Manifolds - Category Framework

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# 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

        sage: from sage.categories.manifolds import Manifolds
        sage: Manifolds(RR).super_categories()
        [Category of topological spaces]

    Python

        >>> 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)

  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

      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]

  Python

      >>> 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)

    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

          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]

      Python

          >>> 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)

    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

          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]

      Python

          >>> 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)

    Bases: `CategoryWithAxiom_over_base_ring`

    The category of connected manifolds.

    EXAMPLES:

    Sage

        sage: from sage.categories.manifolds import Manifolds
        sage: C = Manifolds(RR).Connected()
        sage: TestSuite(C).run(skip='_test_category_over_bases')

    Python

        >>> 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)

    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

        sage: from sage.categories.manifolds import Manifolds
        sage: C = Manifolds(RR).FiniteDimensional()
        sage: TestSuite(C).run(skip='_test_category_over_bases')

    Python

        >>> from sage.all import *
        >>> from sage.categories.manifolds import Manifolds
        >>> C = Manifolds(RR).FiniteDimensional()
        >>> TestSuite(C).run(skip='_test_category_over_bases')

  class ParentMethods

    Bases: `object`

    dimension()

      Return the dimension of `self`.

      EXAMPLES:

      Sage

          sage: from sage.categories.manifolds import Manifolds
          sage: M = Manifolds(RR).example()
          sage: M.dimension()
          3

      Python

          >>> from sage.all import *
          >>> from sage.categories.manifolds import Manifolds
          >>> M = Manifolds(RR).example()
          >>> 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

          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]

      Python

          >>> 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

    Bases: `object`

    AlmostComplex()

      Return the subcategory of the almost complex objects of `self`.

      EXAMPLES:

      Sage

          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

      Python

          >>> 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()

      Return the subcategory of the analytic objects of `self`.

      EXAMPLES:

      Sage

          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

      Python

          >>> 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()

      Return the subcategory of manifolds over \(\CC\) of `self`.

      EXAMPLES:

      Sage

          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

      Python

          >>> 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()

      Return the full subcategory of the connected objects of `self`.

      EXAMPLES:

      Sage

          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

      Python

          >>> 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()

      Return the subcategory of the differentiable objects of `self`.

      EXAMPLES:

      Sage

          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

      Python

          >>> 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()

      Return the full subcategory of the finite dimensional objects of `self`.

      EXAMPLES:

      Sage

          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

      Python

          >>> 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()

      Return the subcategory of the smooth objects of `self`.

      EXAMPLES:

      Sage

          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

      Python

          >>> 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()

    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

        sage: from sage.categories.manifolds import Manifolds
        sage: Manifolds(RR).additional_structure()

    Python

        >>> from sage.all import *
        >>> from sage.categories.manifolds import Manifolds
        >>> Manifolds(RR).additional_structure()

  super_categories()

    EXAMPLES:

    Sage

        sage: from sage.categories.manifolds import Manifolds
        sage: Manifolds(RR).super_categories()
        [Category of topological spaces]

    Python

        >>> from sage.all import *
        >>> from sage.categories.manifolds import Manifolds
        >>> Manifolds(RR).super_categories()
        [Category of topological spaces]