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