Submanifolds of differentiable manifolds#

Given two differentiable manifolds \(N\) and \(M\), an immersion \(\phi\) is a differentiable map \(N\to M\) whose differential is everywhere injective. One then says that \(N\) is an immersed submanifold of \(M\), via \(\phi\).

If in addition, \(\phi\) is a differentiable embedding (i.e. \(\phi\) is an immersion that is a homeomorphism onto its image), then \(N\) is called an embedded submanifold of \(M\) (or simply a submanifold).

\(\phi\) can also depend on one or multiple parameters. As long as the differential of \(\phi\) 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

  • Eric Gourgoulhon (2018-2019): add documentation

  • Matthias Koeppe (2021): open subsets of submanifolds

REFERENCES:

  • J. M. Lee: Introduction to Smooth Manifolds [Lee2013]

class sage.manifolds.differentiable.differentiable_submanifold.DifferentiableSubmanifold(n, name, field, structure, ambient=None, base_manifold=None, diff_degree=+Infinity, latex_name=None, start_index=0, category=None, unique_tag=None)[source]#

Bases: DifferentiableManifold, TopologicalSubmanifold

Submanifold of a differentiable manifold.

Given two differentiable manifolds \(N\) and \(M\), an immersion \(\phi\) is a differentiable map \(N\to M\) whose differential is everywhere injective. One then says that \(N\) is an immersed submanifold of \(M\), via \(\phi\).

If in addition, \(\phi\) is a differentiable embedding (i.e. \(\phi\) is an immersion that is a homeomorphism onto its image), then \(N\) is called an embedded submanifold of \(M\) (or simply a submanifold).

\(\phi\) can also depend on one or multiple parameters. As long as the differential of \(\phi\) 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 sub 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) codomain \(M\) of the immersion \(\phi\); must be a differentiable manifold. If None, it is set to self

  • base_manifold – (default: None) if not None, must be a differentiable manifold; the created object is then an open subset of base_manifold

  • diff_degree – (default: infinity) degree of differentiability

  • 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).Differentiable() (or Manifolds(field).Smooth() if diff_degree = infinity) 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 DifferentiableManifold 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')
sage: N = Manifold(2, 'N', ambient=M)
sage: N
2-dimensional differentiable submanifold N immersed in the
 3-dimensional differentiable manifold M
sage: CM.<x,y,z> = M.chart()
sage: CN.<u,v> = N.chart()

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M')
>>> N = Manifold(Integer(2), 'N', ambient=M)
>>> N
2-dimensional differentiable submanifold N immersed in the
 3-dimensional differentiable 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)

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)

>>> from sage.all import *
>>> t = var('t')
>>> phi = N.continuous_map(M, {(CN,CM): [u, v, t+u**Integer(2)+v**Integer(2)]})
>>> 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

>>> from sage.all import *
>>> phi_inv = M.continuous_map(N, {(CM, CN): [x, y]})
>>> phi_inv.display()
M → N
   (x, y, z) ↦ (u, v) = (x, y)
>>> phi_inv_t = M.scalar_field({CM: z-x**Integer(2)-y**Integer(2)})
>>> 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})

>>> from sage.all import *
>>> 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, ie 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

>>> from sage.all import *
>>> N.adapted_chart()
[Chart (M, (u_M, v_M, t_M))]
>>> M.atlas()
[Chart (M, (x, y, z)), Chart (M, (u_M, v_M, t_M))]
>>> len(M.coord_changes())
2

See also

manifold and topological_submanifold

open_subset(name, latex_name=None, coord_def={}, supersets=None)[source]#

Create an open subset of the manifold.

An open subset is a set that is (i) included in the manifold and (ii) open with respect to the manifold’s topology. It is a differentiable manifold by itself.

As self is a submanifold of its ambient manifold, the new open subset is also considered a submanifold of that. Hence the returned object is an instance of DifferentiableSubmanifold.

INPUT:

  • name – name given to the open subset

  • latex_name – (default: None) LaTeX symbol to denote the subset; if none is provided, it is set to name

  • coord_def – (default: {}) definition of the subset in terms of coordinates; coord_def must a be dictionary with keys charts in the manifold’s atlas and values the symbolic expressions formed by the coordinates to define the subset.

  • supersets – (default: only self) list of sets that the new open subset is a subset of

OUTPUT:

EXAMPLES:

sage: M = Manifold(3, 'M', structure="differentiable")
sage: N = Manifold(2, 'N', ambient=M, structure="differentiable"); N
2-dimensional differentiable submanifold N immersed in the
 3-dimensional differentiable manifold M
sage: S = N.subset('S'); S
Subset S of the
 2-dimensional differentiable submanifold N immersed in the
  3-dimensional differentiable manifold M
sage: O = N.subset('O', is_open=True); O  # indirect doctest
Open subset O of the
 2-dimensional differentiable submanifold N immersed in the
  3-dimensional differentiable manifold M

sage: phi = N.diff_map(M)
sage: N.set_embedding(phi)
sage: N
2-dimensional differentiable submanifold N embedded in the
 3-dimensional differentiable manifold M
sage: S = N.subset('S'); S
Subset S of the
 2-dimensional differentiable submanifold N embedded in the
  3-dimensional differentiable manifold M
sage: O = N.subset('O', is_open=True); O  # indirect doctest
Open subset O of the
 2-dimensional differentiable submanifold N embedded in the
  3-dimensional differentiable manifold M

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure="differentiable")
>>> N = Manifold(Integer(2), 'N', ambient=M, structure="differentiable"); N
2-dimensional differentiable submanifold N immersed in the
 3-dimensional differentiable manifold M
>>> S = N.subset('S'); S
Subset S of the
 2-dimensional differentiable submanifold N immersed in the
  3-dimensional differentiable manifold M
>>> O = N.subset('O', is_open=True); O  # indirect doctest
Open subset O of the
 2-dimensional differentiable submanifold N immersed in the
  3-dimensional differentiable manifold M

>>> phi = N.diff_map(M)
>>> N.set_embedding(phi)
>>> N
2-dimensional differentiable submanifold N embedded in the
 3-dimensional differentiable manifold M
>>> S = N.subset('S'); S
Subset S of the
 2-dimensional differentiable submanifold N embedded in the
  3-dimensional differentiable manifold M
>>> O = N.subset('O', is_open=True); O  # indirect doctest
Open subset O of the
 2-dimensional differentiable submanifold N embedded in the
  3-dimensional differentiable manifold M