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]

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
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
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
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
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
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
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
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

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

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

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

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

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

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


additional_structure
()¶ Return
None
.Indeed, the category of manifolds defines no new structure: a morphism of topological spaces between manifolds is a manifold morphism.
See also
EXAMPLES:
sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).additional_structure()

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

class