Manifolds#
- class sage.categories.manifolds.ComplexManifolds(base, name=None)#
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()#
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:
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]
- class AlmostComplex(base_category)#
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()#
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]
- class Analytic(base_category)#
Bases:
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 # needs sage.rings.real_mpfr [Category of smooth manifolds over Real Field with 53 bits of precision]
- class Connected(base_category)#
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")
- class Differentiable(base_category)#
Bases:
CategoryWithAxiom_over_base_ring
The category of differentiable manifolds.
A differentiable manifold is a manifold with a differentiable atlas.
- class FiniteDimensional(base_category)#
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")
- 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
- class Smooth(base_category)#
Bases:
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 # needs sage.rings.real_mpfr [Category of differentiable manifolds over Real Field with 53 bits of precision]
- 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() # needs sage.rings.real_mpfr 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() # needs sage.rings.real_mpfr 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() # needs sage.rings.real_mpfr 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() # needs sage.rings.real_mpfr 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() # needs sage.rings.real_mpfr 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 # needs sage.rings.real_mpfr 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() # needs sage.rings.real_mpfr 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]