Manifolds

class sage.categories.manifolds.ComplexManifolds(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring

The category of complex manifolds.

A \(d\)-dimensional complex manifold is a manifold whose underlying vector space is \(\CC^d\) and has a holomorphic atlas.

super_categories()

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).super_categories()
[Category of topological spaces]
class sage.categories.manifolds.Manifolds(base, name=None)

Bases: sage.categories.category_types.Category_over_base_ring

The category of manifolds over any topological field.

Let \(k\) be a topological field. A \(d\)-dimensional \(k\)-manifold \(M\) is a second countable Hausdorff space such that the neighborhood of any point \(x \in M\) is homeomorphic to \(k^d\).

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: C = Manifolds(RR); C
Category of manifolds over Real Field with 53 bits of precision
sage: C.super_categories()
[Category of topological spaces]

TESTS:

sage: TestSuite(C).run(skip="_test_category_over_bases")
class AlmostComplex(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

The category of almost complex manifolds.

An almost complex manifold \(M\) is a manifold with a smooth tensor field \(J\) of rank \((1, 1)\) such that \(J^2 = -1\) when regarded as a vector bundle isomorphism \(J : TM \to TM\) on the tangent bundle. The tensor field \(J\) is called the almost complex structure of \(M\).

extra_super_categories()

Return the extra super categories of self.

An almost complex manifold is smooth.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).AlmostComplex().super_categories() # indirect doctest
[Category of smooth manifolds
 over Real Field with 53 bits of precision]
class Manifolds.Analytic(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

The category of complex manifolds.

An analytic manifold is a manifold with an analytic atlas.

extra_super_categories()

Return the extra super categories of self.

An analytic manifold is smooth.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).Analytic().super_categories() # indirect doctest
[Category of smooth manifolds
 over Real Field with 53 bits of precision]
class Manifolds.Connected(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

The category of connected manifolds.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: C = Manifolds(RR).Connected()
sage: TestSuite(C).run(skip="_test_category_over_bases")
class Manifolds.Differentiable(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

The category of differentiable manifolds.

A differentiable manifold is a manifold with a differentiable atlas.

class Manifolds.FiniteDimensional(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

Category of finite dimensional manifolds.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: C = Manifolds(RR).FiniteDimensional()
sage: TestSuite(C).run(skip="_test_category_over_bases")
class Manifolds.ParentMethods
dimension()

Return the dimension of self.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: M = Manifolds(RR).example()
sage: M.dimension()
3
class Manifolds.Smooth(base_category)

Bases: sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring

The category of smooth manifolds.

A smooth manifold is a manifold with a smooth atlas.

extra_super_categories()

Return the extra super categories of self.

A smooth manifold is differentiable.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).Smooth().super_categories() # indirect doctest
[Category of differentiable manifolds
 over Real Field with 53 bits of precision]
class Manifolds.SubcategoryMethods
AlmostComplex()

Return the subcategory of the almost complex objects of self.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).AlmostComplex()
Category of almost complex manifolds
 over Real Field with 53 bits of precision

TESTS:

sage: TestSuite(Manifolds(RR).AlmostComplex()).run()
sage: Manifolds(RR).AlmostComplex.__module__
'sage.categories.manifolds'
Analytic()

Return the subcategory of the analytic objects of self.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).Analytic()
Category of analytic manifolds
 over Real Field with 53 bits of precision

TESTS:

sage: TestSuite(Manifolds(RR).Analytic()).run()
sage: Manifolds(RR).Analytic.__module__
'sage.categories.manifolds'
Complex()

Return the subcategory of manifolds over \(\CC\) of self.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(CC).Complex()
Category of complex manifolds over
 Complex Field with 53 bits of precision

TESTS:

sage: TestSuite(Manifolds(CC).Complex()).run()
sage: Manifolds(CC).Complex.__module__
'sage.categories.manifolds'
Connected()

Return the full subcategory of the connected objects of self.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).Connected()
Category of connected manifolds
 over Real Field with 53 bits of precision

TESTS:

sage: Manifolds(RR).Connected.__module__
'sage.categories.manifolds'
Differentiable()

Return the subcategory of the differentiable objects of self.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).Differentiable()
Category of differentiable manifolds
 over Real Field with 53 bits of precision

TESTS:

sage: TestSuite(Manifolds(RR).Differentiable()).run()
sage: Manifolds(RR).Differentiable.__module__
'sage.categories.manifolds'
FiniteDimensional()

Return the full subcategory of the finite dimensional objects of self.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: C = Manifolds(RR).Connected().FiniteDimensional(); C
Category of finite dimensional connected manifolds
 over Real Field with 53 bits of precision

TESTS:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).Connected().FiniteDimensional.__module__
'sage.categories.manifolds'
Smooth()

Return the subcategory of the smooth objects of self.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).Smooth()
Category of smooth manifolds
 over Real Field with 53 bits of precision

TESTS:

sage: TestSuite(Manifolds(RR).Smooth()).run()
sage: Manifolds(RR).Smooth.__module__
'sage.categories.manifolds'
Manifolds.additional_structure()

Return None.

Indeed, the category of manifolds defines no new structure: a morphism of topological spaces between manifolds is a manifold morphism.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).additional_structure()
Manifolds.super_categories()

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds
sage: Manifolds(RR).super_categories()
[Category of topological spaces]