Manifold Structures#
These classes encode the structure of a manifold.
AUTHORS:
Travis Scrimshaw (2015-11-25): Initial version
Eric Gourgoulhon (2015): add
DifferentialStructure
andRealDifferentialStructure
Eric Gourgoulhon (2018): add
PseudoRiemannianStructure
,RiemannianStructure
andLorentzianStructure
- class sage.manifolds.structure.DegenerateStructure#
Bases:
Singleton
The structure of a degenerate manifold.
- chart#
alias of
RealDiffChart
- homset#
alias of
DifferentiableManifoldHomset
- name = 'degenerate_metric'#
- scalar_field_algebra#
alias of
DiffScalarFieldAlgebra
- subcategory(cat)#
Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import DegenerateStructure sage: from sage.categories.manifolds import Manifolds sage: DegenerateStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
- class sage.manifolds.structure.DifferentialStructure#
Bases:
Singleton
The structure of a differentiable manifold over a general topological field.
- homset#
alias of
DifferentiableManifoldHomset
- name = 'differentiable'#
- scalar_field_algebra#
alias of
DiffScalarFieldAlgebra
- subcategory(cat)#
Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import DifferentialStructure sage: from sage.categories.manifolds import Manifolds sage: DifferentialStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
- class sage.manifolds.structure.LorentzianStructure#
Bases:
Singleton
The structure of a Lorentzian manifold.
- chart#
alias of
RealDiffChart
- homset#
alias of
DifferentiableManifoldHomset
- name = 'Lorentzian'#
- scalar_field_algebra#
alias of
DiffScalarFieldAlgebra
- subcategory(cat)#
Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import LorentzianStructure sage: from sage.categories.manifolds import Manifolds sage: LorentzianStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
- class sage.manifolds.structure.PseudoRiemannianStructure#
Bases:
Singleton
The structure of a pseudo-Riemannian manifold.
- chart#
alias of
RealDiffChart
- homset#
alias of
DifferentiableManifoldHomset
- name = 'pseudo-Riemannian'#
- scalar_field_algebra#
alias of
DiffScalarFieldAlgebra
- subcategory(cat)#
Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import PseudoRiemannianStructure sage: from sage.categories.manifolds import Manifolds sage: PseudoRiemannianStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
- class sage.manifolds.structure.RealDifferentialStructure#
Bases:
Singleton
The structure of a differentiable manifold over \(\RR\).
- chart#
alias of
RealDiffChart
- homset#
alias of
DifferentiableManifoldHomset
- name = 'differentiable'#
- scalar_field_algebra#
alias of
DiffScalarFieldAlgebra
- subcategory(cat)#
Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import RealDifferentialStructure sage: from sage.categories.manifolds import Manifolds sage: RealDifferentialStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
- class sage.manifolds.structure.RealTopologicalStructure#
Bases:
Singleton
The structure of a topological manifold over \(\RR\).
- homset#
alias of
TopologicalManifoldHomset
- name = 'topological'#
- scalar_field_algebra#
alias of
ScalarFieldAlgebra
- subcategory(cat)#
Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import RealTopologicalStructure sage: from sage.categories.manifolds import Manifolds sage: RealTopologicalStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
- class sage.manifolds.structure.RiemannianStructure#
Bases:
Singleton
The structure of a Riemannian manifold.
- chart#
alias of
RealDiffChart
- homset#
alias of
DifferentiableManifoldHomset
- name = 'Riemannian'#
- scalar_field_algebra#
alias of
DiffScalarFieldAlgebra
- subcategory(cat)#
Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import RiemannianStructure sage: from sage.categories.manifolds import Manifolds sage: RiemannianStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision
- class sage.manifolds.structure.TopologicalStructure#
Bases:
Singleton
The structure of a topological manifold over a general topological field.
- homset#
alias of
TopologicalManifoldHomset
- name = 'topological'#
- scalar_field_algebra#
alias of
ScalarFieldAlgebra
- subcategory(cat)#
Return the subcategory of
cat
corresponding to the structure ofself
.EXAMPLES:
sage: from sage.manifolds.structure import TopologicalStructure sage: from sage.categories.manifolds import Manifolds sage: TopologicalStructure().subcategory(Manifolds(RR)) Category of manifolds over Real Field with 53 bits of precision