Manifolds¶
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class
sage.categories.manifolds.ComplexManifolds(base, name=None)¶ Bases:
sage.categories.category_types.Category_over_base_ringThe category of complex manifolds.
A \(d\)-dimensional complex manifold is a manifold whose underlying vector space is \(\CC^d\) and has a holomorphic atlas.
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super_categories()¶ EXAMPLES:
sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).super_categories() [Category of topological spaces]
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class
sage.categories.manifolds.Manifolds(base, name=None)¶ Bases:
sage.categories.category_types.Category_over_base_ringThe 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]
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class
AlmostComplex(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ringThe 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 \(M\).
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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]
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class
Analytic(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ringThe category of complex manifolds.
An analytic manifold is a manifold with an analytic atlas.
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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]
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class
Connected(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ringThe 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")
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class
Differentiable(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ringThe category of differentiable manifolds.
A differentiable manifold is a manifold with a differentiable atlas.
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class
FiniteDimensional(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ringCategory 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")
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class
ParentMethods¶ Bases:
object-
dimension()¶ Return the dimension of
self.EXAMPLES:
sage: from sage.categories.manifolds import Manifolds sage: M = Manifolds(RR).example() sage: M.dimension() 3
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class
Smooth(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ringThe category of smooth manifolds.
A smooth manifold is a manifold with a smooth atlas.
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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]
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class
SubcategoryMethods¶ Bases:
object-
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
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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
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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
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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
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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
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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
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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
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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()
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super_categories()¶ EXAMPLES:
sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).super_categories() [Category of topological spaces]
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class