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:

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)¶ Bases:
sage.manifolds.manifold.TopologicalManifold
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 manifoldname
– string; name (symbol) given to the manifoldfield
– field \(K\) on which the manifold is defined; allowed values are'real'
or an object of typeRealField
(e.g.,RR
) for a manifold over \(\RR\)'complex'
or an object of typeComplexField
(e.g.,CC
) for a manifold over \(\CC\) an object in the category of topological fields (see
Fields
andTopologicalSpaces
) for other types of manifolds
structure
– manifold structure (seeTopologicalStructure
orRealTopologicalStructure
)ambient
– (default:None
) manifold of destination of the immersion. IfNone
, set toself
base_manifold
– (default:None
) if notNone
, must be a topological manifold; the created object is then an open subset ofbase_manifold
latex_name
– (default:None
) string; LaTeX symbol to denote the manifold; if none are provided, it is set toname
start_index
– (default: 0) integer; lower value of the range of indices used for “indexed objects” on the manifold, e.g., coordinates in a chartcategory
– (default:None
) to specify the category; ifNone
,Manifolds(field)
is assumed (see the categoryManifolds
)unique_tag
– (default:None
) tag used to force the construction of a new object when all the other arguments have been used previously (withoutunique_tag
, theUniqueRepresentation
behavior inherited fromManifoldSubset
would return the previously constructed object corresponding to these arguments)
EXAMPLES:
Let \(N\) be a 2dimensional submanifold of a 3dimensional manifold \(M\):
sage: M = Manifold(3, 'M', structure="topological") sage: N = Manifold(2, 'N', ambient=M, structure="topological") sage: N 2dimensional submanifold N embedded in 3dimensional manifold M sage: CM.<x,y,z> = M.chart() sage: CN.<u,v> = N.chart()
Let us define a 1dimensional foliation indexed by \(t\). The inverse map is needed in order to compute the adapted chart in the ambient manifold:
sage: t = var('t') sage: phi = N.continuous_map(M, {(CN,CM):[u, v, t+u**2+v**2]}); phi Continuous map from the 2dimensional submanifold N embedded in 3dimensional manifold M to the 3dimensional topological manifold M sage: phi_inv = M.continuous_map(N, {(CM, CN):[x, y]}) sage: phi_inv_t = M.scalar_field({CM: zx**2y**2})
\(\phi\) 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: len(M.coord_changes()) 2
The foliations parameters are always added as the last coordinates.
See also

adapted_chart
(index='', latex_index='')¶ 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 \(mn\) free parameters to describe the whole manifold.
A set of coordinates adapted to a foliation is a set of coordinates \((x_1,...,x_n,t_1,...t_{mn})\) such that \((x_1,...x_n)\) are coordinates of \(N\) and \((t_1,...t_{mn})\) are the \(mn\) 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:
index
– (default:""
) string defining the name of the coordinates in the new chart. This string will be added at the end of the names of the old coordinates. By default, it is replaced by"_"+self._ambient._name
latex_index
– (default:""
) string defining the latex name of the coordinates in the new chart. This string will be added at the end of the latex names of the old coordinates. By default, it is replaced by"_"+self._ambient._latex_()
OUTPUT:
 list of charts created from the charts of
self
EXAMPLES:
sage: M = Manifold(3, 'M', structure="topological") sage: N = Manifold(2, 'N', ambient=M, structure="topological") sage: N 2dimensional submanifold N embedded in 3dimensional 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:zx**2y**2}) sage: N.set_immersion(phi, inverse=phi_inv, var=t, ....: t_inverse={t:phi_inv_t}) sage: N.declare_embedding() sage: N.adapted_chart() [Chart (M, (u_M, v_M, t_M))]

ambient
()¶ Return the ambient 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() 3dimensional 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 2dimensional submanifold N embedded in 3dimensional 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]}); phi Continuous map from the 2dimensional submanifold N embedded in 3dimensional manifold M to the 3dimensional topological manifold M sage: phi_inv = M.continuous_map(N,{(CM,CN):[x,y]}) sage: phi_inv_t = M.scalar_field({CM:zx**2y**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 the submanifold.
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:zx**2y**2}) sage: N.set_embedding(phi, inverse=phi_inv, var=t, ....: t_inverse={t: phi_inv_t}) sage: N.embedding() Homeomorphism from the 2dimensional submanifold N embedded in 3dimensional manifold M to the 3dimensional topological manifold M

immersion
()¶ Return the immersion of the submanifold.
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:zx**2y**2}) sage: N.set_immersion(phi, inverse=phi_inv, var=t, ....: t_inverse={t: phi_inv_t}) sage: N.immersion() Continuous map from the 2dimensional submanifold N embedded in 3dimensional manifold M to the 3dimensional 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 2dimensional surfaces embedded in 3dimensional 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 plotv
– iterable of the values taken by the second coordinate of the surface to plotchart1
– (default:None
) chart in whichu
andv
are considered. By default, the default chart of the submanifold is usedchart1
– (default:None
) destination chart. By default, the default chart of the manifold is used**kwargs
– other arguments as used inParametricSurface
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:zx**2y**2}) sage: N.set_immersion(phi, inverse=phi_inv, var=t, ....: t_inverse = {t:phi_inv_t}) sage: N.declare_embedding() sage: N.adapted_chart() [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: show(P0+P1+P2+P3)
See also

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.
INPUTS:
phi
– continuous map \(\phi\) from self to self._ambientinverse
– (default:None
) inverse of \(\phi\) onto its image, used for computing changes of chart from or to adapted charts. No verification is madevar
– (default:None
) list of parameters appearing in \(\phi\)t_inverse
– (default:None
) dictionary of scalar field on self._ambient indexed by elements ofvar
representing the missing information ininverse
EXAMPLES:
sage: M = Manifold(3, 'M', structure="topological") sage: N = Manifold(2, 'N', ambient=M, structure="topological") sage: N 2dimensional submanifold N embedded in 3dimensional 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]}); phi Continuous map from the 2dimensional submanifold N embedded in 3dimensional manifold M to the 3dimensional topological manifold M sage: phi_inv = M.continuous_map(N,{(CM,CN):[x,y]}) sage: phi_inv_t = M.scalar_field({CM:zx**2y**2}) sage: N.set_embedding(phi, inverse=phi_inv, var=t, ....: t_inverse={t: phi_inv_t})

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.
INPUTS:
phi
– continuous map \(\phi\) from self to self._ambientinverse
– (default:None
) inverse of \(\phi\) onto its image, used for computing changes of chart from or to adapted charts. No verification is madevar
– (default:None
) list of parameters appearing in \(\phi\)t_inverse
– (default:None
) dictionary of scalar field on self._ambient indexed by elements ofvar
representing the missing information ininverse
EXAMPLES:
sage: M = Manifold(3, 'M', structure="topological") sage: N = Manifold(2, 'N', ambient=M, structure="topological") sage: N 2dimensional submanifold N embedded in 3dimensional 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]}); phi Continuous map from the 2dimensional submanifold N embedded in 3dimensional manifold M to the 3dimensional topological manifold M sage: phi_inv = M.continuous_map(N,{(CM,CN):[x,y]}) sage: phi_inv_t = M.scalar_field({CM:zx**2y**2}) sage: N.set_immersion(phi, inverse=phi_inv, var=t, ....: t_inverse={t: phi_inv_t})