [r*cos(ph), r*sin(ph)])\n", "pol_to_hyp" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} X & = & r \\cos\\left({\\varphi}\\right) \\\\ Y & = & r \\sin\\left({\\varphi}\\right) \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} X & = & r \\cos\\left({\\varphi}\\right) \\\\ Y & = & r \\sin\\left({\\varphi}\\right) \\end{array}\\right.$$" ], "text/plain": [ "X = r*cos(ph)\n", "Y = r*sin(ph)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_hyp.display()" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " r == r *passed*\n", " ph == arctan2(r*sin(ph), r*cos(ph)) **failed**\n", " X == X *passed*\n", " Y == Y *passed*\n", "NB: a failed report can reflect a mere lack of simplification.\n" ] } ], "source": [ "pol_to_hyp.set_inverse(sqrt(X^2+Y^2), atan2(Y, X)) " ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} r & = & \\sqrt{X^{2} + Y^{2}} \\\\ {\\varphi} & = & \\arctan\\left(Y, X\\right) \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} r & = & \\sqrt{X^{2} + Y^{2}} \\\\ {\\varphi} & = & \\arctan\\left(Y, X\\right) \\end{array}\\right.$$" ], "text/plain": [ "r = sqrt(X^2 + Y^2)\n", "ph = arctan2(Y, X)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_hyp.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The restriction of the embedding $\\Phi_1$ to $U$ has then two coordinate expressions:" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_1:& U & \\longrightarrow & \\mathbb{R}^3 \\\\ & \\left(X, Y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(X, Y, \\sqrt{X^{2} + Y^{2} + 1}\\right) \\\\ & \\left(r, {\\varphi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(r \\cos\\left({\\varphi}\\right), r \\sin\\left({\\varphi}\\right), \\sqrt{r^{2} + 1}\\right) \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_1:& U & \\longrightarrow & \\mathbb{R}^3 \\\\ & \\left(X, Y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(X, Y, \\sqrt{X^{2} + Y^{2} + 1}\\right) \\\\ & \\left(r, {\\varphi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(r \\cos\\left({\\varphi}\\right), r \\sin\\left({\\varphi}\\right), \\sqrt{r^{2} + 1}\\right) \\end{array}$$" ], "text/plain": [ "Phi_1: U → R3\n", " (X, Y) ↦ (X, Y, Z) = (X, Y, sqrt(X^2 + Y^2 + 1))\n", " (r, ph) ↦ (X, Y, Z) = (r*cos(ph), r*sin(ph), sqrt(r^2 + 1))" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi1.restrict(U).display()" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_hyp = X_pol.plot(X3, mapping=Phi1.restrict(U), number_values=15, ranges={r: (0,3)}, \n", " color='blue')\n", "show(graph_hyp, aspect_ratio=1, figsize=7)" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(\\mathbb{H}^2,(X, Y)\\right), \\left(\\mathbb{R}^3,(X, Y, Z)\\right)\\right) : \\left(X, Y, \\sqrt{X^{2} + Y^{2} + 1}\\right)\\right\\}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\left(\\left(\\mathbb{H}^2,(X, Y)\\right), \\left(\\mathbb{R}^3,(X, Y, Z)\\right)\\right) : \\left(X, Y, \\sqrt{X^{2} + Y^{2} + 1}\\right)\\right\\}$$" ], "text/plain": [ "{(Chart (H2, (X, Y)),\n", " Chart (R3, (X, Y, Z))): Coordinate functions (X, Y, sqrt(X^2 + Y^2 + 1)) on the Chart (H2, (X, Y))}" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi1._coord_expression" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Metric and curvature\n", "\n", "The metric on $\\mathbb{H}^2$ is that induced by the Minkowksy metric on $\\mathbb{R}^3$: \n", "$$ \\eta = \\mathrm{d}X\\otimes\\mathrm{d}X + \\mathrm{d}Y\\otimes\\mathrm{d}Y - \\mathrm{d}Z\\otimes\\mathrm{d}Z $$" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\eta = \\mathrm{d} X\\otimes \\mathrm{d} X+\\mathrm{d} Y\\otimes \\mathrm{d} Y-\\mathrm{d} Z\\otimes \\mathrm{d} Z\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\eta = \\mathrm{d} X\\otimes \\mathrm{d} X+\\mathrm{d} Y\\otimes \\mathrm{d} Y-\\mathrm{d} Z\\otimes \\mathrm{d} Z$$" ], "text/plain": [ "eta = dX⊗dX + dY⊗dY - dZ⊗dZ" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eta = R3.lorentzian_metric('eta', latex_name=r'\\eta')\n", "eta[1,1] = 1 ; eta[2,2] = 1 ; eta[3,3] = -1\n", "eta.display()" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{Y^{2} + 1}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} X + \\left( -\\frac{X Y}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} Y + \\left( -\\frac{X Y}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} X + \\left( \\frac{X^{2} + 1}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} Y\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{Y^{2} + 1}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} X + \\left( -\\frac{X Y}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} Y + \\left( -\\frac{X Y}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} X + \\left( \\frac{X^{2} + 1}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} Y$$" ], "text/plain": [ "g = (Y^2 + 1)/(X^2 + Y^2 + 1) dX⊗dX - X*Y/(X^2 + Y^2 + 1) dX⊗dY - X*Y/(X^2 + Y^2 + 1) dY⊗dX + (X^2 + 1)/(X^2 + Y^2 + 1) dY⊗dY" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g = H2.metric('g')\n", "g.set( Phi1.pullback(eta) )\n", "g.display() " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression of the metric tensor in terms of the polar coordinates is" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{1}{r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{1}{r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$$" ], "text/plain": [ "g = 1/(r^2 + 1) dr⊗dr + r^2 dph⊗dph" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_pol.frame(), X_pol)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Riemann curvature tensor associated with $g$ is" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Tensor field Riem(g) of type (1,3) on the 2-dimensional differentiable manifold H2\n" ] } ], "source": [ "Riem = g.riemann()\n", "print(Riem)" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(g\\right) = -r^{2} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\varphi} + r^{2} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r + \\left( \\frac{1}{r^{2} + 1} \\right) \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\varphi} + \\left( -\\frac{1}{r^{2} + 1} \\right) \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Riem}\\left(g\\right) = -r^{2} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\varphi} + r^{2} \\frac{\\partial}{\\partial r }\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r + \\left( \\frac{1}{r^{2} + 1} \\right) \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\varphi} + \\left( -\\frac{1}{r^{2} + 1} \\right) \\frac{\\partial}{\\partial {\\varphi} }\\otimes \\mathrm{d} r\\otimes \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} r$$" ], "text/plain": [ "Riem(g) = -r^2 ∂/∂r⊗dph⊗dr⊗dph + r^2 ∂/∂r⊗dph⊗dph⊗dr + 1/(r^2 + 1) ∂/∂ph⊗dr⊗dr⊗dph - 1/(r^2 + 1) ∂/∂ph⊗dr⊗dph⊗dr" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Riem.display(X_pol.frame(), X_pol)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Ricci tensor and the Ricci scalar:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Field of symmetric bilinear forms Ric(g) on the 2-dimensional differentiable manifold H2\n" ] } ], "source": [ "Ric = g.ricci()\n", "print(Ric)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = \\left( -\\frac{1}{r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r -r^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{Ric}\\left(g\\right) = \\left( -\\frac{1}{r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r -r^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$$" ], "text/plain": [ "Ric(g) = -1/(r^2 + 1) dr⊗dr - r^2 dph⊗dph" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric.display(X_pol.frame(), X_pol)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Scalar field r(g) on the 2-dimensional differentiable manifold H2\n" ] } ], "source": [ "Rscal = g.ricci_scalar()\n", "print(Rscal)" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R} \\\\ & \\left(X, Y\\right) & \\longmapsto & -2 \\\\ \\mbox{on}\\ U : & \\left(r, {\\varphi}\\right) & \\longmapsto & -2 \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\mathrm{r}\\left(g\\right):& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R} \\\\ & \\left(X, Y\\right) & \\longmapsto & -2 \\\\ \\mbox{on}\\ U : & \\left(r, {\\varphi}\\right) & \\longmapsto & -2 \\end{array}$$" ], "text/plain": [ "r(g): H2 → ℝ\n", " (X, Y) ↦ -2\n", "on U: (r, ph) ↦ -2" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Rscal.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Hence we recover the fact that $(\\mathbb{H}^2,g)$ is a space of **constant negative curvature**." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In dimension 2, the Riemann curvature tensor is entirely determined by the Ricci scalar $R$ according to\n", "\n", "$$R^i_{\\ \\, jlk} = \\frac{R}{2} \\left( \\delta^i_{\\ \\, k} g_{jl} - \\delta^i_{\\ \\, l} g_{jk} \\right)$$\n", "\n", "Let us check this formula here, under the form $R^i_{\\ \\, jlk} = -R g_{j[k} \\delta^i_{\\ \\, l]}$:" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "delta = H2.tangent_identity_field()\n", "Riem == - Rscal*(g*delta).antisymmetrize(2,3) # 2,3 = last positions of the type-(1,3) tensor g*delta " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similarly the relation $\\mathrm{Ric} = (R/2)\\; g$ must hold:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}$$" ], "text/plain": [ "True" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric == (Rscal/2)*g" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Poincaré disk model\n", "\n", "The Poincaré disk model of $\\mathbb{H}^2$ is obtained by stereographic projection from the point $S=(0,0,-1)$ of the hyperboloid model to the plane $Z=0$. The radial coordinate $R$ of the image of a point of polar coordinate $(r,\\varphi)$ is\n", "$$ R = \\frac{r}{1+\\sqrt{1+r^2}}.$$\n", "Hence we define the Poincaré disk chart on $\\mathbb{H}^2$ by" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right)$$" ], "text/plain": [ "Chart (U, (R, ph))" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_Pdisk. = U.chart(r'R:(0,1) ph:(0,2*pi):\\varphi')\n", "X_Pdisk" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}R :\\ \\left( 0 , 1 \\right) ;\\quad {\\varphi} :\\ \\left( 0 , 2 \\, \\pi \\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}R :\\ \\left( 0 , 1 \\right) ;\\quad {\\varphi} :\\ \\left( 0 , 2 \\, \\pi \\right)$$" ], "text/plain": [ "R: (0, 1); ph: (0, 2*pi)" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_Pdisk.coord_range()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "and relate it to the hyperboloid polar chart by" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right) \\rightarrow \\left(U,(R, {\\varphi})\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right) \\rightarrow \\left(U,(R, {\\varphi})\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (U, (r, ph)) to Chart (U, (R, ph))" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk = X_pol.transition_map(X_Pdisk, [r/(1+sqrt(1+r^2)), ph])\n", "pol_to_Pdisk" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} R & = & \\frac{r}{\\sqrt{r^{2} + 1} + 1} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} R & = & \\frac{r}{\\sqrt{r^{2} + 1} + 1} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$$" ], "text/plain": [ "R = r/(sqrt(r^2 + 1) + 1)\n", "ph = ph" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk.display()" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} r & = & -\\frac{2 \\, R}{R^{2} - 1} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} r & = & -\\frac{2 \\, R}{R^{2} - 1} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$$" ], "text/plain": [ "r = -2*R/(R^2 - 1)\n", "ph = ph" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk.set_inverse(2*R/(1-R^2), ph)\n", "pol_to_Pdisk.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A view of the Poincaré disk chart via the embedding $\\Phi_1$:" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(X_Pdisk.plot(X3, mapping=Phi1.restrict(U), ranges={R: (0,0.9)}, color='blue',\n", " number_values=15), \n", " aspect_ratio=1, figsize=7)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression of the metric tensor in terms of coordinates $(R,\\varphi)$:" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{4}{R^{4} - 2 \\, R^{2} + 1} \\right) \\mathrm{d} R\\otimes \\mathrm{d} R + \\left( \\frac{4 \\, R^{2}}{R^{4} - 2 \\, R^{2} + 1} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{4}{R^{4} - 2 \\, R^{2} + 1} \\right) \\mathrm{d} R\\otimes \\mathrm{d} R + \\left( \\frac{4 \\, R^{2}}{R^{4} - 2 \\, R^{2} + 1} \\right) \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$$" ], "text/plain": [ "g = 4/(R^4 - 2*R^2 + 1) dR⊗dR + 4*R^2/(R^4 - 2*R^2 + 1) dph⊗dph" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_Pdisk.frame(), X_Pdisk)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may factorize each metric component:" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{4}{{\\left(R + 1\\right)}^{2} {\\left(R - 1\\right)}^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + \\frac{4 \\, R^{2}}{{\\left(R + 1\\right)}^{2} {\\left(R - 1\\right)}^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{4}{{\\left(R + 1\\right)}^{2} {\\left(R - 1\\right)}^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + \\frac{4 \\, R^{2}}{{\\left(R + 1\\right)}^{2} {\\left(R - 1\\right)}^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$$" ], "text/plain": [ "g = 4/((R + 1)^2*(R - 1)^2) dR⊗dR + 4*R^2/((R + 1)^2*(R - 1)^2) dph⊗dph" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "for i in [1,2]:\n", " g[X_Pdisk.frame(), i, i, X_Pdisk].factor()\n", "g.display(X_Pdisk.frame(), X_Pdisk)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Cartesian coordinates on the Poincaré disk\n", "\n", "Let us introduce Cartesian coordinates $(u,v)$ on the Poincaré disk; since the latter has a unit radius, this amounts to defining the following chart on $\\mathbb{H}^2$:" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(u, v)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(u, v)\\right)$$" ], "text/plain": [ "Chart (H2, (u, v))" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_Pdisk_cart. = H2.chart('u:(-1,1) v:(-1,1)', coord_restrictions=lambda u,v: u^2+v^2 < 1)\n", "X_Pdisk_cart" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On $U$, the Cartesian coordinates $(u,v)$ are related to the polar coordinates $(R,\\varphi)$ by the standard formulas:" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right) \\rightarrow \\left(U,(u, v)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right) \\rightarrow \\left(U,(u, v)\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (U, (R, ph)) to Chart (U, (u, v))" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_Pdisk_cart = X_Pdisk.transition_map(X_Pdisk_cart, [R*cos(ph), R*sin(ph)])\n", "Pdisk_to_Pdisk_cart" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & R \\cos\\left({\\varphi}\\right) \\\\ v & = & R \\sin\\left({\\varphi}\\right) \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & R \\cos\\left({\\varphi}\\right) \\\\ v & = & R \\sin\\left({\\varphi}\\right) \\end{array}\\right.$$" ], "text/plain": [ "u = R*cos(ph)\n", "v = R*sin(ph)" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_Pdisk_cart.display()" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " R == R *passed*\n", " ph == arctan2(R*sin(ph), R*cos(ph)) **failed**\n", " u == u *passed*\n", " v == v *passed*\n", "NB: a failed report can reflect a mere lack of simplification.\n" ] }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} R & = & \\sqrt{u^{2} + v^{2}} \\\\ {\\varphi} & = & \\arctan\\left(v, u\\right) \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} R & = & \\sqrt{u^{2} + v^{2}} \\\\ {\\varphi} & = & \\arctan\\left(v, u\\right) \\end{array}\\right.$$" ], "text/plain": [ "R = sqrt(u^2 + v^2)\n", "ph = arctan2(v, u)" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_Pdisk_cart.set_inverse(sqrt(u^2+v^2), atan2(v, u)) \n", "Pdisk_to_Pdisk_cart.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The embedding of $\\mathbb{H}^2$ in $\\mathbb{R}^3$ associated with the Poincaré disk model is naturally defined as" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_2:& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ & \\left(u, v\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(u, v, 0\\right) \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_2:& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ & \\left(u, v\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(u, v, 0\\right) \\end{array}$$" ], "text/plain": [ "Phi_2: H2 → R3\n", " (u, v) ↦ (X, Y, Z) = (u, v, 0)" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi2 = H2.diff_map(R3, {(X_Pdisk_cart, X3): [u, v, 0]},\n", " name='Phi_2', latex_name=r'\\Phi_2')\n", "Phi2.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us use it to draw the Poincaré disk in $\\mathbb{R}^3$:" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "scrolled": false }, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_disk_uv = X_Pdisk_cart.plot(X3, mapping=Phi2, number_values=15)\n", "show(graph_disk_uv, figsize=7)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "On $U$, the change of coordinates $(r,\\varphi) \\rightarrow (u,v)$ is obtained by combining the changes $(r,\\varphi) \\rightarrow (R,\\varphi)$ and $(R,\\varphi) \\rightarrow (u,v)$:" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right) \\rightarrow \\left(U,(u, v)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right) \\rightarrow \\left(U,(u, v)\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (U, (r, ph)) to Chart (U, (u, v))" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk_cart = Pdisk_to_Pdisk_cart * pol_to_Pdisk\n", "pol_to_Pdisk_cart" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & \\frac{r \\cos\\left({\\varphi}\\right)}{\\sqrt{r^{2} + 1} + 1} \\\\ v & = & \\frac{r \\sin\\left({\\varphi}\\right)}{\\sqrt{r^{2} + 1} + 1} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & \\frac{r \\cos\\left({\\varphi}\\right)}{\\sqrt{r^{2} + 1} + 1} \\\\ v & = & \\frac{r \\sin\\left({\\varphi}\\right)}{\\sqrt{r^{2} + 1} + 1} \\end{array}\\right.$$" ], "text/plain": [ "u = r*cos(ph)/(sqrt(r^2 + 1) + 1)\n", "v = r*sin(ph)/(sqrt(r^2 + 1) + 1)" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pol_to_Pdisk_cart.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Still on $U$, the change of coordinates $(X,Y) \\rightarrow (u,v)$ is obtained by combining the changes $(X,Y) \\rightarrow (r,\\varphi)$ with $(r,\\varphi) \\rightarrow (u,v)$:" ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(X, Y)\\right) \\rightarrow \\left(U,(u, v)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(X, Y)\\right) \\rightarrow \\left(U,(u, v)\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (U, (X, Y)) to Chart (U, (u, v))" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart_U = pol_to_Pdisk_cart * pol_to_hyp.inverse()\n", "hyp_to_Pdisk_cart_U" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & \\frac{X}{\\sqrt{X^{2} + Y^{2} + 1} + 1} \\\\ v & = & \\frac{Y}{\\sqrt{X^{2} + Y^{2} + 1} + 1} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & \\frac{X}{\\sqrt{X^{2} + Y^{2} + 1} + 1} \\\\ v & = & \\frac{Y}{\\sqrt{X^{2} + Y^{2} + 1} + 1} \\end{array}\\right.$$" ], "text/plain": [ "u = X/(sqrt(X^2 + Y^2 + 1) + 1)\n", "v = Y/(sqrt(X^2 + Y^2 + 1) + 1)" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart_U.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We use the above expression to extend the change of coordinates $(X,Y) \\rightarrow (u,v)$ from $U$ to the whole manifold $\\mathbb{H}^2$:" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(X, Y)\\right) \\rightarrow \\left(\\mathbb{H}^2,(u, v)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(X, Y)\\right) \\rightarrow \\left(\\mathbb{H}^2,(u, v)\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (H2, (X, Y)) to Chart (H2, (u, v))" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart = X_hyp.transition_map(X_Pdisk_cart, hyp_to_Pdisk_cart_U(X,Y))\n", "hyp_to_Pdisk_cart" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & \\frac{X}{\\sqrt{X^{2} + Y^{2} + 1} + 1} \\\\ v & = & \\frac{Y}{\\sqrt{X^{2} + Y^{2} + 1} + 1} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & \\frac{X}{\\sqrt{X^{2} + Y^{2} + 1} + 1} \\\\ v & = & \\frac{Y}{\\sqrt{X^{2} + Y^{2} + 1} + 1} \\end{array}\\right.$$" ], "text/plain": [ "u = X/(sqrt(X^2 + Y^2 + 1) + 1)\n", "v = Y/(sqrt(X^2 + Y^2 + 1) + 1)" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart.display()" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Check of the inverse coordinate transformation:\n", " X == X *passed*\n", " Y == Y *passed*\n", " u == -2*u*abs(u^2 + v^2 - 1)/(u^4 + 2*u^2*v^2 + v^4 + (u^2 + v^2 - 1)*abs(u^2 + v^2 - 1) - 1) **failed**\n", " v == -2*v*abs(u^2 + v^2 - 1)/(u^4 + 2*u^2*v^2 + v^4 + (u^2 + v^2 - 1)*abs(u^2 + v^2 - 1) - 1) **failed**\n", "NB: a failed report can reflect a mere lack of simplification.\n" ] }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} X & = & -\\frac{2 \\, u}{u^{2} + v^{2} - 1} \\\\ Y & = & -\\frac{2 \\, v}{u^{2} + v^{2} - 1} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} X & = & -\\frac{2 \\, u}{u^{2} + v^{2} - 1} \\\\ Y & = & -\\frac{2 \\, v}{u^{2} + v^{2} - 1} \\end{array}\\right.$$" ], "text/plain": [ "X = -2*u/(u^2 + v^2 - 1)\n", "Y = -2*v/(u^2 + v^2 - 1)" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "hyp_to_Pdisk_cart.set_inverse(2*u/(1-u^2-v^2), 2*v/(1-u^2-v^2))\n", "hyp_to_Pdisk_cart.inverse().display()" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_Pdisk = X_pol.plot(X3, mapping=Phi2.restrict(U), ranges={r: (0, 20)}, number_values=15, \n", " label_axes=False)\n", "show(graph_hyp + graph_Pdisk, aspect_ratio=1, figsize=7)" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 30 graphics primitives" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_pol.plot(X_Pdisk_cart, ranges={r: (0, 20)}, number_values=15)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Metric tensor in Poincaré disk coordinates $(u,v)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From now on, we are using the Poincaré disk chart $(\\mathbb{H}^2,(u,v))$ as the default one on $\\mathbb{H}^2$:" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [], "source": [ "H2.set_default_chart(X_Pdisk_cart)\n", "H2.set_default_frame(X_Pdisk_cart.frame())" ] }, { "cell_type": "code", "execution_count": 54, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} - 2 \\, u^{2} + 1}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} + 2 \\, u^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} X + \\left( -\\frac{4 \\, u v}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} + 2 \\, u^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} Y + \\left( -\\frac{4 \\, u v}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} + 2 \\, u^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} X + \\left( \\frac{u^{4} + v^{4} + 2 \\, {\\left(u^{2} - 1\\right)} v^{2} + 2 \\, u^{2} + 1}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} + 2 \\, u^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} Y\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} - 2 \\, u^{2} + 1}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} + 2 \\, u^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} X + \\left( -\\frac{4 \\, u v}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} + 2 \\, u^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} Y + \\left( -\\frac{4 \\, u v}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} + 2 \\, u^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} X + \\left( \\frac{u^{4} + v^{4} + 2 \\, {\\left(u^{2} - 1\\right)} v^{2} + 2 \\, u^{2} + 1}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} + 1\\right)} v^{2} + 2 \\, u^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} Y$$" ], "text/plain": [ "g = (u^4 + v^4 + 2*(u^2 + 1)*v^2 - 2*u^2 + 1)/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dX⊗dX - 4*u*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dX⊗dY - 4*u*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dY⊗dX + (u^4 + v^4 + 2*(u^2 - 1)*v^2 + 2*u^2 + 1)/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dY⊗dY" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_hyp.frame())" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{4}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} - 1\\right)} v^{2} - 2 \\, u^{2} + 1} \\right) \\mathrm{d} u\\otimes \\mathrm{d} u + \\left( \\frac{4}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} - 1\\right)} v^{2} - 2 \\, u^{2} + 1} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{4}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} - 1\\right)} v^{2} - 2 \\, u^{2} + 1} \\right) \\mathrm{d} u\\otimes \\mathrm{d} u + \\left( \\frac{4}{u^{4} + v^{4} + 2 \\, {\\left(u^{2} - 1\\right)} v^{2} - 2 \\, u^{2} + 1} \\right) \\mathrm{d} v\\otimes \\mathrm{d} v$$" ], "text/plain": [ "g = 4/(u^4 + v^4 + 2*(u^2 - 1)*v^2 - 2*u^2 + 1) du⊗du + 4/(u^4 + v^4 + 2*(u^2 - 1)*v^2 - 2*u^2 + 1) dv⊗dv" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display()" ] }, { "cell_type": "code", "execution_count": 56, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{4}{{\\left(u^{2} + v^{2} - 1\\right)}^{2}} \\mathrm{d} u\\otimes \\mathrm{d} u + \\frac{4}{{\\left(u^{2} + v^{2} - 1\\right)}^{2}} \\mathrm{d} v\\otimes \\mathrm{d} v\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{4}{{\\left(u^{2} + v^{2} - 1\\right)}^{2}} \\mathrm{d} u\\otimes \\mathrm{d} u + \\frac{4}{{\\left(u^{2} + v^{2} - 1\\right)}^{2}} \\mathrm{d} v\\otimes \\mathrm{d} v$$" ], "text/plain": [ "g = 4/(u^2 + v^2 - 1)^2 du⊗du + 4/(u^2 + v^2 - 1)^2 dv⊗dv" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g[1,1].factor() ; g[2,2].factor()\n", "g.display()" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## Hemispherical model\n", "\n", "The **hemispherical model of $\\mathbb{H}^2$** is obtained by the inverse stereographic projection from the point $S = (0,0,-1)$ of the Poincaré disk to the unit sphere $X^2+Y^2+Z^2=1$. This induces a spherical coordinate chart on $U$:" ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,({\\theta}, {\\varphi})\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,({\\theta}, {\\varphi})\\right)$$" ], "text/plain": [ "Chart (U, (th, ph))" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_spher. = U.chart(r'th:(0,pi/2):\\theta ph:(0,2*pi):\\varphi')\n", "X_spher" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From the stereographic projection from $S$, we obtain that\n", "\\begin{equation}\n", "\\sin\\theta = \\frac{2R}{1+R^2}\n", "\\end{equation}\n", "Hence the transition map:" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right) \\rightarrow \\left(U,({\\theta}, {\\varphi})\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right) \\rightarrow \\left(U,({\\theta}, {\\varphi})\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (U, (R, ph)) to Chart (U, (th, ph))" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_spher = X_Pdisk.transition_map(X_spher, [arcsin(2*R/(1+R^2)), ph])\n", "Pdisk_to_spher" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\theta} & = & \\arcsin\\left(\\frac{2 \\, R}{R^{2} + 1}\\right) \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\theta} & = & \\arcsin\\left(\\frac{2 \\, R}{R^{2} + 1}\\right) \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$$" ], "text/plain": [ "th = arcsin(2*R/(R^2 + 1))\n", "ph = ph" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_spher.display()" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} R & = & \\frac{\\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right) + 1} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} R & = & \\frac{\\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right) + 1} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.$$" ], "text/plain": [ "R = sin(th)/(cos(th) + 1)\n", "ph = ph" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_spher.set_inverse(sin(th)/(1+cos(th)), ph)\n", "Pdisk_to_spher.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the spherical coordinates $(\\theta,\\varphi)$, the metric takes the following form:" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{1}{\\cos\\left({\\theta}\\right)^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{\\sin\\left({\\theta}\\right)^{2}}{\\cos\\left({\\theta}\\right)^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{1}{\\cos\\left({\\theta}\\right)^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{\\sin\\left({\\theta}\\right)^{2}}{\\cos\\left({\\theta}\\right)^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$$" ], "text/plain": [ "g = cos(th)^(-2) dth⊗dth + sin(th)^2/cos(th)^2 dph⊗dph" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_spher.frame(), X_spher)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The embedding of $\\mathbb{H}^2$ in $\\mathbb{R}^3$ associated with the hemispherical model is naturally:" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_3:& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\mbox{on}\\ U : & \\left(R, {\\varphi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, R \\cos\\left({\\varphi}\\right)}{R^{2} + 1}, \\frac{2 \\, R \\sin\\left({\\varphi}\\right)}{R^{2} + 1}, -\\frac{R^{2} - 1}{R^{2} + 1}\\right) \\\\ \\mbox{on}\\ U : & \\left({\\theta}, {\\varphi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\cos\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right), \\sin\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right), \\cos\\left({\\theta}\\right)\\right) \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_3:& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ \\mbox{on}\\ U : & \\left(R, {\\varphi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\frac{2 \\, R \\cos\\left({\\varphi}\\right)}{R^{2} + 1}, \\frac{2 \\, R \\sin\\left({\\varphi}\\right)}{R^{2} + 1}, -\\frac{R^{2} - 1}{R^{2} + 1}\\right) \\\\ \\mbox{on}\\ U : & \\left({\\theta}, {\\varphi}\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(\\cos\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right), \\sin\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right), \\cos\\left({\\theta}\\right)\\right) \\end{array}$$" ], "text/plain": [ "Phi_3: H2 → R3\n", "on U: (R, ph) ↦ (X, Y, Z) = (2*R*cos(ph)/(R^2 + 1), 2*R*sin(ph)/(R^2 + 1), -(R^2 - 1)/(R^2 + 1))\n", "on U: (th, ph) ↦ (X, Y, Z) = (cos(ph)*sin(th), sin(ph)*sin(th), cos(th))" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi3 = H2.diff_map(R3, {(X_spher, X3): [sin(th)*cos(ph), sin(th)*sin(ph), cos(th)]},\n", " name='Phi_3', latex_name=r'\\Phi_3')\n", "Phi3.display()" ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_spher = X_pol.plot(X3, mapping=Phi3, ranges={r: (0, 20)}, number_values=15, \n", " color='orange', label_axes=False)\n", "show(graph_hyp + graph_Pdisk + graph_spher, aspect_ratio=1, \n", " figsize=7)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Poincaré half-plane model\n", "\n", "The **Poincaré half-plane model of $\\mathbb{H}^2$** is obtained by stereographic projection from the point $W=(-1,0,0)$ of the hemispherical model to the plane $X=1$. This induces a new coordinate chart on $\\mathbb{H}^2$ by setting $(x,y)=(Y,Z)$ in the plane $X=1$:" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(x, y)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(x, y)\\right)$$" ], "text/plain": [ "Chart (H2, (x, y))" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_hplane. = H2.chart('x y:(0,+oo)')\n", "X_hplane" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The coordinate transformation $(\\theta,\\varphi)\\rightarrow (x,y)$ is easily deduced from the stereographic projection from the point $W$:" ] }, { "cell_type": "code", "execution_count": 65, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,({\\theta}, {\\varphi})\\right) \\rightarrow \\left(U,(x, y)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,({\\theta}, {\\varphi})\\right) \\rightarrow \\left(U,(x, y)\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (U, (th, ph)) to Chart (U, (x, y))" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher_to_hplane = X_spher.transition_map(X_hplane, [2*sin(th)*sin(ph)/(1+sin(th)*cos(ph)),\n", " 2*cos(th)/(1+sin(th)*cos(ph))])\n", "spher_to_hplane" ] }, { "cell_type": "code", "execution_count": 66, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & \\frac{2 \\, \\sin\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right)}{\\cos\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right) + 1} \\\\ y & = & \\frac{2 \\, \\cos\\left({\\theta}\\right)}{\\cos\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right) + 1} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & \\frac{2 \\, \\sin\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right)}{\\cos\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right) + 1} \\\\ y & = & \\frac{2 \\, \\cos\\left({\\theta}\\right)}{\\cos\\left({\\varphi}\\right) \\sin\\left({\\theta}\\right) + 1} \\end{array}\\right.$$" ], "text/plain": [ "x = 2*sin(ph)*sin(th)/(cos(ph)*sin(th) + 1)\n", "y = 2*cos(th)/(cos(ph)*sin(th) + 1)" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spher_to_hplane.display()" ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right) \\rightarrow \\left(U,(x, y)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right) \\rightarrow \\left(U,(x, y)\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (U, (R, ph)) to Chart (U, (x, y))" ] }, "execution_count": 67, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_hplane = spher_to_hplane * Pdisk_to_spher\n", "Pdisk_to_hplane" ] }, { "cell_type": "code", "execution_count": 68, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & \\frac{4 \\, R \\sin\\left({\\varphi}\\right)}{R^{2} + 2 \\, R \\cos\\left({\\varphi}\\right) + 1} \\\\ y & = & -\\frac{2 \\, {\\left(R^{2} - 1\\right)}}{R^{2} + 2 \\, R \\cos\\left({\\varphi}\\right) + 1} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & \\frac{4 \\, R \\sin\\left({\\varphi}\\right)}{R^{2} + 2 \\, R \\cos\\left({\\varphi}\\right) + 1} \\\\ y & = & -\\frac{2 \\, {\\left(R^{2} - 1\\right)}}{R^{2} + 2 \\, R \\cos\\left({\\varphi}\\right) + 1} \\end{array}\\right.$$" ], "text/plain": [ "x = 4*R*sin(ph)/(R^2 + 2*R*cos(ph) + 1)\n", "y = -2*(R^2 - 1)/(R^2 + 2*R*cos(ph) + 1)" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_to_hplane.display()" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(u, v)\\right) \\rightarrow \\left(U,(x, y)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(u, v)\\right) \\rightarrow \\left(U,(x, y)\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (U, (u, v)) to Chart (U, (x, y))" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_cart_to_hplane_U = Pdisk_to_hplane * Pdisk_to_Pdisk_cart.inverse()\n", "Pdisk_cart_to_hplane_U" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & \\frac{4 \\, v}{u^{2} + v^{2} + 2 \\, u + 1} \\\\ y & = & -\\frac{2 \\, {\\left(u^{2} + v^{2} - 1\\right)}}{u^{2} + v^{2} + 2 \\, u + 1} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & \\frac{4 \\, v}{u^{2} + v^{2} + 2 \\, u + 1} \\\\ y & = & -\\frac{2 \\, {\\left(u^{2} + v^{2} - 1\\right)}}{u^{2} + v^{2} + 2 \\, u + 1} \\end{array}\\right.$$" ], "text/plain": [ "x = 4*v/(u^2 + v^2 + 2*u + 1)\n", "y = -2*(u^2 + v^2 - 1)/(u^2 + v^2 + 2*u + 1)" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_cart_to_hplane_U.display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us use the above formula to define the transition map $(u,v)\\rightarrow (x,y)$ on the whole manifold $\\mathbb{H}^2$ (and not only on $U$):" ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(u, v)\\right) \\rightarrow \\left(\\mathbb{H}^2,(x, y)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(u, v)\\right) \\rightarrow \\left(\\mathbb{H}^2,(x, y)\\right)$$" ], "text/plain": [ "Change of coordinates from Chart (H2, (u, v)) to Chart (H2, (x, y))" ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_cart_to_hplane = X_Pdisk_cart.transition_map(X_hplane, Pdisk_cart_to_hplane_U(u,v))\n", "Pdisk_cart_to_hplane" ] }, { "cell_type": "code", "execution_count": 72, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & \\frac{4 \\, v}{u^{2} + v^{2} + 2 \\, u + 1} \\\\ y & = & -\\frac{2 \\, {\\left(u^{2} + v^{2} - 1\\right)}}{u^{2} + v^{2} + 2 \\, u + 1} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} x & = & \\frac{4 \\, v}{u^{2} + v^{2} + 2 \\, u + 1} \\\\ y & = & -\\frac{2 \\, {\\left(u^{2} + v^{2} - 1\\right)}}{u^{2} + v^{2} + 2 \\, u + 1} \\end{array}\\right.$$" ], "text/plain": [ "x = 4*v/(u^2 + v^2 + 2*u + 1)\n", "y = -2*(u^2 + v^2 - 1)/(u^2 + v^2 + 2*u + 1)" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_cart_to_hplane.display()" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & -\\frac{x^{2} + y^{2} - 4}{x^{2} + {\\left(y + 2\\right)}^{2}} \\\\ v & = & \\frac{4 \\, x}{x^{2} + {\\left(y + 2\\right)}^{2}} \\end{array}\\right.\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & -\\frac{x^{2} + y^{2} - 4}{x^{2} + {\\left(y + 2\\right)}^{2}} \\\\ v & = & \\frac{4 \\, x}{x^{2} + {\\left(y + 2\\right)}^{2}} \\end{array}\\right.$$" ], "text/plain": [ "u = -(x^2 + y^2 - 4)/(x^2 + (y + 2)^2)\n", "v = 4*x/(x^2 + (y + 2)^2)" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Pdisk_cart_to_hplane.set_inverse((4-x^2-y^2)/(x^2+(2+y)^2), 4*x/(x^2+(2+y)^2))\n", "Pdisk_cart_to_hplane.inverse().display()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since the coordinates $(x,y)$ correspond to $(Y,Z)$ in the plane $X=1$, the embedding of $\\mathbb{H}^2$ in $\\mathbb{R}^3$ naturally associated with the Poincaré half-plane model is" ] }, { "cell_type": "code", "execution_count": 74, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_4:& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ & \\left(u, v\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(1, \\frac{4 \\, v}{u^{2} + v^{2} + 2 \\, u + 1}, -\\frac{2 \\, {\\left(u^{2} + v^{2} - 1\\right)}}{u^{2} + v^{2} + 2 \\, u + 1}\\right) \\\\ & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(1, x, y\\right) \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} \\Phi_4:& \\mathbb{H}^2 & \\longrightarrow & \\mathbb{R}^3 \\\\ & \\left(u, v\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(1, \\frac{4 \\, v}{u^{2} + v^{2} + 2 \\, u + 1}, -\\frac{2 \\, {\\left(u^{2} + v^{2} - 1\\right)}}{u^{2} + v^{2} + 2 \\, u + 1}\\right) \\\\ & \\left(x, y\\right) & \\longmapsto & \\left(X, Y, Z\\right) = \\left(1, x, y\\right) \\end{array}$$" ], "text/plain": [ "Phi_4: H2 → R3\n", " (u, v) ↦ (X, Y, Z) = (1, 4*v/(u^2 + v^2 + 2*u + 1), -2*(u^2 + v^2 - 1)/(u^2 + v^2 + 2*u + 1))\n", " (x, y) ↦ (X, Y, Z) = (1, x, y)" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Phi4 = H2.diff_map(R3, {(X_hplane, X3): [1, x, y]}, name='Phi_4', latex_name=r'\\Phi_4')\n", "Phi4.display()" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n" ], "text/plain": [ "Graphics3d Object" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "graph_hplane = X_pol.plot(X3, mapping=Phi4.restrict(U), ranges={r: (0, 1.5)}, \n", " number_values=15, color='brown', label_axes=False)\n", "show(graph_hyp + graph_Pdisk + graph_spher + graph_hplane, \n", " aspect_ratio=1, figsize=8)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us draw the grid of the hyperboloidal coordinates $(r,\\varphi)$ in terms of the half-plane coordinates $(x,y)$:" ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [], "source": [ "pol_to_hplane = Pdisk_to_hplane * pol_to_Pdisk" ] }, { "cell_type": "code", "execution_count": 77, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 30 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(X_pol.plot(X_hplane, ranges={r: (0,24)}, style={r: '-', ph: '--'}, number_values=15, \n", " plot_points=200, color='brown'), xmin=-5, xmax=5, ymin=0, ymax=5, \n", " aspect_ratio=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The solid curves are those along which $r$ varies while $\\varphi$ is kept constant. Conversely, the dashed curves are those along which $\\varphi$ varies, while $r$ is kept constant. We notice that the former curves are arcs of circles orthogonal to the half-plane boundary $y=0$, hence they are geodesics of $(\\mathbb{H}^2,g)$. This is not surprising since they correspond to the intersections of the hyperboloid with planes through the origin (namely the plane $\\varphi=\\mathrm{const}$). The point $(x,y) = (0,2)$ corresponds to $r=0$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We may also depict the Poincaré disk coordinates $(u,v)$ in terms of the half-plane coordinates $(x,y)$:" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 64 graphics primitives" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X_Pdisk_cart.plot(ranges={u: (-1, 0), v: (-1, 0)}, \n", " style={u: '-', v: '--'}) + \\\n", "X_Pdisk_cart.plot(ranges={u: (-1, 0), v: (0., 1)}, \n", " style={u: '-', v: '--'}, color='orange') + \\\n", "X_Pdisk_cart.plot(ranges={u: (0, 1), v: (-1, 0)}, \n", " style={u: '-', v: '--'}, color='pink') + \\\n", "X_Pdisk_cart.plot(ranges={u: (0, 1), v: (0, 1)}, \n", " style={u: '-', v: '--'}, color='violet')" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "Graphics object consisting of 64 graphics primitives" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show(X_Pdisk_cart.plot(X_hplane, ranges={u: (-1, 0), v: (-1, 0)}, \n", " style={u: '-', v: '--'}) + \\\n", " X_Pdisk_cart.plot(X_hplane, ranges={u: (-1, 0), v: (0, 1)}, \n", " style={u: '-', v: '--'}, color='orange') + \\\n", " X_Pdisk_cart.plot(X_hplane, ranges={u: (0, 1), v: (-1, 0)}, \n", " style={u: '-', v: '--'}, color='pink') + \\\n", " X_Pdisk_cart.plot(X_hplane, ranges={u: (0, 1), v: (0, 1)}, \n", " style={u: '-', v: '--'}, color='violet'),\n", " xmin=-5, xmax=5, ymin=0, ymax=5, aspect_ratio=1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression of the metric tensor in the half-plane coordinates $(x,y)$ is" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{1}{y^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x + \\frac{1}{y^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{1}{y^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x + \\frac{1}{y^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y$$" ], "text/plain": [ "g = y^(-2) dx⊗dx + y^(-2) dy⊗dy" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g.display(X_hplane.frame(), X_hplane)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "## Summary\n", "\n", "9 charts have been defined on $\\mathbb{H}^2$:" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{H}^2,(X, Y)\\right), \\left(U,(X, Y)\\right), \\left(U,(r, {\\varphi})\\right), \\left(U,(R, {\\varphi})\\right), \\left(\\mathbb{H}^2,(u, v)\\right), \\left(U,(u, v)\\right), \\left(U,({\\theta}, {\\varphi})\\right), \\left(\\mathbb{H}^2,(x, y)\\right), \\left(U,(x, y)\\right)\\right]\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{H}^2,(X, Y)\\right), \\left(U,(X, Y)\\right), \\left(U,(r, {\\varphi})\\right), \\left(U,(R, {\\varphi})\\right), \\left(\\mathbb{H}^2,(u, v)\\right), \\left(U,(u, v)\\right), \\left(U,({\\theta}, {\\varphi})\\right), \\left(\\mathbb{H}^2,(x, y)\\right), \\left(U,(x, y)\\right)\\right]$$" ], "text/plain": [ "[Chart (H2, (X, Y)),\n", " Chart (U, (X, Y)),\n", " Chart (U, (r, ph)),\n", " Chart (U, (R, ph)),\n", " Chart (H2, (u, v)),\n", " Chart (U, (u, v)),\n", " Chart (U, (th, ph)),\n", " Chart (H2, (x, y)),\n", " Chart (U, (x, y))]" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H2.atlas()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "There are actually 6 main (top) charts, the other ones being subcharts:" ] }, { "cell_type": "code", "execution_count": 82, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{H}^2,(X, Y)\\right), \\left(U,(r, {\\varphi})\\right), \\left(U,(R, {\\varphi})\\right), \\left(\\mathbb{H}^2,(u, v)\\right), \\left(U,({\\theta}, {\\varphi})\\right), \\left(\\mathbb{H}^2,(x, y)\\right)\\right]\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left(\\mathbb{H}^2,(X, Y)\\right), \\left(U,(r, {\\varphi})\\right), \\left(U,(R, {\\varphi})\\right), \\left(\\mathbb{H}^2,(u, v)\\right), \\left(U,({\\theta}, {\\varphi})\\right), \\left(\\mathbb{H}^2,(x, y)\\right)\\right]$$" ], "text/plain": [ "[Chart (H2, (X, Y)),\n", " Chart (U, (r, ph)),\n", " Chart (U, (R, ph)),\n", " Chart (H2, (u, v)),\n", " Chart (U, (th, ph)),\n", " Chart (H2, (x, y))]" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "H2.top_charts()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The expression of the metric tensor in each of these charts is" ] }, { "cell_type": "code", "execution_count": 83, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{Y^{2} + 1}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} X + \\left( -\\frac{X Y}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} Y + \\left( -\\frac{X Y}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} X + \\left( \\frac{X^{2} + 1}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} Y\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{Y^{2} + 1}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} X + \\left( -\\frac{X Y}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} X\\otimes \\mathrm{d} Y + \\left( -\\frac{X Y}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} X + \\left( \\frac{X^{2} + 1}{X^{2} + Y^{2} + 1} \\right) \\mathrm{d} Y\\otimes \\mathrm{d} Y$$" ], "text/plain": [ "g = (Y^2 + 1)/(X^2 + Y^2 + 1) dX⊗dX - X*Y/(X^2 + Y^2 + 1) dX⊗dY - X*Y/(X^2 + Y^2 + 1) dY⊗dX + (X^2 + 1)/(X^2 + Y^2 + 1) dY⊗dY" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{1}{r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( \\frac{1}{r^{2} + 1} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + r^{2} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$$" ], "text/plain": [ "g = 1/(r^2 + 1) dr⊗dr + r^2 dph⊗dph" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{4}{{\\left(R + 1\\right)}^{2} {\\left(R - 1\\right)}^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + \\frac{4 \\, R^{2}}{{\\left(R + 1\\right)}^{2} {\\left(R - 1\\right)}^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{4}{{\\left(R + 1\\right)}^{2} {\\left(R - 1\\right)}^{2}} \\mathrm{d} R\\otimes \\mathrm{d} R + \\frac{4 \\, R^{2}}{{\\left(R + 1\\right)}^{2} {\\left(R - 1\\right)}^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$$" ], "text/plain": [ "g = 4/((R + 1)^2*(R - 1)^2) dR⊗dR + 4*R^2/((R + 1)^2*(R - 1)^2) dph⊗dph" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{4}{{\\left(u^{2} + v^{2} - 1\\right)}^{2}} \\mathrm{d} u\\otimes \\mathrm{d} u + \\frac{4}{{\\left(u^{2} + v^{2} - 1\\right)}^{2}} \\mathrm{d} v\\otimes \\mathrm{d} v\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{4}{{\\left(u^{2} + v^{2} - 1\\right)}^{2}} \\mathrm{d} u\\otimes \\mathrm{d} u + \\frac{4}{{\\left(u^{2} + v^{2} - 1\\right)}^{2}} \\mathrm{d} v\\otimes \\mathrm{d} v$$" ], "text/plain": [ "g = 4/(u^2 + v^2 - 1)^2 du⊗du + 4/(u^2 + v^2 - 1)^2 dv⊗dv" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{1}{\\cos\\left({\\theta}\\right)^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{\\sin\\left({\\theta}\\right)^{2}}{\\cos\\left({\\theta}\\right)^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{1}{\\cos\\left({\\theta}\\right)^{2}} \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\frac{\\sin\\left({\\theta}\\right)^{2}}{\\cos\\left({\\theta}\\right)^{2}} \\mathrm{d} {\\varphi}\\otimes \\mathrm{d} {\\varphi}$$" ], "text/plain": [ "g = cos(th)^(-2) dth⊗dth + sin(th)^2/cos(th)^2 dph⊗dph" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{1}{y^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x + \\frac{1}{y^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\frac{1}{y^{2}} \\mathrm{d} x\\otimes \\mathrm{d} x + \\frac{1}{y^{2}} \\mathrm{d} y\\otimes \\mathrm{d} y$$" ], "text/plain": [ "g = y^(-2) dx⊗dx + y^(-2) dy⊗dy" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for chart in H2.top_charts():\n", " show(g.display(chart.frame(), chart))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For each of these charts, the non-vanishing (and non-redundant w.r.t. the symmetry on the last 2 indices) **Christoffel symbols of $g$** are" ] }, { "cell_type": "code", "execution_count": 84, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(X, Y)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(X, Y)\\right)$$" ], "text/plain": [ "Chart (H2, (X, Y))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, X} \\, X \\, X }^{ \\, X \\phantom{\\, X} \\phantom{\\, X} } & = & -\\frac{X Y^{2} + X}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, X} \\, X \\, Y }^{ \\, X \\phantom{\\, X} \\phantom{\\, Y} } & = & \\frac{X^{2} Y}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, X} \\, Y \\, Y }^{ \\, X \\phantom{\\, Y} \\phantom{\\, Y} } & = & -\\frac{X^{3} + X}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, Y} \\, X \\, X }^{ \\, Y \\phantom{\\, X} \\phantom{\\, X} } & = & -\\frac{Y^{3} + Y}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, Y} \\, X \\, Y }^{ \\, Y \\phantom{\\, X} \\phantom{\\, Y} } & = & \\frac{X Y^{2}}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, Y} \\, Y \\, Y }^{ \\, Y \\phantom{\\, Y} \\phantom{\\, Y} } & = & -\\frac{{\\left(X^{2} + 1\\right)} Y}{X^{2} + Y^{2} + 1} \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, X} \\, X \\, X }^{ \\, X \\phantom{\\, X} \\phantom{\\, X} } & = & -\\frac{X Y^{2} + X}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, X} \\, X \\, Y }^{ \\, X \\phantom{\\, X} \\phantom{\\, Y} } & = & \\frac{X^{2} Y}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, X} \\, Y \\, Y }^{ \\, X \\phantom{\\, Y} \\phantom{\\, Y} } & = & -\\frac{X^{3} + X}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, Y} \\, X \\, X }^{ \\, Y \\phantom{\\, X} \\phantom{\\, X} } & = & -\\frac{Y^{3} + Y}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, Y} \\, X \\, Y }^{ \\, Y \\phantom{\\, X} \\phantom{\\, Y} } & = & \\frac{X Y^{2}}{X^{2} + Y^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, Y} \\, Y \\, Y }^{ \\, Y \\phantom{\\, Y} \\phantom{\\, Y} } & = & -\\frac{{\\left(X^{2} + 1\\right)} Y}{X^{2} + Y^{2} + 1} \\end{array}$$" ], "text/plain": [ "Gam^X_XX = -(X*Y^2 + X)/(X^2 + Y^2 + 1) \n", "Gam^X_XY = X^2*Y/(X^2 + Y^2 + 1) \n", "Gam^X_YY = -(X^3 + X)/(X^2 + Y^2 + 1) \n", "Gam^Y_XX = -(Y^3 + Y)/(X^2 + Y^2 + 1) \n", "Gam^Y_XY = X*Y^2/(X^2 + Y^2 + 1) \n", "Gam^Y_YY = -(X^2 + 1)*Y/(X^2 + Y^2 + 1) " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right)$$" ], "text/plain": [ "Chart (U, (r, ph))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & -\\frac{r}{r^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\varphi} \\, {\\varphi} }^{ \\, r \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & -r^{3} - r \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, r \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, r} \\phantom{\\, {\\varphi}} } & = & \\frac{1}{r} \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, r} \\, r \\, r }^{ \\, r \\phantom{\\, r} \\phantom{\\, r} } & = & -\\frac{r}{r^{2} + 1} \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\varphi} \\, {\\varphi} }^{ \\, r \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & -r^{3} - r \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, r \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, r} \\phantom{\\, {\\varphi}} } & = & \\frac{1}{r} \\end{array}$$" ], "text/plain": [ "Gam^r_r,r = -r/(r^2 + 1) \n", "Gam^r_ph,ph = -r^3 - r \n", "Gam^ph_r,ph = 1/r " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right)$$" ], "text/plain": [ "Chart (U, (R, ph))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, R} \\, R \\, R }^{ \\, R \\phantom{\\, R} \\phantom{\\, R} } & = & -\\frac{2 \\, R}{R^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, R} \\, {\\varphi} \\, {\\varphi} }^{ \\, R \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & \\frac{R^{3} + R}{R^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, R \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, R} \\phantom{\\, {\\varphi}} } & = & -\\frac{R^{2} + 1}{R^{3} - R} \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, R} \\, R \\, R }^{ \\, R \\phantom{\\, R} \\phantom{\\, R} } & = & -\\frac{2 \\, R}{R^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, R} \\, {\\varphi} \\, {\\varphi} }^{ \\, R \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & \\frac{R^{3} + R}{R^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, R \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, R} \\phantom{\\, {\\varphi}} } & = & -\\frac{R^{2} + 1}{R^{3} - R} \\end{array}$$" ], "text/plain": [ "Gam^R_R,R = -2*R/(R^2 - 1) \n", "Gam^R_ph,ph = (R^3 + R)/(R^2 - 1) \n", "Gam^ph_R,ph = -(R^2 + 1)/(R^3 - R) " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(u, v)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(u, v)\\right)$$" ], "text/plain": [ "Chart (H2, (u, v))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, u} \\, u \\, u }^{ \\, u \\phantom{\\, u} \\phantom{\\, u} } & = & -\\frac{2 \\, u}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, u} \\, u \\, v }^{ \\, u \\phantom{\\, u} \\phantom{\\, v} } & = & -\\frac{2 \\, v}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, u} \\, v \\, v }^{ \\, u \\phantom{\\, v} \\phantom{\\, v} } & = & \\frac{2 \\, u}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, v} \\, u \\, u }^{ \\, v \\phantom{\\, u} \\phantom{\\, u} } & = & \\frac{2 \\, v}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, v} \\, u \\, v }^{ \\, v \\phantom{\\, u} \\phantom{\\, v} } & = & -\\frac{2 \\, u}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, v} \\, v \\, v }^{ \\, v \\phantom{\\, v} \\phantom{\\, v} } & = & -\\frac{2 \\, v}{u^{2} + v^{2} - 1} \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, u} \\, u \\, u }^{ \\, u \\phantom{\\, u} \\phantom{\\, u} } & = & -\\frac{2 \\, u}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, u} \\, u \\, v }^{ \\, u \\phantom{\\, u} \\phantom{\\, v} } & = & -\\frac{2 \\, v}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, u} \\, v \\, v }^{ \\, u \\phantom{\\, v} \\phantom{\\, v} } & = & \\frac{2 \\, u}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, v} \\, u \\, u }^{ \\, v \\phantom{\\, u} \\phantom{\\, u} } & = & \\frac{2 \\, v}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, v} \\, u \\, v }^{ \\, v \\phantom{\\, u} \\phantom{\\, v} } & = & -\\frac{2 \\, u}{u^{2} + v^{2} - 1} \\\\ \\Gamma_{ \\phantom{\\, v} \\, v \\, v }^{ \\, v \\phantom{\\, v} \\phantom{\\, v} } & = & -\\frac{2 \\, v}{u^{2} + v^{2} - 1} \\end{array}$$" ], "text/plain": [ "Gam^u_uu = -2*u/(u^2 + v^2 - 1) \n", "Gam^u_uv = -2*v/(u^2 + v^2 - 1) \n", "Gam^u_vv = 2*u/(u^2 + v^2 - 1) \n", "Gam^v_uu = 2*v/(u^2 + v^2 - 1) \n", "Gam^v_uv = -2*u/(u^2 + v^2 - 1) \n", "Gam^v_vv = -2*v/(u^2 + v^2 - 1) " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,({\\theta}, {\\varphi})\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,({\\theta}, {\\varphi})\\right)$$" ], "text/plain": [ "Chart (U, (th, ph))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & \\frac{\\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\varphi} \\, {\\varphi} }^{ \\, {\\theta} \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & -\\frac{\\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, {\\theta} \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\varphi}} } & = & \\frac{1}{\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)} \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & \\frac{\\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\varphi} \\, {\\varphi} }^{ \\, {\\theta} \\phantom{\\, {\\varphi}} \\phantom{\\, {\\varphi}} } & = & -\\frac{\\sin\\left({\\theta}\\right)}{\\cos\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\varphi}} \\, {\\theta} \\, {\\varphi} }^{ \\, {\\varphi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\varphi}} } & = & \\frac{1}{\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right)} \\end{array}$$" ], "text/plain": [ "Gam^th_th,th = sin(th)/cos(th) \n", "Gam^th_ph,ph = -sin(th)/cos(th) \n", "Gam^ph_th,ph = 1/(cos(th)*sin(th)) " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(x, y)\\right)\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(x, y)\\right)$$" ], "text/plain": [ "Chart (H2, (x, y))" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "\\[\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, x} \\, x \\, y }^{ \\, x \\phantom{\\, x} \\phantom{\\, y} } & = & -\\frac{1}{y} \\\\ \\Gamma_{ \\phantom{\\, y} \\, x \\, x }^{ \\, y \\phantom{\\, x} \\phantom{\\, x} } & = & \\frac{1}{y} \\\\ \\Gamma_{ \\phantom{\\, y} \\, y \\, y }^{ \\, y \\phantom{\\, y} \\phantom{\\, y} } & = & -\\frac{1}{y} \\end{array}\\]" ], "text/latex": [ "$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{lcl} \\Gamma_{ \\phantom{\\, x} \\, x \\, y }^{ \\, x \\phantom{\\, x} \\phantom{\\, y} } & = & -\\frac{1}{y} \\\\ \\Gamma_{ \\phantom{\\, y} \\, x \\, x }^{ \\, y \\phantom{\\, x} \\phantom{\\, x} } & = & \\frac{1}{y} \\\\ \\Gamma_{ \\phantom{\\, y} \\, y \\, y }^{ \\, y \\phantom{\\, y} \\phantom{\\, y} } & = & -\\frac{1}{y} \\end{array}$$" ], "text/plain": [ "Gam^x_xy = -1/y \n", "Gam^y_xx = 1/y \n", "Gam^y_yy = -1/y " ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "for chart in H2.top_charts():\n", " show(chart)\n", " show(g.christoffel_symbols_display(chart=chart))" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.4.rc2", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 1 }