Manifold Subsets Defined as Pullbacks of Subsets under Continuous Maps

class sage.manifolds.subsets.pullback.ManifoldSubsetPullback(map, codomain_subset, inverse, name, latex_name)

Bases: sage.manifolds.subset.ManifoldSubset

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

INPUT:

EXAMPLES:

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

Pulling back a real interval under a scalar field:

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

Pulling back an open real interval gives an open subset:

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

Pulling back a polytope under a chart:

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

Pulling back the interior of a polytope under a chart:

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

Using the embedding map of a submanifold:

sage: M = Manifold(3, 'M', structure="topological")
sage: N = Manifold(2, 'N', ambient=M, structure="topological")
sage: 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()]}); 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
closure(name=None, latex_name=None)

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: 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
is_closed()

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

EXAMPLES:

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

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

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

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

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

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

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

The pullback of point lattices under a chart is closed:

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

The pullback of finite sets is closed:

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
is_open()

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: 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: 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
some_elements()

Generate some elements of self.

EXAMPLES:

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