# Manifold Structures¶

These classes encode the structure of a manifold.

AUTHORS:

class sage.manifolds.structure.DifferentialStructure

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

chart
homset
scalar_field_algebra
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

The structure of a Lorentzian manifold.

chart
homset
scalar_field_algebra
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

The structure of a pseudo-Riemannian manifold.

chart
homset
scalar_field_algebra
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

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

chart
homset
scalar_field_algebra
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

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

chart
homset
scalar_field_algebra
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

The structure of a Riemannian manifold.

chart
homset
scalar_field_algebra
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

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

chart
homset
scalar_field_algebra
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