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]

from sage.categories.manifolds import Manifolds
Manifolds(RR).super_categories()
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]

# needs sage.rings.real_mpfr
from sage.categories.manifolds import Manifolds
C = Manifolds(RR); C
C.super_categories()
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]

from sage.categories.manifolds import Manifolds
Manifolds(RR).AlmostComplex().super_categories()  # indirect doctest          # needs sage.rings.real_mpfr
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]

from sage.categories.manifolds import Manifolds
Manifolds(RR).Analytic().super_categories()  # indirect doctest   # needs sage.rings.real_mpfr
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')

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')

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

from sage.categories.manifolds import Manifolds
M = Manifolds(RR).example()
M.dimension()
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]

from sage.categories.manifolds import Manifolds
Manifolds(RR).Smooth().super_categories()  # indirect doctest     # needs sage.rings.real_mpfr
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

from sage.categories.manifolds import Manifolds
Manifolds(RR).AlmostComplex()                                     # needs sage.rings.real_mpfr
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

from sage.categories.manifolds import Manifolds
Manifolds(RR).Analytic()                                          # needs sage.rings.real_mpfr
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

from sage.categories.manifolds import Manifolds
Manifolds(CC).Complex()                                           # needs sage.rings.real_mpfr
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

from sage.categories.manifolds import Manifolds
Manifolds(RR).Connected()                                         # needs sage.rings.real_mpfr
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

from sage.categories.manifolds import Manifolds
Manifolds(RR).Differentiable()                                    # needs sage.rings.real_mpfr
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

from sage.categories.manifolds import Manifolds
C = Manifolds(RR).Connected().FiniteDimensional(); C              # needs sage.rings.real_mpfr
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

from sage.categories.manifolds import Manifolds
Manifolds(RR).Smooth()                                            # needs sage.rings.real_mpfr
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

Category.additional_structure()

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()

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]

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