Submanifolds of topological manifolds¶

Given a topological manifold $$M$$ over a topological field $$K$$, a topological submanifold of $$M$$ is defined by a topological manifold $$N$$ over the same field $$K$$ of dimension lower than the dimension of $$M$$ and a topological embedding $$\phi$$ from $$N$$ to $$M$$ (i.e. $$\phi$$ is a homeomorphism onto its image).

In the case where the map $$\phi$$ is only an embedding locally, it is called an topological immersion, and defines an immersed submanifold.

The global embedding property cannot be checked in sage, so the immersed or embedded aspect of the manifold must be declared by the user, by calling either set_embedding() or set_immersion() while declaring the map $$\phi$$.

The map $$\phi: N\to M$$ can also depend on one or multiple parameters. As long as $$\phi$$ remains injective in these parameters, it represents a foliation. The dimension of the foliation is defined as the number of parameters.

AUTHORS:

• Florentin Jaffredo (2018): initial version

REFERENCES:

• J. M. Lee: Introduction to Smooth Manifolds [Lee2013]
class sage.manifolds.topological_submanifold.TopologicalSubmanifold(n, name, field, structure, ambient=None, base_manifold=None, latex_name=None, start_index=0, category=None, unique_tag=None)

Submanifold of a topological manifold.

Given a topological manifold $$M$$ over a topological field $$K$$, a topological submanifold of $$M$$ is defined by a topological manifold $$N$$ over the same field $$K$$ of dimension lower than the dimension of $$M$$ and a topological embedding $$\phi$$ from $$N$$ to $$M$$ (i.e. $$\phi$$ is an homeomorphism onto its image).

In the case where $$\phi$$ is only an topological immersion (i.e. is only locally an embedding), one says that $$N$$ is an immersed submanifold.

The map $$\phi$$ can also depend on one or multiple parameters. As long as $$\phi$$ remains injective in these parameters, it represents a foliation. The dimension of the foliation is defined as the number of parameters.

INPUT:

• n – positive integer; dimension of the submanifold
• name – string; name (symbol) given to the submanifold
• field – field $$K$$ on which the submanifold is defined; allowed values are
• 'real' or an object of type RealField (e.g., RR) for a manifold over $$\RR$$
• 'complex' or an object of type ComplexField (e.g., CC) for a manifold over $$\CC$$
• an object in the category of topological fields (see Fields and TopologicalSpaces) for other types of manifolds
• structure – manifold structure (see TopologicalStructure or RealTopologicalStructure)
• ambient – (default: None) codomain $$M$$ of the immersion $$\phi$$; must be a topological manifold. If None, it is set to self
• base_manifold – (default: None) if not None, must be a topological manifold; the created object is then an open subset of base_manifold
• latex_name – (default: None) string; LaTeX symbol to denote the submanifold; if none are provided, it is set to name
• start_index – (default: 0) integer; lower value of the range of indices used for “indexed objects” on the submanifold, e.g., coordinates in a chart
• category – (default: None) to specify the category; if None, Manifolds(field) is assumed (see the category Manifolds)
• unique_tag – (default: None) tag used to force the construction of a new object when all the other arguments have been used previously (without unique_tag, the UniqueRepresentation behavior inherited from ManifoldSubset via TopologicalManifold would return the previously constructed object corresponding to these arguments)

EXAMPLES:

Let $$N$$ be a 2-dimensional submanifold of a 3-dimensional manifold $$M$$:

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()


Let us define a 1-dimensional foliation indexed by $$t$$:

sage: t = var('t')
sage: phi = N.continuous_map(M, {(CN,CM): [u, v, t+u^2+v^2]})
sage: phi.display()
N --> M
(u, v) |--> (x, y, z) = (u, v, u^2 + v^2 + t)


The foliation inverse maps are needed for computing the adapted chart on the ambient manifold:

sage: phi_inv = M.continuous_map(N, {(CM, CN): [x, y]})
sage: phi_inv.display()
M --> N
(x, y, z) |--> (u, v) = (x, y)
sage: phi_inv_t = M.scalar_field({CM: z-x^2-y^2})
sage: phi_inv_t.display()
M --> R
(x, y, z) |--> -x^2 - y^2 + z


$$\phi$$ can then be declared as an embedding $$N\to M$$:

sage: N.set_embedding(phi, inverse=phi_inv, var=t,
....:                 t_inverse={t: phi_inv_t})


The foliation can also be used to find new charts on the ambient manifold that are adapted to the foliation, i.e. in which the expression of the immersion is trivial. At the same time, the appropriate coordinate changes are computed:

sage: N.adapted_chart()
[Chart (M, (u_M, v_M, t_M))]
sage: M.atlas()
[Chart (M, (x, y, z)), Chart (M, (u_M, v_M, t_M))]
sage: len(M.coord_changes())
2


The foliation parameters are always added as the last coordinates.

adapted_chart(postscript=None, latex_postscript=None)

Create charts and changes of charts in the ambient manifold adapted to the foliation.

A manifold $$M$$ of dimension $$m$$ can be foliated by submanifolds $$N$$ of dimension $$n$$. The corresponding embedding needs $$m-n$$ free parameters to describe the whole manifold.

A chart adapted to the foliation is a set of coordinates $$(x_1,\ldots,x_n,t_1,\ldots,t_{m-n})$$ on $$M$$ such that $$(x_1,\ldots,x_n)$$ are coordinates on $$N$$ and $$(t_1,\ldots,t_{m-n})$$ are the $$m-n$$ free parameters of the foliation.

Provided that an embedding with free variables is already defined, this function constructs such charts and coordinates changes whenever it is possible.

If there are restrictions of the coordinates on the starting chart, these restrictions are also propagated.

INPUT:

• postscript – (default: None) string defining the name of the coordinates of the adapted chart. This string will be appended to the names of the coordinates $$(x_1,\ldots,x_n)$$ and of the parameters $$(t_1,\ldots,t_{m-n})$$. If None, "_" + self.ambient()._name is used
• latex_postscript – (default: None) string defining the LaTeX name of the coordinates of the adapted chart. This string will be appended to the LaTeX names of the coordinates $$(x_1,\ldots,x_n)$$ and of the parameters $$(t_1,\ldots,t_{m-n})$$, If None, "_" + self.ambient()._latex_() is used

OUTPUT:

• list of adapted charts on $$M$$ created from the charts of self

EXAMPLES:

sage: M = Manifold(3, 'M', structure="topological",
....:              latex_name=r"\mathcal{M}")
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_embedding(phi, inverse=phi_inv, var=t,
....:                 t_inverse={t:phi_inv_t})
[Chart (M, (u_M, v_M, t_M))]
sage: latex(_)
\left[\left(\mathcal{M},({{u}_{\mathcal{M}}}, {{v}_{\mathcal{M}}},
{{t}_{\mathcal{M}}})\right)\right]


The adapted chart has been added to the atlas of M:

sage: M.atlas()
[Chart (M, (x, y, z)), Chart (M, (u_M, v_M, t_M))]
sage: N.atlas()
[Chart (N, (u, v))]


The names of the adapted coordinates can be customized:

sage: N.adapted_chart(postscript='1', latex_postscript='_1')
[Chart (M, (u1, v1, t1))]
sage: latex(_)
\left[\left(\mathcal{M},({{u}_1}, {{v}_1}, {{t}_1})\right)\right]

ambient()

Return the manifold in which self is immersed or embedded.

EXAMPLES:

sage: M = Manifold(3, 'M', structure="topological")
sage: N = Manifold(2, 'N', ambient=M, structure="topological")
sage: N.ambient()
3-dimensional topological manifold M

declare_embedding()

Declare that the immersion provided by set_immersion() is in fact an embedding.

A topological embedding is a continuous map that is a homeomorphism onto its image. A differentiable embedding is a topological embedding that is also a differentiable immersion.

EXAMPLES:

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._immersed
True
sage: N._embedded
False
sage: N.declare_embedding()
sage: N._immersed
True
sage: N._embedded
True

embedding()

Return the embedding of self into the ambient manifold.

EXAMPLES:

sage: M = Manifold(3, 'M', structure="topological")
sage: N = Manifold(2, 'N', ambient=M, structure="topological")
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_embedding(phi, inverse=phi_inv, var=t,
....:                 t_inverse={t: phi_inv_t})
sage: N.embedding()
Continuous map from the 2-dimensional topological submanifold N
embedded in the 3-dimensional topological manifold M to the
3-dimensional topological manifold M

immersion()

Return the immersion of self into the ambient manifold.

EXAMPLES:

sage: M = Manifold(3, 'M', structure="topological")
sage: N = Manifold(2, 'N', ambient=M, structure="topological")
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.immersion()
Continuous map from the 2-dimensional topological submanifold N
immersed in the 3-dimensional topological manifold M to the
3-dimensional topological manifold M

plot(param, u, v, chart1=None, chart2=None, **kwargs)

Plot an embedding.

Plot the embedding defined by the foliation and a set of values for the free parameters. This function can only plot 2-dimensional surfaces embedded in 3-dimensional manifolds. It ultimately calls ParametricSurface.

INPUT:

• param – dictionary of values indexed by the free variables appearing in the foliation.
• u – iterable of the values taken by the first coordinate of the surface to plot
• v – iterable of the values taken by the second coordinate of the surface to plot
• chart1 – (default: None) chart in which u and v are considered. By default, the default chart of the submanifold is used
• chart2 – (default: None) chart in the codomain of the embedding. By default, the default chart of the codomain is used
• **kwargs – other arguments as used in ParametricSurface

EXAMPLES:

sage: M = Manifold(3, 'M', structure="topological")
sage: N = Manifold(2, 'N', ambient = M, structure="topological")
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_embedding(phi, inverse=phi_inv, var=t,
....:                 t_inverse = {t:phi_inv_t})
[Chart (M, (u_M, v_M, t_M))]
sage: P0 = N.plot({t:0}, srange(-1, 1, 0.1), srange(-1, 1, 0.1),
....:             CN, CM, opacity=0.3, mesh=True)
sage: P1 = N.plot({t:1}, srange(-1, 1, 0.1), srange(-1, 1, 0.1),
....:             CN, CM, opacity=0.3, mesh=True)
sage: P2 = N.plot({t:2}, srange(-1, 1, 0.1), srange(-1, 1, 0.1),
....:             CN, CM, opacity=0.3, mesh=True)
sage: P3 = N.plot({t:3}, srange(-1, 1, 0.1), srange(-1, 1, 0.1),
....:             CN, CM, opacity=0.3, mesh=True)
sage: P0 + P1 + P2 + P3
Graphics3d Object

set_embedding(phi, inverse=None, var=None, t_inverse=None)

Register the embedding of an embedded submanifold.

A topological embedding is a continuous map that is a homeomorphism onto its image. A differentiable embedding is a topological embedding that is also a differentiable immersion.

INPUT:

• phi – continuous map $$\phi$$ from self to self.ambient()
• inverse – (default: None) continuous map from self.ambient() to self, which once restricted to the image of $$\phi$$ is the inverse of $$\phi$$ onto its image (NB: no check of this is performed)
• var – (default: None) list of parameters involved in the definition of $$\phi$$ (case of foliation); if $$\phi$$ depends on a single parameter t, one can write var=t as a shortcut for var=[t]
• t_inverse – (default: None) dictionary of scalar fields on self.ambient() providing the values of the parameters involved in the definition of $$\phi$$ (case of foliation), the keys being the parameters

EXAMPLES:

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.display()
N --> M
(u, v) |--> (x, y, z) = (u, v, u^2 + v^2 + t)
sage: phi_inv = M.continuous_map(N, {(CM,CN): [x,y]})
sage: phi_inv.display()
M --> N
(x, y, z) |--> (u, v) = (x, y)
sage: phi_inv_t = M.scalar_field({CM: z-x^2-y^2})
sage: phi_inv_t.display()
M --> R
(x, y, z) |--> -x^2 - y^2 + z
sage: N.set_embedding(phi, inverse=phi_inv, var=t,
....:                 t_inverse={t: phi_inv_t})


Now N appears as an embbeded submanifold:

sage: N
2-dimensional topological submanifold N embedded in the
3-dimensional topological manifold M

set_immersion(phi, inverse=None, var=None, t_inverse=None)

Register the immersion of the immersed submanifold.

A topological immersion is a continuous map that is locally a topological embedding (i.e. a homeomorphism onto its image). A differentiable immersion is a differentiable map whose differential is injective at each point.

If an inverse of the immersion onto its image exists, it can be registered at the same time. If the immersion depends on parameters, they must also be declared here.

INPUT:

• phi – continuous map $$\phi$$ from self to self.ambient()
• inverse – (default: None) continuous map from self.ambient() to self, which once restricted to the image of $$\phi$$ is the inverse of $$\phi$$ onto its image if the latter exists (NB: no check of this is performed)
• var – (default: None) list of parameters involved in the definition of $$\phi$$ (case of foliation); if $$\phi$$ depends on a single parameter t, one can write var=t as a shortcut for var=[t]
• t_inverse – (default: None) dictionary of scalar fields on self.ambient() providing the values of the parameters involved in the definition of $$\phi$$ (case of foliation), the keys being the parameters

EXAMPLES:

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.display()
N --> M
(u, v) |--> (x, y, z) = (u, v, u^2 + v^2 + t)
sage: phi_inv = M.continuous_map(N, {(CM,CN): [x,y]})
sage: phi_inv.display()
M --> N
(x, y, z) |--> (u, v) = (x, y)
sage: phi_inv_t = M.scalar_field({CM: z-x^2-y^2})
sage: phi_inv_t.display()
M --> R
(x, y, z) |--> -x^2 - y^2 + z
sage: N.set_immersion(phi, inverse=phi_inv, var=t,
....:                 t_inverse={t: phi_inv_t})