{ "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 7.5 is required to run this notebook:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'SageMath version 9.2, Release Date: 2020-10-24'" ] }, "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|real|\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathbb{H}^2\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{R}^3,(X, Y, Z)\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(X, Y)\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}r :\\ \\left( 0 , +\\infty \\right) ;\\quad {\\varphi} :\\ \\left( 0 , 2 \\, \\pi \\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right) \\rightarrow \\left(U,(X, Y)\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} X & = & r \\cos\\left({\\varphi}\\right) \\\\ Y & = & r \\sin\\left({\\varphi}\\right) \\end{array}\\right.\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} r & = & \\sqrt{X^{2} + Y^{2}} \\\\ {\\varphi} & = & \\arctan\\left(Y, X\\right) \\end{array}\\right.\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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\\}\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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\n", "\\end{math}" ], "text/plain": [ "Riem(g) = -r^2 d/dr*dph*dr*dph + r^2 d/dr*dph*dph*dr + 1/(r^2 + 1) d/dph*dr*dr*dph - 1/(r^2 + 1) d/dph*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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\mathrm{True}\n", "\\end{math}" ], "text/plain": [ "True" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Ric == (Rscal/2)*g" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## PoincarĂ© disk model\n", "\n", "The PoincarĂ© 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 PoincarĂ© disk chart on $\\mathbb{H}^2$ by" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}R :\\ \\left( 0 , 1 \\right) ;\\quad {\\varphi} :\\ \\left( 0 , 2 \\, \\pi \\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right) \\rightarrow \\left(U,(R, {\\varphi})\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} R & = & \\frac{r}{\\sqrt{r^{2} + 1} + 1} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} r & = & -\\frac{2 \\, R}{R^{2} - 1} \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "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 PoincarĂ© 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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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 PoincarĂ© disk\n", "\n", "Let us introduce Cartesian coordinates $(u,v)$ on the PoincarĂ© disk; since the latter has a unit radius, this amounts to define the following chart on $\\mathbb{H}^2$:" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(u, v)\\right)\n", "\\end{math}" ], "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)')\n", "X_Pdisk_cart.add_restrictions(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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right) \\rightarrow \\left(U,(u, v)\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} u & = & R \\cos\\left({\\varphi}\\right) \\\\ v & = & R \\sin\\left({\\varphi}\\right) \\end{array}\\right.\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} R & = & \\sqrt{u^{2} + v^{2}} \\\\ {\\varphi} & = & \\arctan\\left(v, u\\right) \\end{array}\\right.\n", "\\end{math}" ], "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 PoincarĂ© disk model is naturally defined as" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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 PoincarĂ© 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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(r, {\\varphi})\\right) \\rightarrow \\left(U,(u, v)\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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.\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(X, Y)\\right) \\rightarrow \\left(U,(u, v)\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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.\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{H}^2,(X, Y)\\right) \\rightarrow \\left(\\mathbb{H}^2,(u, v)\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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.\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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.\n", "\\end{math}" ], "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 PoincarĂ© disk coordinates $(u,v)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From now on, we are using the PoincarĂ© 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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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\n", "\\end{math}" ], "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 PoincarĂ© 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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,({\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "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", "\$$\n", "\\sin\\theta = \\frac{2R}{1+R^2}\n", "\$$\n", "Hence the transition map:" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(U,(R, {\\varphi})\\right) \\rightarrow \\left(U,({\\theta}, {\\varphi})\\right)\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left\\{\\begin{array}{lcl} {\\theta} & = & \\arcsin\\left(\\frac{2 \\, R}{R^{2} + 1}\\right) \\\\ {\\varphi} & = & {\\varphi} \\end{array}\\right.\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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.\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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": [ "" ], "text/latex": [ "\\begin{math}\n", "\\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}\n", "\\end{math}" ], "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", "