Manifold Subsets Defined as Pullbacks of Subsets under Continuous Maps¶

class sage.manifolds.subsets.pullback.ManifoldSubsetPullback(map, codomain_subset, inverse, name, latex_name)[source]¶

Bases: ManifoldSubset

Manifold subset defined as a pullback of a subset under a continuous map.

INPUT:

map – an instance of ContinuousMap, ScalarField, or Chart
codomain_subset – an instance of ManifoldSubset, RealSet, or ConvexSet_base

EXAMPLES:

Sage

sage: from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2

Python

>>> from sage.all import *
>>> from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
>>> M = Manifold(Integer(2), 'R^2', structure='topological')
>>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2)# Cartesian coordinates on R^2

Pulling back a real interval under a scalar field:

Sage

sage: r_squared = M.scalar_field(x^2+y^2)
sage: r_squared.set_immutable()
sage: cl_I = RealSet([1, 4]); cl_I
[1, 4]
sage: cl_O = ManifoldSubsetPullback(r_squared, cl_I); cl_O
Subset f_inv_[1, 4] of the 2-dimensional topological manifold R^2
sage: M.point((0, 0)) in cl_O
False
sage: M.point((0, 1)) in cl_O
True

Python

>>> from sage.all import *
>>> r_squared = M.scalar_field(x**Integer(2)+y**Integer(2))
>>> r_squared.set_immutable()
>>> cl_I = RealSet([Integer(1), Integer(4)]); cl_I
[1, 4]
>>> cl_O = ManifoldSubsetPullback(r_squared, cl_I); cl_O
Subset f_inv_[1, 4] of the 2-dimensional topological manifold R^2
>>> M.point((Integer(0), Integer(0))) in cl_O
False
>>> M.point((Integer(0), Integer(1))) in cl_O
True

Pulling back an open real interval gives an open subset:

Sage

sage: I = RealSet((1, 4)); I
(1, 4)
sage: O = ManifoldSubsetPullback(r_squared, I); O
Open subset f_inv_(1, 4) of the 2-dimensional topological manifold R^2
sage: M.point((1, 0)) in O
False
sage: M.point((1, 1)) in O
True

Python

>>> from sage.all import *
>>> I = RealSet((Integer(1), Integer(4))); I
(1, 4)
>>> O = ManifoldSubsetPullback(r_squared, I); O
Open subset f_inv_(1, 4) of the 2-dimensional topological manifold R^2
>>> M.point((Integer(1), Integer(0))) in O
False
>>> M.point((Integer(1), Integer(1))) in O
True

Pulling back a polytope under a chart:

Sage

sage: # needs sage.geometry.polyhedron
sage: P = Polyhedron(vertices=[[0, 0], [1, 2], [2, 1]]); P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
sage: S = ManifoldSubsetPullback(c_cart, P); S
Subset x_y_inv_P of the 2-dimensional topological manifold R^2
sage: M((1, 2)) in S
True
sage: M((2, 0)) in S
False

Python

>>> from sage.all import *
>>> # needs sage.geometry.polyhedron
>>> P = Polyhedron(vertices=[[Integer(0), Integer(0)], [Integer(1), Integer(2)], [Integer(2), Integer(1)]]); P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
>>> S = ManifoldSubsetPullback(c_cart, P); S
Subset x_y_inv_P of the 2-dimensional topological manifold R^2
>>> M((Integer(1), Integer(2))) in S
True
>>> M((Integer(2), Integer(0))) in S
False

Pulling back the interior of a polytope under a chart:

Sage

sage: # needs sage.geometry.polyhedron
sage: int_P = P.interior(); int_P
Relative interior of a
 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
sage: int_S = ManifoldSubsetPullback(c_cart, int_P, name='int_S'); int_S
Open subset int_S of the 2-dimensional topological manifold R^2
sage: M((0, 0)) in int_S
False
sage: M((1, 1)) in int_S
True

Python

>>> from sage.all import *
>>> # needs sage.geometry.polyhedron
>>> int_P = P.interior(); int_P
Relative interior of a
 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
>>> int_S = ManifoldSubsetPullback(c_cart, int_P, name='int_S'); int_S
Open subset int_S of the 2-dimensional topological manifold R^2
>>> M((Integer(0), Integer(0))) in int_S
False
>>> M((Integer(1), Integer(1))) in int_S
True

Using the embedding map of a submanifold:

Sage

sage: M = Manifold(3, 'M', structure='topological')
sage: N = Manifold(2, 'N', ambient=M, structure='topological'); N
2-dimensional topological submanifold N
 immersed in the 3-dimensional topological manifold M
sage: CM.<x,y,z> = M.chart()
sage: CN.<u,v> = N.chart()
sage: t = var('t')
sage: phi = N.continuous_map(M, {(CN,CM): [u,v,t+u^2+v^2]})
sage: phi_inv = M.continuous_map(N, {(CM,CN): [x,y]})
sage: phi_inv_t = M.scalar_field({CM: z-x^2-y^2})
sage: N.set_immersion(phi, inverse=phi_inv, var=t,
....:                 t_inverse={t: phi_inv_t})
sage: N.declare_embedding()

sage: from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
sage: S = M.open_subset('S', coord_def={CM: z<1})
sage: phi_without_t = N.continuous_map(M, {(CN, CM): [expr.subs(t=0)
....:                                                 for expr in phi.expr()]})
sage: phi_without_t
Continuous map
 from the 2-dimensional topological submanifold N
  embedded in the 3-dimensional topological manifold M
 to the 3-dimensional topological manifold M
sage: phi_without_t.expr()
(u, v, u^2 + v^2)
sage: D = ManifoldSubsetPullback(phi_without_t, S); D
Subset f_inv_S of the 2-dimensional topological submanifold N
 embedded in the 3-dimensional topological manifold M
sage: N.point((2,0)) in D
False

Python

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='topological')
>>> N = Manifold(Integer(2), 'N', ambient=M, structure='topological'); N
2-dimensional topological submanifold N
 immersed in the 3-dimensional topological manifold M
>>> CM = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = CM._first_ngens(3)
>>> CN = N.chart(names=('u', 'v',)); (u, v,) = CN._first_ngens(2)
>>> t = var('t')
>>> phi = N.continuous_map(M, {(CN,CM): [u,v,t+u**Integer(2)+v**Integer(2)]})
>>> phi_inv = M.continuous_map(N, {(CM,CN): [x,y]})
>>> phi_inv_t = M.scalar_field({CM: z-x**Integer(2)-y**Integer(2)})
>>> N.set_immersion(phi, inverse=phi_inv, var=t,
...                 t_inverse={t: phi_inv_t})
>>> N.declare_embedding()

>>> from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
>>> S = M.open_subset('S', coord_def={CM: z<Integer(1)})
>>> phi_without_t = N.continuous_map(M, {(CN, CM): [expr.subs(t=Integer(0))
...                                                 for expr in phi.expr()]})
>>> phi_without_t
Continuous map
 from the 2-dimensional topological submanifold N
  embedded in the 3-dimensional topological manifold M
 to the 3-dimensional topological manifold M
>>> phi_without_t.expr()
(u, v, u^2 + v^2)
>>> D = ManifoldSubsetPullback(phi_without_t, S); D
Subset f_inv_S of the 2-dimensional topological submanifold N
 embedded in the 3-dimensional topological manifold M
>>> N.point((Integer(2),Integer(0))) in D
False

closure(name=None, latex_name=None)[source]¶

Return the topological closure of self in the manifold.

Because self is a pullback of some subset under a continuous map, the closure of self is the pullback of the closure.

EXAMPLES:

Sage

sage: from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: r_squared = M.scalar_field(x^2+y^2)
sage: r_squared.set_immutable()
sage: I = RealSet.open_closed(1, 2); I
(1, 2]
sage: O = ManifoldSubsetPullback(r_squared, I); O
Subset f_inv_(1, 2] of the 2-dimensional topological manifold R^2
sage: latex(O)
f^{-1}((1, 2])
sage: cl_O = O.closure(); cl_O
Subset f_inv_[1, 2] of the 2-dimensional topological manifold R^2
sage: cl_O.is_closed()
True

Python

>>> from sage.all import *
>>> from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
>>> M = Manifold(Integer(2), 'R^2', structure='topological')
>>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2)# Cartesian coordinates on R^2
>>> r_squared = M.scalar_field(x**Integer(2)+y**Integer(2))
>>> r_squared.set_immutable()
>>> I = RealSet.open_closed(Integer(1), Integer(2)); I
(1, 2]
>>> O = ManifoldSubsetPullback(r_squared, I); O
Subset f_inv_(1, 2] of the 2-dimensional topological manifold R^2
>>> latex(O)
f^{-1}((1, 2])
>>> cl_O = O.closure(); cl_O
Subset f_inv_[1, 2] of the 2-dimensional topological manifold R^2
>>> cl_O.is_closed()
True

is_closed()[source]¶

Return if self is (known to be) a closed subset of the manifold.

EXAMPLES:

Sage

sage: from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2

Python

>>> from sage.all import *
>>> from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
>>> M = Manifold(Integer(2), 'R^2', structure='topological')
>>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2)# Cartesian coordinates on R^2

The pullback of a closed real interval under a scalar field is closed:

Sage

sage: r_squared = M.scalar_field(x^2+y^2)
sage: r_squared.set_immutable()
sage: cl_I = RealSet([1, 2]); cl_I
[1, 2]
sage: cl_O = ManifoldSubsetPullback(r_squared, cl_I); cl_O
Subset f_inv_[1, 2] of the 2-dimensional topological manifold R^2
sage: cl_O.is_closed()
True

Python

>>> from sage.all import *
>>> r_squared = M.scalar_field(x**Integer(2)+y**Integer(2))
>>> r_squared.set_immutable()
>>> cl_I = RealSet([Integer(1), Integer(2)]); cl_I
[1, 2]
>>> cl_O = ManifoldSubsetPullback(r_squared, cl_I); cl_O
Subset f_inv_[1, 2] of the 2-dimensional topological manifold R^2
>>> cl_O.is_closed()
True

The pullback of a (closed convex) polyhedron under a chart is closed:

Sage

sage: # needs sage.geometry.polyhedron
sage: P = Polyhedron(vertices=[[0, 0], [1, 2], [3, 4]]); P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
sage: McP = ManifoldSubsetPullback(c_cart, P, name='McP'); McP
Subset McP of the 2-dimensional topological manifold R^2
sage: McP.is_closed()
True

Python

>>> from sage.all import *
>>> # needs sage.geometry.polyhedron
>>> P = Polyhedron(vertices=[[Integer(0), Integer(0)], [Integer(1), Integer(2)], [Integer(3), Integer(4)]]); P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
>>> McP = ManifoldSubsetPullback(c_cart, P, name='McP'); McP
Subset McP of the 2-dimensional topological manifold R^2
>>> McP.is_closed()
True

The pullback of real vector subspaces under a chart is closed:

Sage

sage: V = span([[1, 2]], RR); V
Vector space of degree 2 and dimension 1 over Real Field with 53 bits of precision
Basis matrix:
[1.00000000000000 2.00000000000000]
sage: McV = ManifoldSubsetPullback(c_cart, V, name='McV'); McV
Subset McV of the 2-dimensional topological manifold R^2
sage: McV.is_closed()
True

Python

>>> from sage.all import *
>>> V = span([[Integer(1), Integer(2)]], RR); V
Vector space of degree 2 and dimension 1 over Real Field with 53 bits of precision
Basis matrix:
[1.00000000000000 2.00000000000000]
>>> McV = ManifoldSubsetPullback(c_cart, V, name='McV'); McV
Subset McV of the 2-dimensional topological manifold R^2
>>> McV.is_closed()
True

The pullback of point lattices under a chart is closed:

Sage

sage: W = span([[1, 0], [3, 5]], ZZ); W
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1 0]
[0 5]
sage: McW = ManifoldSubsetPullback(c_cart, W, name='McW'); McW
Subset McW of the 2-dimensional topological manifold R^2
sage: McW.is_closed()
True

Python

>>> from sage.all import *
>>> W = span([[Integer(1), Integer(0)], [Integer(3), Integer(5)]], ZZ); W
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1 0]
[0 5]
>>> McW = ManifoldSubsetPullback(c_cart, W, name='McW'); McW
Subset McW of the 2-dimensional topological manifold R^2
>>> McW.is_closed()
True

The pullback of finite sets is closed:

Sage

sage: F = Family([vector(QQ, [1, 2], immutable=True), vector(QQ, [2, 3], immutable=True)])
sage: McF = ManifoldSubsetPullback(c_cart, F, name='McF'); McF
Subset McF of the 2-dimensional topological manifold R^2
sage: McF.is_closed()
True

Python

>>> from sage.all import *
>>> F = Family([vector(QQ, [Integer(1), Integer(2)], immutable=True), vector(QQ, [Integer(2), Integer(3)], immutable=True)])
>>> McF = ManifoldSubsetPullback(c_cart, F, name='McF'); McF
Subset McF of the 2-dimensional topological manifold R^2
>>> McF.is_closed()
True

is_open()[source]¶

Return if self is (known to be) an open set.

This version of the method always returns False.

Because the map is continuous, the pullback is open if the codomain_subset is open.

However, the design of ManifoldSubset requires that open subsets are instances of the subclass sage.manifolds.manifold.TopologicalManifold. The constructor of ManifoldSubsetPullback delegates to a subclass of sage.manifolds.manifold.TopologicalManifold for some open subsets.

EXAMPLES:

Sage

sage: from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
sage: M = Manifold(2, 'R^2', structure='topological')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2

sage: # needs sage.geometry.polyhedron
sage: P = Polyhedron(vertices=[[0, 0], [1, 2], [3, 4]]); P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
sage: P.is_open()
False
sage: McP = ManifoldSubsetPullback(c_cart, P, name='McP'); McP
Subset McP of the 2-dimensional topological manifold R^2
sage: McP.is_open()
False

Python

>>> from sage.all import *
>>> from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
>>> M = Manifold(Integer(2), 'R^2', structure='topological')
>>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2)# Cartesian coordinates on R^2

>>> # needs sage.geometry.polyhedron
>>> P = Polyhedron(vertices=[[Integer(0), Integer(0)], [Integer(1), Integer(2)], [Integer(3), Integer(4)]]); P
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
>>> P.is_open()
False
>>> McP = ManifoldSubsetPullback(c_cart, P, name='McP'); McP
Subset McP of the 2-dimensional topological manifold R^2
>>> McP.is_open()
False

some_elements()[source]¶

Generate some elements of self.

EXAMPLES:

Sage

sage: # needs sage.geometry.polyhedron
sage: from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
sage: M = Manifold(3, 'R^3', structure='topological')
sage: c_cart.<x,y,z> = M.chart() # Cartesian coordinates on R^3
sage: Cube = polytopes.cube(); Cube
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices
sage: McCube = ManifoldSubsetPullback(c_cart, Cube, name='McCube'); McCube
Subset McCube of the 3-dimensional topological manifold R^3
sage: L = list(McCube.some_elements()); L
[Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3]
sage: list(p.coordinates(c_cart) for p in L)
[(0, 0, 0),
 (1, -1, -1),
 (1, 0, -1),
 (1, 1/2, 0),
 (1, -1/4, 1/2),
 (0, -5/8, 3/4)]

sage: # needs sage.geometry.polyhedron
sage: Empty = Polyhedron(ambient_dim=3)
sage: McEmpty = ManifoldSubsetPullback(c_cart, Empty, name='McEmpty')
sage: McEmpty
Subset McEmpty of the 3-dimensional topological manifold R^3
sage: list(McEmpty.some_elements())
[]

Python

>>> from sage.all import *
>>> # needs sage.geometry.polyhedron
>>> from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
>>> M = Manifold(Integer(3), 'R^3', structure='topological')
>>> c_cart = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = c_cart._first_ngens(3)# Cartesian coordinates on R^3
>>> Cube = polytopes.cube(); Cube
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices
>>> McCube = ManifoldSubsetPullback(c_cart, Cube, name='McCube'); McCube
Subset McCube of the 3-dimensional topological manifold R^3
>>> L = list(McCube.some_elements()); L
[Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3,
 Point on the 3-dimensional topological manifold R^3]
>>> list(p.coordinates(c_cart) for p in L)
[(0, 0, 0),
 (1, -1, -1),
 (1, 0, -1),
 (1, 1/2, 0),
 (1, -1/4, 1/2),
 (0, -5/8, 3/4)]

>>> # needs sage.geometry.polyhedron
>>> Empty = Polyhedron(ambient_dim=Integer(3))
>>> McEmpty = ManifoldSubsetPullback(c_cart, Empty, name='McEmpty')
>>> McEmpty
Subset McEmpty of the 3-dimensional topological manifold R^3
>>> list(McEmpty.some_elements())
[]