Coverage for local/lib/python2.7/site-packages/sage/categories/manifolds.py : 98%

r""" 

Manifolds 

""" 

#***************************************************************************** 

# Copyright (C) 2015 Travis Scrimshaw <tscrim at ucdavis.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#****************************************************************************** 

from sage.misc.abstract_method import abstract_method 

from sage.misc.cachefunc import cached_method 

from sage.categories.category_types import Category_over_base_ring 

from sage.categories.category_with_axiom import CategoryWithAxiom_over_base_ring 

from sage.categories.sets_cat import Sets 

from sage.categories.fields import Fields 

class Manifolds(Category_over_base_ring): 

r""" 

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: 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] 

TESTS:: 

sage: TestSuite(C).run(skip="_test_category_over_bases") 

""" 

def __init__(self, base, name=None): 

r""" 

Initialize ``self``. 

EXAMPLES:: 

sage: from sage.categories.manifolds import Manifolds 

sage: C = Manifolds(RR) 

sage: TestSuite(C).run(skip="_test_category_over_bases") 

""" 

if base not in Fields().Topological(): 

raise ValueError("base must be a topological field") 

Category_over_base_ring.__init__(self, base, name) 

@cached_method 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: from sage.categories.manifolds import Manifolds 

sage: Manifolds(RR).super_categories() 

[Category of topological spaces] 

""" 

return [Sets().Topological()] 

def additional_structure(self): 

r""" 

Return ``None``. 

Indeed, the category of manifolds defines no new 

structure: a morphism of topological spaces between 

manifolds is a manifold morphism. 

.. SEEALSO:: :meth:`Category.additional_structure` 

EXAMPLES:: 

sage: from sage.categories.manifolds import Manifolds 

sage: Manifolds(RR).additional_structure() 

""" 

return None 

class ParentMethods: 

@abstract_method 

def dimension(self): 

""" 

Return the dimension of ``self``. 

EXAMPLES:: 

sage: from sage.categories.manifolds import Manifolds 

sage: M = Manifolds(RR).example() 

sage: M.dimension() 

3 

""" 

class SubcategoryMethods: 

@cached_method 

def Connected(self): 

""" 

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 

TESTS:: 

sage: Manifolds(RR).Connected.__module__ 

'sage.categories.manifolds' 

""" 

return self._with_axiom('Connected') 

@cached_method 

def FiniteDimensional(self): 

""" 

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 

TESTS:: 

sage: from sage.categories.manifolds import Manifolds 

sage: Manifolds(RR).Connected().FiniteDimensional.__module__ 

'sage.categories.manifolds' 

""" 

return self._with_axiom('FiniteDimensional') 

@cached_method 

def Differentiable(self): 

""" 

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 

TESTS:: 

sage: TestSuite(Manifolds(RR).Differentiable()).run() 

sage: Manifolds(RR).Differentiable.__module__ 

'sage.categories.manifolds' 

""" 

return self._with_axiom('Differentiable') 

@cached_method 

def Smooth(self): 

""" 

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 

TESTS:: 

sage: TestSuite(Manifolds(RR).Smooth()).run() 

sage: Manifolds(RR).Smooth.__module__ 

'sage.categories.manifolds' 

""" 

return self._with_axiom('Smooth') 

@cached_method 

def Analytic(self): 

""" 

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 

TESTS:: 

sage: TestSuite(Manifolds(RR).Analytic()).run() 

sage: Manifolds(RR).Analytic.__module__ 

'sage.categories.manifolds' 

""" 

return self._with_axiom('Analytic') 

@cached_method 

def AlmostComplex(self): 

""" 

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 

TESTS:: 

sage: TestSuite(Manifolds(RR).AlmostComplex()).run() 

sage: Manifolds(RR).AlmostComplex.__module__ 

'sage.categories.manifolds' 

""" 

return self._with_axiom('AlmostComplex') 

@cached_method 

def Complex(self): 

""" 

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 

TESTS:: 

sage: TestSuite(Manifolds(CC).Complex()).run() 

sage: Manifolds(CC).Complex.__module__ 

'sage.categories.manifolds' 

""" 

return ComplexManifolds(self.base())._with_axioms(self.axioms()) 

class Differentiable(CategoryWithAxiom_over_base_ring): 

""" 

The category of differentiable manifolds. 

A differentiable manifold is a manifold with a differentiable atlas. 

""" 

class Smooth(CategoryWithAxiom_over_base_ring): 

""" 

The category of smooth manifolds. 

A smooth manifold is a manifold with a smooth atlas. 

""" 

def extra_super_categories(self): 

""" 

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] 

""" 

return [Manifolds(self.base()).Differentiable()] 

class Analytic(CategoryWithAxiom_over_base_ring): 

r""" 

The category of complex manifolds. 

An analytic manifold is a manifold with an analytic atlas. 

""" 

def extra_super_categories(self): 

""" 

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] 

""" 

return [Manifolds(self.base()).Smooth()] 

class AlmostComplex(CategoryWithAxiom_over_base_ring): 

r""" 

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`. 

""" 

def extra_super_categories(self): 

""" 

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] 

""" 

return [Manifolds(self.base()).Smooth()] 

class FiniteDimensional(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 Connected(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 ComplexManifolds(Category_over_base_ring): 

r""" 

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. 

""" 

@cached_method 

def super_categories(self): 

""" 

EXAMPLES:: 

sage: from sage.categories.manifolds import Manifolds 

sage: Manifolds(RR).super_categories() 

[Category of topological spaces] 

""" 

return [Manifolds(self.base()).Analytic()]