# Manifold Structures#

These classes encode the structure of a manifold.

AUTHORS:

class sage.manifolds.structure.DegenerateStructure#

Bases: Singleton

The structure of a degenerate manifold.

chart#

alias of RealDiffChart

homset#
name = 'degenerate_metric'#
scalar_field_algebra#

alias of 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: Singleton

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

chart#

alias of DiffChart

homset#
name = 'differentiable'#
scalar_field_algebra#

alias of 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: Singleton

The structure of a Lorentzian manifold.

chart#

alias of RealDiffChart

homset#
name = 'Lorentzian'#
scalar_field_algebra#

alias of 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: Singleton

The structure of a pseudo-Riemannian manifold.

chart#

alias of RealDiffChart

homset#
name = 'pseudo-Riemannian'#
scalar_field_algebra#

alias of 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: Singleton

The structure of a differentiable manifold over $$\RR$$.

chart#

alias of RealDiffChart

homset#
name = 'differentiable'#
scalar_field_algebra#

alias of 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: Singleton

The structure of a topological manifold over $$\RR$$.

chart#

alias of RealChart

homset#
name = 'topological'#
scalar_field_algebra#

alias of 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: Singleton

The structure of a Riemannian manifold.

chart#

alias of RealDiffChart

homset#
name = 'Riemannian'#
scalar_field_algebra#

alias of 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: Singleton

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

chart#

alias of Chart

homset#
name = 'topological'#
scalar_field_algebra#

alias of 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