# Manifolds¶

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

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)

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)

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)

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)

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)

The category of differentiable manifolds.

A differentiable manifold is a manifold with a differentiable atlas.

class FiniteDimensional(base_category)

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)

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.

EXAMPLES:

sage: from sage.categories.manifolds import Manifolds

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