Manifolds¶
- class sage.categories.manifolds.ComplexManifolds(base, name=None)[source]¶
- Bases: - 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()[source]¶
- EXAMPLES: - sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).super_categories() [Category of topological spaces] - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).super_categories() [Category of topological spaces] 
 
- class sage.categories.manifolds.Manifolds(base, name=None)[source]¶
- Bases: - 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: # needs sage.rings.real_mpfr 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] - >>> from sage.all import * >>> # needs sage.rings.real_mpfr >>> from sage.categories.manifolds import Manifolds >>> C = Manifolds(RR); C Category of manifolds over Real Field with 53 bits of precision >>> C.super_categories() [Category of topological spaces] - class AlmostComplex(base_category)[source]¶
- Bases: - 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()[source]¶
- 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 # needs sage.rings.real_mpfr [Category of smooth manifolds over Real Field with 53 bits of precision] - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).AlmostComplex().super_categories() # indirect doctest # needs sage.rings.real_mpfr [Category of smooth manifolds over Real Field with 53 bits of precision] 
 
 - class Analytic(base_category)[source]¶
- Bases: - CategoryWithAxiom_over_base_ring- The category of complex manifolds. - An analytic manifold is a manifold with an analytic atlas. - extra_super_categories()[source]¶
- 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 # needs sage.rings.real_mpfr [Category of smooth manifolds over Real Field with 53 bits of precision] - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).Analytic().super_categories() # indirect doctest # needs sage.rings.real_mpfr [Category of smooth manifolds over Real Field with 53 bits of precision] 
 
 - class Connected(base_category)[source]¶
- Bases: - 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') - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> C = Manifolds(RR).Connected() >>> TestSuite(C).run(skip='_test_category_over_bases') 
 - class Differentiable(base_category)[source]¶
- Bases: - CategoryWithAxiom_over_base_ring- The category of differentiable manifolds. - A differentiable manifold is a manifold with a differentiable atlas. 
 - class FiniteDimensional(base_category)[source]¶
- Bases: - 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') - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> C = Manifolds(RR).FiniteDimensional() >>> TestSuite(C).run(skip='_test_category_over_bases') 
 - class ParentMethods[source]¶
- Bases: - object- dimension()[source]¶
- Return the dimension of - self.- EXAMPLES: - sage: from sage.categories.manifolds import Manifolds sage: M = Manifolds(RR).example() sage: M.dimension() 3 - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> M = Manifolds(RR).example() >>> M.dimension() 3 
 
 - class Smooth(base_category)[source]¶
- Bases: - CategoryWithAxiom_over_base_ring- The category of smooth manifolds. - A smooth manifold is a manifold with a smooth atlas. - extra_super_categories()[source]¶
- 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 # needs sage.rings.real_mpfr [Category of differentiable manifolds over Real Field with 53 bits of precision] - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).Smooth().super_categories() # indirect doctest # needs sage.rings.real_mpfr [Category of differentiable manifolds over Real Field with 53 bits of precision] 
 
 - class SubcategoryMethods[source]¶
- Bases: - object- AlmostComplex()[source]¶
- Return the subcategory of the almost complex objects of - self.- EXAMPLES: - sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).AlmostComplex() # needs sage.rings.real_mpfr Category of almost complex manifolds over Real Field with 53 bits of precision - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).AlmostComplex() # needs sage.rings.real_mpfr Category of almost complex manifolds over Real Field with 53 bits of precision 
 - Analytic()[source]¶
- Return the subcategory of the analytic objects of - self.- EXAMPLES: - sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).Analytic() # needs sage.rings.real_mpfr Category of analytic manifolds over Real Field with 53 bits of precision - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).Analytic() # needs sage.rings.real_mpfr Category of analytic manifolds over Real Field with 53 bits of precision 
 - Complex()[source]¶
- Return the subcategory of manifolds over \(\CC\) of - self.- EXAMPLES: - sage: from sage.categories.manifolds import Manifolds sage: Manifolds(CC).Complex() # needs sage.rings.real_mpfr Category of complex manifolds over Complex Field with 53 bits of precision - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(CC).Complex() # needs sage.rings.real_mpfr Category of complex manifolds over Complex Field with 53 bits of precision 
 - Connected()[source]¶
- Return the full subcategory of the connected objects of - self.- EXAMPLES: - sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).Connected() # needs sage.rings.real_mpfr Category of connected manifolds over Real Field with 53 bits of precision - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).Connected() # needs sage.rings.real_mpfr Category of connected manifolds over Real Field with 53 bits of precision 
 - Differentiable()[source]¶
- Return the subcategory of the differentiable objects of - self.- EXAMPLES: - sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).Differentiable() # needs sage.rings.real_mpfr Category of differentiable manifolds over Real Field with 53 bits of precision - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).Differentiable() # needs sage.rings.real_mpfr Category of differentiable manifolds over Real Field with 53 bits of precision 
 - FiniteDimensional()[source]¶
- 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 # needs sage.rings.real_mpfr Category of finite dimensional connected manifolds over Real Field with 53 bits of precision - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> C = Manifolds(RR).Connected().FiniteDimensional(); C # needs sage.rings.real_mpfr Category of finite dimensional connected manifolds over Real Field with 53 bits of precision 
 - Smooth()[source]¶
- Return the subcategory of the smooth objects of - self.- EXAMPLES: - sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).Smooth() # needs sage.rings.real_mpfr Category of smooth manifolds over Real Field with 53 bits of precision - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).Smooth() # needs sage.rings.real_mpfr Category of smooth manifolds over Real Field with 53 bits of precision 
 
 - additional_structure()[source]¶
- 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() - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).additional_structure() 
 - super_categories()[source]¶
- EXAMPLES: - sage: from sage.categories.manifolds import Manifolds sage: Manifolds(RR).super_categories() [Category of topological spaces] - >>> from sage.all import * >>> from sage.categories.manifolds import Manifolds >>> Manifolds(RR).super_categories() [Category of topological spaces]