Manifold Structures#

These classes encode the structure of a manifold.

AUTHORS:

Travis Scrimshaw (2015-11-25): Initial version
Eric Gourgoulhon (2015): add DifferentialStructure and RealDifferentialStructure
Eric Gourgoulhon (2018): add PseudoRiemannianStructure, RiemannianStructure and 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 cat corresponding to the structure of self.

EXAMPLES:

Sage

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

Python

>>> 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.

chart[source]#: alias of DiffChart

homset[source]#: alias of DifferentiableManifoldHomset

name = 'differentiable'#

scalar_field_algebra[source]#: alias of DiffScalarFieldAlgebra

subcategory(cat)[source]#

Return the subcategory of cat corresponding to the structure of self.

EXAMPLES:

Sage

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

Python

>>> 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 cat corresponding to the structure of self.

EXAMPLES:

Sage

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

Python

>>> 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 cat corresponding to the structure of self.

EXAMPLES:

Sage

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

Python

>>> 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 cat corresponding to the structure of self.

EXAMPLES:

Sage

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

Python

>>> 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 \(\RR\).

chart[source]#: alias of RealChart

homset[source]#: alias of TopologicalManifoldHomset

name = 'topological'#

scalar_field_algebra[source]#: alias of ScalarFieldAlgebra

subcategory(cat)[source]#

Return the subcategory of cat corresponding to the structure of self.

EXAMPLES:

Sage

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

Python

>>> 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 cat corresponding to the structure of self.

EXAMPLES:

Sage

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

Python

>>> 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.

chart[source]#: alias of Chart

homset[source]#: alias of TopologicalManifoldHomset

name = 'topological'#

scalar_field_algebra[source]#: alias of ScalarFieldAlgebra

subcategory(cat)[source]#

Return the subcategory of cat corresponding to the structure of self.

EXAMPLES:

Sage

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

Python

>>> 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