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:

sage.manifolds.topological_submanifold.TopologicalSubmanifold

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 manifold
  • name – string; name (symbol) given to the manifold
  • field – field \(K\) on which the manifold 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) manifold of destination of the immersion. If None, 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 manifold; 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 manifold, 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 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 submanifold N embedded in 3-dimensional manifold M
sage: CM.<x,y,z> = M.chart()
sage: CN.<u,v> = N.chart()

Let us define a 1-dimensional foliation indexed by \(t\). 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 2-dimensional submanifold N embedded in
 3-dimensional manifold M to the 3-dimensional topological manifold M
sage: phi_inv = M.continuous_map(N, {(CM, CN):[x, y]})
sage: phi_inv_t = M.scalar_field({CM: z-x**2-y**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

manifold