Coverage for local/lib/python2.7/site-packages/sage/manifolds/structure.py : 100%

r""" 

Manifold Structures 

These classes encode the structure of a manifold. 

AUTHORS: 

- Travis Scrimshaw (2015-11-25): Initial version 

- Eric Gourgoulhon (2015): add :class:`DifferentialStructure` and 

:class:`RealDifferentialStructure` 

- Eric Gourgoulhon (2018): add :class:`PseudoRiemannianStructure`, 

:class:`RiemannianStructure` and :class:`LorentzianStructure` 

""" 

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

# Copyright (C) 2015, 2018 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr> 

# Copyright (C) 2015 Travis Scrimshaw <tscrimsh at umn.edu> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

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

from sage.misc.fast_methods import Singleton 

from sage.manifolds.chart import Chart, RealChart 

from sage.manifolds.scalarfield_algebra import ScalarFieldAlgebra 

from sage.manifolds.manifold_homset import TopologicalManifoldHomset 

from sage.manifolds.differentiable.chart import DiffChart, RealDiffChart 

from sage.manifolds.differentiable.scalarfield_algebra import \ 

DiffScalarFieldAlgebra 

from sage.manifolds.differentiable.manifold_homset import \ 

DifferentiableManifoldHomset 

# This is a slight abuse by making this a Singleton, but there is no 

# need to have different copies of this object. 

class TopologicalStructure(Singleton): 

""" 

The structure of a topological manifold over a general topological field. 

""" 

chart = Chart 

name = "topological" 

scalar_field_algebra = ScalarFieldAlgebra 

homset = TopologicalManifoldHomset 

def subcategory(self, cat): 

""" 

Return the subcategory of ``cat`` corresponding to the structure 

of ``self``. 

EXAMPLES:: 

sage: from sage.manifolds.structure import TopologicalStructure 

sage: from sage.categories.manifolds import Manifolds 

sage: TopologicalStructure().subcategory(Manifolds(RR)) 

Category of manifolds over Real Field with 53 bits of precision 

""" 

return cat 

class RealTopologicalStructure(Singleton): 

""" 

The structure of a topological manifold over `\RR`. 

""" 

chart = RealChart 

name = "topological" 

scalar_field_algebra = ScalarFieldAlgebra 

homset = TopologicalManifoldHomset 

def subcategory(self, cat): 

""" 

Return the subcategory of ``cat`` corresponding to the structure 

of ``self``. 

EXAMPLES:: 

sage: from sage.manifolds.structure import RealTopologicalStructure 

sage: from sage.categories.manifolds import Manifolds 

sage: RealTopologicalStructure().subcategory(Manifolds(RR)) 

Category of manifolds over Real Field with 53 bits of precision 

""" 

return cat 

class DifferentialStructure(Singleton): 

""" 

The structure of a differentiable manifold over a general topological 

field. 

""" 

chart = DiffChart 

name = "differentiable" 

scalar_field_algebra = DiffScalarFieldAlgebra 

homset = DifferentiableManifoldHomset 

def subcategory(self, cat): 

""" 

Return the subcategory of ``cat`` corresponding to the structure 

of ``self``. 

EXAMPLES:: 

sage: from sage.manifolds.structure import DifferentialStructure 

sage: from sage.categories.manifolds import Manifolds 

sage: DifferentialStructure().subcategory(Manifolds(RR)) 

Category of manifolds over Real Field with 53 bits of precision 

""" 

return cat 

class RealDifferentialStructure(Singleton): 

""" 

The structure of a differentiable manifold over `\RR`. 

""" 

chart = RealDiffChart 

name = "differentiable" 

scalar_field_algebra = DiffScalarFieldAlgebra 

homset = DifferentiableManifoldHomset 

def subcategory(self, cat): 

""" 

Return the subcategory of ``cat`` corresponding to the structure 

of ``self``. 

EXAMPLES:: 

sage: from sage.manifolds.structure import RealDifferentialStructure 

sage: from sage.categories.manifolds import Manifolds 

sage: RealDifferentialStructure().subcategory(Manifolds(RR)) 

Category of manifolds over Real Field with 53 bits of precision 

""" 

return cat 

class PseudoRiemannianStructure(Singleton): 

""" 

The structure of a pseudo-Riemannian manifold. 

""" 

chart = RealDiffChart 

name = "pseudo-Riemannian" 

scalar_field_algebra = DiffScalarFieldAlgebra 

homset = DifferentiableManifoldHomset 

def subcategory(self, cat): 

""" 

Return the subcategory of ``cat`` corresponding to the structure 

of ``self``. 

EXAMPLES:: 

sage: from sage.manifolds.structure import PseudoRiemannianStructure 

sage: from sage.categories.manifolds import Manifolds 

sage: PseudoRiemannianStructure().subcategory(Manifolds(RR)) 

Category of manifolds over Real Field with 53 bits of precision 

""" 

return cat 

class RiemannianStructure(Singleton): 

""" 

The structure of a Riemannian manifold. 

""" 

chart = RealDiffChart 

name = "Riemannian" 

scalar_field_algebra = DiffScalarFieldAlgebra 

homset = DifferentiableManifoldHomset 

def subcategory(self, cat): 

""" 

Return the subcategory of ``cat`` corresponding to the structure 

of ``self``. 

EXAMPLES:: 

sage: from sage.manifolds.structure import RiemannianStructure 

sage: from sage.categories.manifolds import Manifolds 

sage: RiemannianStructure().subcategory(Manifolds(RR)) 

Category of manifolds over Real Field with 53 bits of precision 

""" 

return cat 

class LorentzianStructure(Singleton): 

""" 

The structure of a Lorentzian manifold. 

""" 

chart = RealDiffChart 

name = "Lorentzian" 

scalar_field_algebra = DiffScalarFieldAlgebra 

homset = DifferentiableManifoldHomset 

def subcategory(self, cat): 

""" 

Return the subcategory of ``cat`` corresponding to the structure 

of ``self``. 

EXAMPLES:: 

sage: from sage.manifolds.structure import LorentzianStructure 

sage: from sage.categories.manifolds import Manifolds 

sage: LorentzianStructure().subcategory(Manifolds(RR)) 

Category of manifolds over Real Field with 53 bits of precision 

""" 

return cat