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 - Fieldsand- TopologicalSpaces) for other types of manifolds
 
- structure– manifold structure (see- TopologicalStructureor- 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- UniqueRepresentationbehavior inherited from- ManifoldSubsetvia- DifferentiableManifoldwould 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 - 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 - selfis 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_defmust 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: the open subset, as an instance of - DifferentiableSubmanifold- 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