Hyperbolic plane $\mathbb{H}^2$

This Jupyter notebook illustrates some differential geometry capabilities of SageMath on the example of the hyperbolic plane. The corresponding tools have been developed within the SageManifolds project.

A version of SageMath at least equal to 7.5 is required to run this notebook:

In [1]:
version()
Out[1]:
'SageMath version 9.2, Release Date: 2020-10-24'

First we set up the notebook to display mathematical objects using LaTeX formatting:

In [2]:
%display latex

We also tell Maxima, which is used by SageMath for simplifications of symbolic expressions, that all computations involve real variables:

In [3]:
maxima_calculus.eval("domain: real;")
Out[3]:

We declare $\mathbb{H}^2$ as a 2-dimensional differentiable manifold:

In [4]:
H2 = Manifold(2, 'H2', latex_name=r'\mathbb{H}^2', start_index=1)
print(H2)
H2
2-dimensional differentiable manifold H2
Out[4]:

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 $(X,Y,Z)$:

In [5]:
R3 = Manifold(3, 'R3', latex_name=r'\mathbb{R}^3', start_index=1)
X3.<X,Y,Z> = R3.chart()
X3
Out[5]:

Hyperboloid model

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 $X^2 + Y^2 - Z^2 = -1$:

In [6]:
X_hyp.<X,Y> = H2.chart()
X_hyp
Out[6]:

The corresponding embedding of $\mathbb{H}^2$ in $\mathbb{R}^3$ is

In [7]:
Phi1 = H2.diff_map(R3, [X, Y, sqrt(1+X^2+Y^2)], name='Phi_1', latex_name=r'\Phi_1')
Phi1.display()
Out[7]:

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 $\Phi_1(\mathbb{H}^2)$:

In [8]:
show(X_hyp.plot(X3, mapping=Phi1, number_values=15, color='blue'), 
     aspect_ratio=1, figsize=7)

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 $\{Y=0, x\geq 0\}$:

In [9]:
U = H2.open_subset('U', coord_def={X_hyp: (Y!=0, X<0)})
print(U)
Open subset U of the 2-dimensional differentiable manifold H2

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.

In [10]:
X_pol.<r,ph> = U.chart(r'r:(0,+oo) ph:(0,2*pi):\varphi')
X_pol
Out[10]:
In [11]:
X_pol.coord_range()
Out[11]:

We specify the transition map between the charts $\left(U,(r,\varphi)\right)$ and $\left(\mathbb{H}^2,(X,Y)\right)$ as $X=r\cos\varphi$, $Y=r\sin\varphi$:

In [12]:
pol_to_hyp = X_pol.transition_map(X_hyp, [r*cos(ph), r*sin(ph)])
pol_to_hyp
Out[12]:
In [13]:
pol_to_hyp.display()
Out[13]:
In [14]:
pol_to_hyp.set_inverse(sqrt(X^2+Y^2), atan2(Y, X)) 
Check of the inverse coordinate transformation:
  r == r  *passed*
  ph == arctan2(r*sin(ph), r*cos(ph))  **failed**
  X == X  *passed*
  Y == Y  *passed*
NB: a failed report can reflect a mere lack of simplification.
In [15]:
pol_to_hyp.inverse().display()
Out[15]:

The restriction of the embedding $\Phi_1$ to $U$ has then two coordinate expressions:

In [16]:
Phi1.restrict(U).display()
Out[16]:
In [17]:
graph_hyp = X_pol.plot(X3, mapping=Phi1.restrict(U), number_values=15, ranges={r: (0,3)}, 
                       color='blue')
show(graph_hyp, aspect_ratio=1, figsize=7)
In [18]:
Phi1._coord_expression
Out[18]:

Metric and curvature

The metric on $\mathbb{H}^2$ is that induced by the Minkowksy metric on $\mathbb{R}^3$: $$ \eta = \mathrm{d}X\otimes\mathrm{d}X + \mathrm{d}Y\otimes\mathrm{d}Y - \mathrm{d}Z\otimes\mathrm{d}Z $$

In [19]:
eta = R3.lorentzian_metric('eta', latex_name=r'\eta')
eta[1,1] = 1 ; eta[2,2] = 1 ; eta[3,3] = -1
eta.display()
Out[19]:
In [20]:
g = H2.metric('g')
g.set( Phi1.pullback(eta) )
g.display() 
Out[20]:

The expression of the metric tensor in terms of the polar coordinates is

In [21]:
g.display(X_pol.frame(), X_pol)
Out[21]:

The Riemann curvature tensor associated with $g$ is

In [22]:
Riem = g.riemann()
print(Riem)
Tensor field Riem(g) of type (1,3) on the 2-dimensional differentiable manifold H2
In [23]:
Riem.display(X_pol.frame(), X_pol)
Out[23]:

The Ricci tensor and the Ricci scalar:

In [24]:
Ric = g.ricci()
print(Ric)
Field of symmetric bilinear forms Ric(g) on the 2-dimensional differentiable manifold H2
In [25]:
Ric.display(X_pol.frame(), X_pol)
Out[25]:
In [26]:
Rscal = g.ricci_scalar()
print(Rscal)
Scalar field r(g) on the 2-dimensional differentiable manifold H2
In [27]:
Rscal.display()
Out[27]:

Hence we recover the fact that $(\mathbb{H}^2,g)$ is a space of constant negative curvature.

In dimension 2, the Riemann curvature tensor is entirely determined by the Ricci scalar $R$ according to

$$R^i_{\ \, jlk} = \frac{R}{2} \left( \delta^i_{\ \, k} g_{jl} - \delta^i_{\ \, l} g_{jk} \right)$$

Let us check this formula here, under the form $R^i_{\ \, jlk} = -R g_{j[k} \delta^i_{\ \, l]}$:

In [28]:
delta = H2.tangent_identity_field()
Riem == - Rscal*(g*delta).antisymmetrize(2,3)  # 2,3 = last positions of the type-(1,3) tensor g*delta 
Out[28]:

Similarly the relation $\mathrm{Ric} = (R/2)\; g$ must hold:

In [29]:
Ric == (Rscal/2)*g
Out[29]:

Poincaré disk model

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 $$ R = \frac{r}{1+\sqrt{1+r^2}}.$$ Hence we define the Poincaré disk chart on $\mathbb{H}^2$ by

In [30]:
X_Pdisk.<R,ph> = U.chart(r'R:(0,1) ph:(0,2*pi):\varphi')
X_Pdisk
Out[30]:
In [31]:
X_Pdisk.coord_range()
Out[31]:

and relate it to the hyperboloid polar chart by

In [32]:
pol_to_Pdisk = X_pol.transition_map(X_Pdisk, [r/(1+sqrt(1+r^2)), ph])
pol_to_Pdisk
Out[32]:
In [33]:
pol_to_Pdisk.display()
Out[33]:
In [34]:
pol_to_Pdisk.set_inverse(2*R/(1-R^2), ph)
pol_to_Pdisk.inverse().display()
Out[34]:

A view of the Poincaré disk chart via the embedding $\Phi_1$:

In [35]:
show(X_Pdisk.plot(X3, mapping=Phi1.restrict(U), ranges={R: (0,0.9)}, color='blue',
                  number_values=15), 
     aspect_ratio=1, figsize=7)

The expression of the metric tensor in terms of coordinates $(R,\varphi)$:

In [36]:
g.display(X_Pdisk.frame(), X_Pdisk)
Out[36]:

We may factorize each metric component:

In [37]:
for i in [1,2]:
    g[X_Pdisk.frame(), i, i, X_Pdisk].factor()
g.display(X_Pdisk.frame(), X_Pdisk)
Out[37]:

Cartesian coordinates on the Poincaré disk

Let us introduce Cartesian coordinates $(u,v)$ on the Poincaré disk; since the latter has a unit radius, this amounts to define the following chart on $\mathbb{H}^2$:

In [38]:
X_Pdisk_cart.<u,v> = H2.chart('u:(-1,1) v:(-1,1)')
X_Pdisk_cart.add_restrictions(u^2+v^2 < 1)
X_Pdisk_cart
Out[38]:

On $U$, the Cartesian coordinates $(u,v)$ are related to the polar coordinates $(R,\varphi)$ by the standard formulas:

In [39]:
Pdisk_to_Pdisk_cart = X_Pdisk.transition_map(X_Pdisk_cart, [R*cos(ph), R*sin(ph)])
Pdisk_to_Pdisk_cart
Out[39]:
In [40]:
Pdisk_to_Pdisk_cart.display()
Out[40]:
In [41]:
Pdisk_to_Pdisk_cart.set_inverse(sqrt(u^2+v^2), atan2(v, u)) 
Pdisk_to_Pdisk_cart.inverse().display()
Check of the inverse coordinate transformation:
  R == R  *passed*
  ph == arctan2(R*sin(ph), R*cos(ph))  **failed**
  u == u  *passed*
  v == v  *passed*
NB: a failed report can reflect a mere lack of simplification.
Out[41]:

The embedding of $\mathbb{H}^2$ in $\mathbb{R}^3$ associated with the Poincaré disk model is naturally defined as

In [42]:
Phi2 = H2.diff_map(R3, {(X_Pdisk_cart, X3): [u, v, 0]},
                   name='Phi_2', latex_name=r'\Phi_2')
Phi2.display()
Out[42]:

Let us use it to draw the Poincaré disk in $\mathbb{R}^3$:

In [43]:
graph_disk_uv = X_Pdisk_cart.plot(X3, mapping=Phi2, number_values=15)
show(graph_disk_uv, figsize=7)

On $U$, the change of coordinates $(r,\varphi) \rightarrow (u,v)$ is obtained by combining the changes $(r,\varphi) \rightarrow (R,\varphi)$ and $(R,\varphi) \rightarrow (u,v)$:

In [44]:
pol_to_Pdisk_cart = Pdisk_to_Pdisk_cart * pol_to_Pdisk
pol_to_Pdisk_cart
Out[44]:
In [45]:
pol_to_Pdisk_cart.display()
Out[45]:

Still on $U$, the change of coordinates $(X,Y) \rightarrow (u,v)$ is obtained by combining the changes $(X,Y) \rightarrow (r,\varphi)$ with $(r,\varphi) \rightarrow (u,v)$:

In [46]:
hyp_to_Pdisk_cart_U = pol_to_Pdisk_cart * pol_to_hyp.inverse()
hyp_to_Pdisk_cart_U
Out[46]:
In [47]:
hyp_to_Pdisk_cart_U.display()
Out[47]:

We use the above expression to extend the change of coordinates $(X,Y) \rightarrow (u,v)$ from $U$ to the whole manifold $\mathbb{H}^2$:

In [48]:
hyp_to_Pdisk_cart = X_hyp.transition_map(X_Pdisk_cart, hyp_to_Pdisk_cart_U(X,Y))
hyp_to_Pdisk_cart
Out[48]:
In [49]:
hyp_to_Pdisk_cart.display()
Out[49]:
In [50]:
hyp_to_Pdisk_cart.set_inverse(2*u/(1-u^2-v^2), 2*v/(1-u^2-v^2))
hyp_to_Pdisk_cart.inverse().display()
Check of the inverse coordinate transformation:
  X == X  *passed*
  Y == Y  *passed*
  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**
  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**
NB: a failed report can reflect a mere lack of simplification.
Out[50]:
In [51]:
graph_Pdisk = X_pol.plot(X3, mapping=Phi2.restrict(U), ranges={r: (0, 20)}, number_values=15, 
                         label_axes=False)
show(graph_hyp + graph_Pdisk, aspect_ratio=1, figsize=7)
In [52]:
X_pol.plot(X_Pdisk_cart, ranges={r: (0, 20)}, number_values=15)
Out[52]:

Metric tensor in Poincaré disk coordinates $(u,v)$

From now on, we are using the Poincaré disk chart $(\mathbb{H}^2,(u,v))$ as the default one on $\mathbb{H}^2$:

In [53]:
H2.set_default_chart(X_Pdisk_cart)
H2.set_default_frame(X_Pdisk_cart.frame())
In [54]:
g.display(X_hyp.frame())
Out[54]:
In [55]:
g.display()
Out[55]:
In [56]:
g[1,1].factor() ; g[2,2].factor()
g.display()
Out[56]:

Hemispherical model

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 $U$:

In [57]:
X_spher.<th,ph> = U.chart(r'th:(0,pi/2):\theta ph:(0,2*pi):\varphi')
X_spher
Out[57]:

From the stereographic projection from $S$, we obtain that \begin{equation} \sin\theta = \frac{2R}{1+R^2} \end{equation} Hence the transition map:

In [58]:
Pdisk_to_spher = X_Pdisk.transition_map(X_spher, [arcsin(2*R/(1+R^2)), ph])
Pdisk_to_spher
Out[58]:
In [59]:
Pdisk_to_spher.display()
Out[59]:
In [60]:
Pdisk_to_spher.set_inverse(sin(th)/(1+cos(th)), ph)
Pdisk_to_spher.inverse().display()
Out[60]:

In the spherical coordinates $(\theta,\varphi)$, the metric takes the following form:

In [61]:
g.display(X_spher.frame(), X_spher)
Out[61]:

The embedding of $\mathbb{H}^2$ in $\mathbb{R}^3$ associated with the hemispherical model is naturally:

In [62]:
Phi3 = H2.diff_map(R3, {(X_spher, X3): [sin(th)*cos(ph), sin(th)*sin(ph), cos(th)]},
                   name='Phi_3', latex_name=r'\Phi_3')
Phi3.display()
Out[62]:
In [63]:
graph_spher = X_pol.plot(X3, mapping=Phi3, ranges={r: (0, 20)}, number_values=15, 
                         color='orange', label_axes=False)
show(graph_hyp + graph_Pdisk + graph_spher, aspect_ratio=1,  
     figsize=7)