This worksheet demonstrates a few capabilities of SageManifolds (version 0.8) in computations regarding the real projective plane.
It is released under the GNU General Public License version 3.
(c) Eric Gourgoulhon, Michal Bejger (2015)
The corresponding worksheet file can be downloaded from here
We start by declaring the real projective plane as a 2-dimensional differentiable manifold:
This worksheet demonstrates a few capabilities of SageManifolds (version 0.8) in computations regarding the real projective plane.
\nIt is released under the GNU General Public License version 3.
\n(c) Eric Gourgoulhon, Michal Bejger (2015)
\nThe corresponding worksheet file can be downloaded from here
\nWe start by declaring the real projective plane as a 2-dimensional differentiable manifold:
\nThen we provide $\mathbb{RP}^2$ with some atlas. A minimal atlas on $\mathbb{RP}^2$ must have at least three charts. Such an atlas is easy to infer from the common interpretation of $\mathbb{RP}^2$ as the set of lines of $\mathbb{R}^3$ passing through the origin $(x,y,z)=(0,0,0)$. Let $U_1$ be the subset of lines that are not contained in the plane $z=0$; this is an open set of $\mathbb{RP}^2$, so that we declare it as:
︡c6bfef80-1dfe-49c7-bc2b-0abe0b2f187f︡︡{"done":true,"html":"Then we provide $\\mathbb{RP}^2$ with some atlas. A minimal atlas on $\\mathbb{RP}^2$ must have at least three charts. Such an atlas is easy to infer from the common interpretation of $\\mathbb{RP}^2$ as the set of lines of $\\mathbb{R}^3$ passing through the origin $(x,y,z)=(0,0,0)$. Let $U_1$ be the subset of lines that are not contained in the plane $z=0$; this is an open set of $\\mathbb{RP}^2$, so that we declare it as:
"} ︠1c7e5243-684e-489e-aba0-24ca6e1475dds︠ U1 = RP2.open_subset('U_1') ; U1 ︡64bfa425-d6da-43f6-a32a-c4fb83ee150b︡︡{"html":"Any line in $U_1$ is uniquely determined by its intersection with the plane $z=1$. The Cartesian coordinates $(x,y,1)$ of the intersection point lead to an obvious coordinate system $(x_1,y_1)$ on $U_1$ by setting $(x_1,y_1)=(x,y)$:
︡86f85d73-88a9-40d2-bd28-ab0d35259c2e︡︡{"done":true,"html":"Any line in $U_1$ is uniquely determined by its intersection with the plane $z=1$. The Cartesian coordinates $(x,y,1)$ of the intersection point lead to an obvious coordinate system $(x_1,y_1)$ on $U_1$ by setting $(x_1,y_1)=(x,y)$:
"} ︠8303cd5e-ae86-4a6e-a3c4-9f11c8a02c55s︠ X1.Note that since we have not specified any coordinate range in the arguments of chart(), the range of $(x_1,y_1)$ is $\mathbb{R}^2$.
Similarly, let $U_2$ be the set of lines through the origin of $\mathbb{R}^3$ that are not contained in the plane $x=0$. Any line in $U_2$ is uniquely determined by its intersection $(1,y,z)$ with the plane $x=1$, leading to coordinates $(x_2,y_2)=(y,z)$ on $U_2$:
︡8f832c0a-c8e3-44a0-a10c-dadd8e0e713e︡︡{"done":true,"html":"Note that since we have not specified any coordinate range in the arguments of chart(), the range of $(x_1,y_1)$ is $\\mathbb{R}^2$.
\nSimilarly, let $U_2$ be the set of lines through the origin of $\\mathbb{R}^3$ that are not contained in the plane $x=0$. Any line in $U_2$ is uniquely determined by its intersection $(1,y,z)$ with the plane $x=1$, leading to coordinates $(x_2,y_2)=(y,z)$ on $U_2$:
"} ︠81178c7d-5e8d-475b-b65a-b617aea6550bs︠ U2 = RP2.open_subset('U_2') X2.Finally, let $U_3$ be the set of lines through the origin of $\mathbb{R}^3$ that are not contained in the plane $y=0$. Any line in $U_3$ is uniquely determined by its intersection $(x,1,z)$ with the plane $y=1$, leading to coordinates $(x_3,y_3)=(z,x)$ on $U_3$:
︡881b223f-4e2f-4e91-aeea-ad19d3c76454︡︡{"done":true,"html":"Finally, let $U_3$ be the set of lines through the origin of $\\mathbb{R}^3$ that are not contained in the plane $y=0$. Any line in $U_3$ is uniquely determined by its intersection $(x,1,z)$ with the plane $y=1$, leading to coordinates $(x_3,y_3)=(z,x)$ on $U_3$:
"} ︠81c5159f-e1c8-4e88-9194-ff3895a72dd1s︠ U3 = RP2.open_subset('U_3') X3.We declare that the union of the three (overlapping) open domains $U_1$, $U_2$ and $U_3$ is $\mathbb{RP}^2$:
︡13ce3372-2ccd-42a9-8ff5-560e77e94517︡︡{"done":true,"html":"We declare that the union of the three (overlapping) open domains $U_1$, $U_2$ and $U_3$ is $\\mathbb{RP}^2$:
"} ︠7a3330fc-10ea-4282-9905-ba144adc96f0s︠ RP2.declare_union(U1.union(U2), U3) U1.union(U2).union(U3) ︡fccd0eb3-8972-4550-843b-67af03cbeb33︡︡{"html":"Finally, to fully specify the manifold $\mathbb{RP}^2$, we give the transition maps between the various charts; the transition map between the charts X1=$(U_1,(x_1,y_1))$ and X2=$(U_2,(x_2,y_2))$ is defined on the set $U_{12} := U_1 \cap U_2$ of lines through the origin of $\mathbb{R}^3$ that are neither contained in the plane $x=0$ ($x_1=0$ in $U_1$) nor contained in the plane $z=0$ ($y_2=0$ in $U_2$):
︡c8c6dd47-3b4f-4ac1-9efa-c1de11e0c616︡︡{"done":true,"html":"Finally, to fully specify the manifold $\\mathbb{RP}^2$, we give the transition maps between the various charts; the transition map between the charts X1=$(U_1,(x_1,y_1))$ and X2=$(U_2,(x_2,y_2))$ is defined on the set $U_{12} := U_1 \\cap U_2$ of lines through the origin of $\\mathbb{R}^3$ that are neither contained in the plane $x=0$ ($x_1=0$ in $U_1$) nor contained in the plane $z=0$ ($y_2=0$ in $U_2$):
"} ︠87510d88-1ba7-4581-8e03-70bebf1956cds︠ X1_to_X2 = X1.transition_map(X2, (y1/x1, 1/x1), intersection_name='U_{12}', \ restrictions1= x1!=0, restrictions2= y2!=0) X1_to_X2.display() ︡3cf6df85-708e-4f8b-a05f-59b9d03a9bf7︡︡{"html":"The inverse of this transition map is easily computed by Sage:
︡231d9f9d-0f3f-4f16-a827-2fca938ed7c6︡︡{"done":true,"html":"The inverse of this transition map is easily computed by Sage:
"} ︠ebdbebb4-202b-4c55-b836-adb313ab85bcs︠ X2_to_X1 = X1_to_X2.inverse() X2_to_X1.display() ︡60386d5c-9931-461a-8e1e-624bb1677782︡︡{"html":"The transition map between the charts X1=$(U_1,(x_1,y_1))$ and X3=$(U_3,(x_3,y_3))$ is defined on the set $U_{13} := U_1 \cap U_3$ of lines through the origin of $\mathbb{R}^3$ that are neither contained in the plane $y=0$ ($y_1=0$ in $U_1$) nor contained in the plane $z=0$ ($x_3=0$ in $U_3$):
︡629c8645-6487-40bb-9702-6fc7022e60cf︡︡{"done":true,"html":"The transition map between the charts X1=$(U_1,(x_1,y_1))$ and X3=$(U_3,(x_3,y_3))$ is defined on the set $U_{13} := U_1 \\cap U_3$ of lines through the origin of $\\mathbb{R}^3$ that are neither contained in the plane $y=0$ ($y_1=0$ in $U_1$) nor contained in the plane $z=0$ ($x_3=0$ in $U_3$):
"} ︠6d9035b6-ea98-4789-b13c-3a4b5cfa908as︠ X1_to_X3 = X1.transition_map(X3, (1/y1, x1/y1), intersection_name='U_{13}', \ restrictions1= y1!=0, restrictions2= x3!=0) X1_to_X3.display() ︡5c0fb34d-a196-4620-bd3e-c0a4fcc8526a︡︡{"html":"Finally, the transition map between the charts X2=$(U_2,(x_2,y_2))$ and X3=$(U_3,(x_3,y_3))$ is defined on the set $U_{23} := U_2 \cap U_3$ of lines through the origin of $\mathbb{R}^3$ that are neither contained in the plane $y=0$ ($x_2=0$ in $U_2$) nor contained in the plane $x=0$ ($y_3=0$ in $U_3$):
︡50954b81-2bdc-4717-97e6-f5081968dba7︡︡{"done":true,"html":"Finally, the transition map between the charts X2=$(U_2,(x_2,y_2))$ and X3=$(U_3,(x_3,y_3))$ is defined on the set $U_{23} := U_2 \\cap U_3$ of lines through the origin of $\\mathbb{R}^3$ that are neither contained in the plane $y=0$ ($x_2=0$ in $U_2$) nor contained in the plane $x=0$ ($y_3=0$ in $U_3$):
"} ︠f1a3a15e-3693-4210-b470-98b269355facs︠ X2_to_X3 = X2.transition_map(X3, (y2/x2, 1/x2), intersection_name='U_{23}', \ restrictions1= x2!=0, restrictions2= y3!=0) X2_to_X3.display() ︡8810dbb7-4e4e-46b2-a233-2ca9b279f14e︡︡{"html":"At this stage, the manifold $\mathbb{RP}^2$ is fully constructed. It has been provided with the following atlas:
︡7431a6b5-4b96-4a36-88e4-f5268caaf689︡︡{"done":true,"html":"At this stage, the manifold $\\mathbb{RP}^2$ is fully constructed. It has been provided with the following atlas:
"} ︠aada34be-692f-4636-bc2d-0aa798e8be27s︠ RP2.atlas() ︡e726c702-6804-4dca-b129-9d9736cfba32︡︡{"html":"Note that, in addition to the three chart we have defined, the atlas comprises subcharts on the intersection domains $U_{12}$, $U_{13}$ and $U_{23}$. These charts can be obtained by the method restrict():
︡f7ff38ba-164d-4072-bd55-266eec0b92a0︡︡{"done":true,"html":"Note that, in addition to the three chart we have defined, the atlas comprises subcharts on the intersection domains $U_{12}$, $U_{13}$ and $U_{23}$. These charts can be obtained by the method restrict():
"} ︠d25af259-801c-4079-8f4e-20e8722e6efes︠ U12 = U1.intersection(U2) U13 = U1.intersection(U3) U23 = U2.intersection(U3) X1.restrict(U12) ︡0e571a19-d208-4e32-8736-a18b4050aff3︡︡{"html":"It is well known that $\mathbb{RP}^2$ is not an orientable manifold. To illustrate this, let us make an attempt to construct a global non-vanishing 2-form $\epsilon$ on $\mathbb{RP}^2$. If we succeed, this would provide a volume form and $\mathbb{RP}^2$ would be orientable. We start by declaring $\epsilon$ as a 2-form on $\mathbb{RP}^2$:
︡d8204b19-73fd-4474-a101-c2b6cb114618︡︡{"done":true,"html":"It is well known that $\\mathbb{RP}^2$ is not an orientable manifold. To illustrate this, let us make an attempt to construct a global non-vanishing 2-form $\\epsilon$ on $\\mathbb{RP}^2$. If we succeed, this would provide a volume form and $\\mathbb{RP}^2$ would be orientable. We start by declaring $\\epsilon$ as a 2-form on $\\mathbb{RP}^2$:
"} ︠87ae86cc-7257-48db-9ef2-895529f0b4dds︠ eps = RP2.diff_form(2, name='eps', latex_name=r'\epsilon') print eps ︡10dca47b-16a6-4897-bedd-c086d19a1cf1︡︡{"stdout":"2-form 'eps' on the 2-dimensional manifold 'RP^2'\n","done":false}︡{"done":true} ︠fa056f28-90e6-4b6d-b1a6-f75146a603a1i︠ %htmlWe set the value of $\epsilon$ on domain $U_1$ to be $\mathrm{d}x_1 \wedge \mathrm{d}y_1$ by demanding that the component $\epsilon_{01}$ of $\epsilon$ with respect to coordinates $(x_1,y_1)$ is one, as follows:
︡86733421-0b94-4e9a-a107-b41d18bfaeac︡︡{"done":true,"html":"We set the value of $\\epsilon$ on domain $U_1$ to be $\\mathrm{d}x_1 \\wedge \\mathrm{d}y_1$ by demanding that the component $\\epsilon_{01}$ of $\\epsilon$ with respect to coordinates $(x_1,y_1)$ is one, as follows:
"} ︠a9c5de49-483b-4cdd-aa21-f3f76e0262a3s︠ e1 = X1.frame() ; e1 ︡e66f270c-9d7e-480a-ba92-05ed7295bdef︡︡{"html":"If we ask for the expression of $\epsilon$ in terms of the coframe $(\mathrm{d}x_2, \mathrm{d}y_2)$ associated with the chart X2 on $U_{12} = U_1\cap U_2$, we get
︡dcbcb9d0-97fb-4051-a060-e206e03258c4︡︡{"done":true,"html":"If we ask for the expression of $\\epsilon$ in terms of the coframe $(\\mathrm{d}x_2, \\mathrm{d}y_2)$ associated with the chart X2 on $U_{12} = U_1\\cap U_2$, we get
"} ︠1d8580f5-5b99-42fb-8c76-c70c6dd29f65s︠ eps.display(X2.frame().restrict(U12), chart=X2.restrict(U12)) ︡167a1ee2-11a8-43b8-82a1-fdeedf45f294︡︡{"html":"Now, the complement of $U_{12}$ in $U_2$ is defined by $y_2=0$. The above expression shows that it is not possible to extend smoothly $\epsilon$ to the whole domain $U_2$. We conclude that starting from $\mathrm{d}x_1\wedge\mathrm{d}y_1$ on $U_1$, it is not possible to get a regular non-vanishing 2-form on $\mathbb{RP}^2$. This of course follows from the fact that $\mathbb{RP}^2$ is not orientable.
Let us first define $\mathbb{R}^3$ as a 3-dimensional manifold, with a single-chart atlas (Cartesian coordinates Y):
︡7094becd-0d78-4d55-850f-70c41b9ebe8b︡︡{"done":true,"html":"Now, the complement of $U_{12}$ in $U_2$ is defined by $y_2=0$. The above expression shows that it is not possible to extend smoothly $\\epsilon$ to the whole domain $U_2$. We conclude that starting from $\\mathrm{d}x_1\\wedge\\mathrm{d}y_1$ on $U_1$, it is not possible to get a regular non-vanishing 2-form on $\\mathbb{RP}^2$. This of course follows from the fact that $\\mathbb{RP}^2$ is not orientable.
\n\nLet us first define $\\mathbb{R}^3$ as a 3-dimensional manifold, with a single-chart atlas (Cartesian coordinates Y):
"} ︠5b82f936-2e3f-4937-989d-51a52ed29754s︠ R3 = Manifold(3, 'R^3', r'\mathbb{R}^3') Y.The Steiner map is a map $\mathbb{RP}^2 \rightarrow \mathbb{R}^3$ defined as follows:
︡e0666ef3-fc52-4215-8475-1dbe9d02b371︡︡{"done":true,"html":"The Steiner map is a map $\\mathbb{RP}^2 \\rightarrow \\mathbb{R}^3$ defined as follows:
"} ︠404f144c-6681-44d4-8e61-486273db9795s︠ Phi = RP2.diff_mapping(R3, {(X1,Y): [y1/(1+x1^2+y1^2), x1/(1+x1^2+y1^2), x1*y1/(1+x1^2+y1^2)], \ (X2,Y): [x2*y2/(1+x2^2+y2^2), y2/(1+x2^2+y2^2), x2/(1+x2^2+y2^2)], \ (X3,Y): [x3/(1+x3^2+y3^2), x3*y3/(1+x3^2+y3^2), y3/(1+x3^2+y3^2)]}, \ name='Phi', latex_name=r'\Phi') Phi.display() ︡f8339a34-e398-4828-a8c2-e3d10d4359db︡︡{"html":"$\Phi$ is a topological immersion of $\mathbb{RP}^2$ into $\mathbb{R}^3$, but it is not a smooth immersion (contrary to the Apéry map below): its differential is not injective at $(x_1,y_1)=(0,1)$ and $(x_1,y_1)=(1,0)$. The image of $\Phi$ is a self-intersecting surface of $\mathbb{R}^3$, called the Roman surface:
︡57cfe5ba-6134-4394-88b4-24d0e029ce4e︡︡{"done":true,"html":"$\\Phi$ is a topological immersion of $\\mathbb{RP}^2$ into $\\mathbb{R}^3$, but it is not a smooth immersion (contrary to the Apéry map below): its differential is not injective at $(x_1,y_1)=(0,1)$ and $(x_1,y_1)=(1,0)$. The image of $\\Phi$ is a self-intersecting surface of $\\mathbb{R}^3$, called the Roman surface:
"} ︠27cb34d1-2dc1-41a9-8565-d5f6fa6c795ds︠ g1 = parametric_plot3d(Phi.expr(X1,Y), (x1,-10,10), (y1,-10,10), plot_points=[100,100]) g2 = parametric_plot3d(Phi.expr(X2,Y), (x2,-10,10), (y2,-10,10), plot_points=[100,100]) g3 = parametric_plot3d(Phi.expr(X3,Y), (x3,-10,10), (y3,-10,10), plot_points=[100,100]) show(g1+g2+g3) ︡3568c6d4-00cf-4005-9b34-f05532375a70︡︡{"done":false,"file":{"uuid":"b4e08d97-ac80-4c08-becd-1f3e7e0928f5","filename":"b4e08d97-ac80-4c08-becd-1f3e7e0928f5.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠45a95093-2950-4f2f-ad6e-606a18a8958cs︠ gX1 = X1.plot(Y, mapping=Phi, max_value=16, nb_values=24, plot_points=100) gX2 = X2.plot(Y, mapping=Phi, max_value=16, nb_values=24, plot_points=100, color='green') gX3 = X3.plot(Y, mapping=Phi, max_value=16, nb_values=24, plot_points=100, color='blue') show(gX1) ︡e1656e6c-b298-4c2d-aa4f-84f44a834336︡︡{"done":false,"file":{"uuid":"99a21bd4-3449-419e-ba93-0032aaee1ea1","filename":"99a21bd4-3449-419e-ba93-0032aaee1ea1.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠4840e0a0-e4c4-4b7b-b2b6-71c5df5d0559s︠ show(gX2) ︡9d4734ef-4a07-4708-a8bc-4c6fae5a7081︡︡{"done":false,"file":{"uuid":"9fcd4bad-88bc-423e-bde7-f3b71ec5ef67","filename":"9fcd4bad-88bc-423e-bde7-f3b71ec5ef67.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠ee5172a1-58ed-402c-9ba2-ad1775ab24ads︠ show(gX3) ︡26216bfc-d3d1-44e8-8afe-3dfea83dcde0︡︡{"done":false,"file":{"uuid":"455eaea1-b57f-41df-921e-7034472031b1","filename":"455eaea1-b57f-41df-921e-7034472031b1.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠15df4556-4b31-4600-8222-96ac2de2c543s︠ show(gX1+gX2+gX3) ︡393e5a30-604e-4fe1-a7aa-899848226e2b︡︡{"done":false,"file":{"uuid":"e96564f5-5288-4dce-8715-ba260b914bde","filename":"e96564f5-5288-4dce-8715-ba260b914bde.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠42e7de41-b277-4b27-a5e2-8c703640a295i︠ %htmlThe Apéry map [Apéry, Adv. Math. 61, 185 (1986)] is a smooth immersion $\Psi: \mathbb{RP}^2 \rightarrow \mathbb{R}^3$. In terms of the charts X1, X2, X3 introduced above, it is defined as follows:
︡52f6afe2-7195-4c05-a28c-5e4a30002fa3︡︡{"done":true,"html":"The Apéry map [Apéry, Adv. Math. 61, 185 (1986)] is a smooth immersion $\\Psi: \\mathbb{RP}^2 \\rightarrow \\mathbb{R}^3$. In terms of the charts X1, X2, X3 introduced above, it is defined as follows:
"} ︠4bd23406-fa56-4212-84fa-6fabe787e18cs︠ fx = ((2*x^2-y^2-z^2)*(x^2+y^2+z^2)+2*y*z*(y^2-z^2)+z*x*(x^2-z^2)+x*y*(y^2-z^2))/2 ; fx ︡686d1f47-0709-4ea1-88d0-165f082e6e5a︡︡{"html":"The image of $\Psi$ is a self-intersecting surface of $\mathbb{R}^3$, called the Boy surface, after Werner Boy (1879-1914):
︡4a8581e3-58b8-4168-93e1-6adbef043bfe︡︡{"done":true,"html":"The image of $\\Psi$ is a self-intersecting surface of $\\mathbb{R}^3$, called the Boy surface, after Werner Boy (1879-1914):
"} ︠8ff36b58-e017-494a-bccf-507231460c91s︠ g1 = parametric_plot3d(Psi.expr(X1,Y), (x1,-10,10), (y1,-10,10), plot_points=[100,100]) g2 = parametric_plot3d(Psi.expr(X2,Y), (x2,-10,10), (y2,-10,10), plot_points=[100,100]) g3 = parametric_plot3d(Psi.expr(X3,Y), (x3,-10,10), (y3,-10,10), plot_points=[100,100]) show(g1+g2+g3) ︡72a76901-6145-4a78-bf1e-cb03bccefe90︡︡{"done":false,"file":{"uuid":"0aea9398-912b-4e10-bc3f-c1cf1d84db9c","filename":"0aea9398-912b-4e10-bc3f-c1cf1d84db9c.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠531c40c0-ebb8-4a1e-8d56-f5b7d23c89a4s︠ gX1 = X1.plot(Y, mapping=Psi, nb_values=40, plot_points=100) gX2 = X2.plot(Y, mapping=Psi, nb_values=40, plot_points=100, color='green') gX3 = X3.plot(Y, mapping=Psi, nb_values=40, plot_points=100, color='blue') ︡71dde0a1-bb63-49f0-a06e-f9d83ac51d3b︡︡{"done":true} ︠82bf5453-24c0-4354-a93f-c41043311180s︠ show(gX1) ︡393a4e41-2736-4bbc-b818-e48384da34aa︡{"done":false,"file":{"uuid":"1f97510e-7adb-4d54-990e-778fdfd71b0e","filename":"1f97510e-7adb-4d54-990e-778fdfd71b0e.sage3d"}}︡{"html":"","done":false}︡{"done":true}︡ ︠1d4bf01f-5b93-4108-857f-2c2c9bf3a8e6s︠ show(gX2) ︡b52513c0-f37c-459b-99b5-e06e62fdf6ef︡︡{"done":false,"file":{"uuid":"e6734bfd-de44-4352-a660-ae4719cda070","filename":"e6734bfd-de44-4352-a660-ae4719cda070.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠21c2357f-58af-43d3-a922-41e66b241c6bs︠ show(gX3) ︡1aa9d7b4-146a-4ae4-ab89-53094df96c6b︡︡{"done":false,"file":{"uuid":"25051255-67bd-4525-9cde-12864bde9585","filename":"25051255-67bd-4525-9cde-12864bde9585.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠d4f2e668-d884-4bf8-b8ee-f82b57b42b0cs︠ show(gX1+gX2+gX3) ︡d9255f8c-eefd-4ad5-b503-b55195ae4aa6︡︡{"done":false,"file":{"uuid":"8edf5c46-a9c2-4d6f-87b4-6c88a81c76c7","filename":"8edf5c46-a9c2-4d6f-87b4-6c88a81c76c7.sage3d"}}︡{"html":"","done":false}︡{"done":true} ︠a40b56e7-608f-4d58-83c7-858c5d2b82b0s︠ ︡5835df2a-c171-4a70-8da8-83c946dfca4b︡︡{"done":true}