Manifold Structures

These classes encode the structure of a manifold.

AUTHORS:

class sage.manifolds.structure.DegenerateStructure

Bases: sage.misc.fast_methods.Singleton

The structure of a degenerate manifold.

chart

alias of sage.manifolds.differentiable.chart.RealDiffChart

homset

alias of sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset

scalar_field_algebra

alias of sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra

subcategory(cat)

Return the subcategory of cat corresponding 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
class sage.manifolds.structure.DifferentialStructure

Bases: sage.misc.fast_methods.Singleton

The structure of a differentiable manifold over a general topological field.

chart

alias of sage.manifolds.differentiable.chart.DiffChart

homset

alias of sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset

scalar_field_algebra

alias of sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra

subcategory(cat)

Return the subcategory of cat corresponding 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
class sage.manifolds.structure.LorentzianStructure

Bases: sage.misc.fast_methods.Singleton

The structure of a Lorentzian manifold.

chart

alias of sage.manifolds.differentiable.chart.RealDiffChart

homset

alias of sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset

scalar_field_algebra

alias of sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra

subcategory(cat)

Return the subcategory of cat corresponding 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
class sage.manifolds.structure.PseudoRiemannianStructure

Bases: sage.misc.fast_methods.Singleton

The structure of a pseudo-Riemannian manifold.

chart

alias of sage.manifolds.differentiable.chart.RealDiffChart

homset

alias of sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset

scalar_field_algebra

alias of sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra

subcategory(cat)

Return the subcategory of cat corresponding 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
class sage.manifolds.structure.RealDifferentialStructure

Bases: sage.misc.fast_methods.Singleton

The structure of a differentiable manifold over \(\RR\).

chart

alias of sage.manifolds.differentiable.chart.RealDiffChart

homset

alias of sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset

scalar_field_algebra

alias of sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra

subcategory(cat)

Return the subcategory of cat corresponding 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
class sage.manifolds.structure.RealTopologicalStructure

Bases: sage.misc.fast_methods.Singleton

The structure of a topological manifold over \(\RR\).

chart

alias of sage.manifolds.chart.RealChart

homset

alias of sage.manifolds.manifold_homset.TopologicalManifoldHomset

scalar_field_algebra

alias of sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra

subcategory(cat)

Return the subcategory of cat corresponding 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
class sage.manifolds.structure.RiemannianStructure

Bases: sage.misc.fast_methods.Singleton

The structure of a Riemannian manifold.

chart

alias of sage.manifolds.differentiable.chart.RealDiffChart

homset

alias of sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset

scalar_field_algebra

alias of sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra

subcategory(cat)

Return the subcategory of cat corresponding 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
class sage.manifolds.structure.TopologicalStructure

Bases: sage.misc.fast_methods.Singleton

The structure of a topological manifold over a general topological field.

chart

alias of sage.manifolds.chart.Chart

homset

alias of sage.manifolds.manifold_homset.TopologicalManifoldHomset

scalar_field_algebra

alias of sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra

subcategory(cat)

Return the subcategory of cat corresponding 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