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egourgoulhonegourgoulhon
committedMay 6, 2016
Python3 format for print in morphims of topological manifolds
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Diff for: ‎src/doc/en/reference/manifolds/continuous_map.rst

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Continuous Maps
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===============
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.. toctree::
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:maxdepth: 2
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sage/manifolds/manifold_homset
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sage/manifolds/continuous_map

Diff for: ‎src/doc/en/reference/manifolds/manifold.rst

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@@ -15,3 +15,5 @@ Topological Manifolds
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chart
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scalarfield
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continuous_map

Diff for: ‎src/sage/manifolds/chart.py

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Diff for: ‎src/sage/manifolds/continuous_map.py

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Diff for: ‎src/sage/manifolds/manifold.py

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from sage.categories.fields import Fields
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from sage.categories.manifolds import Manifolds
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from sage.categories.homset import Hom
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from sage.rings.all import CC
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from sage.rings.real_mpfr import RR, RealField_class
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from sage.rings.complex_field import ComplexField_class
@@ -643,6 +644,10 @@ def __init__(self, n, name, field, structure, ambient=None,
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self._zero_scalar_field = self._scalar_field_algebra.zero()
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# The unit scalar field:
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self._one_scalar_field = self._scalar_field_algebra.one()
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# Dictionary of sets of morphisms to over manifolds (keys: codomains):
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self._homsets = {} # to be populated by self._Hom_
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# The identity map on self:
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self._identity_map = Hom(self, self).one()
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def _repr_(self):
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r"""
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global_options = ManifoldOptions
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def _Hom_(self, other, category=None):
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r"""
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Construct the set of morphisms (i.e. continuous maps)
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``self`` --> ``other``.
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INPUT:
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- ``other`` -- an open subset of some topological manifold over the
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same field as ``self``
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- ``category`` -- (default: ``None``) not used here (to ensure
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compatibility with generic hook ``_Hom_``)
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OUTPUT:
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- the homset Hom(U,V), where U is ``self`` and V is ``other``
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See class
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:class:`~sage.manifolds.manifold_homset.TopologicalManifoldHomset`
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for more documentation.
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TESTS::
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sage: M = Manifold(2, 'M', structure='topological')
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sage: N = Manifold(3, 'N', structure='topological')
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sage: H = M._Hom_(N); H
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Set of Morphisms from 2-dimensional topological manifold M to
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3-dimensional topological manifold N in Category of manifolds over
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Real Field with 53 bits of precision
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The result is cached::
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sage: H is Hom(M, N)
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True
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"""
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if other not in self._homsets:
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self._homsets[other] = self._structure.homset(self, other)
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return self._homsets[other]
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def continuous_map(self, codomain, coord_functions=None, chart1=None,
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chart2=None, name=None, latex_name=None):
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r"""
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Define a continuous map between the current topological manifold
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and another topological manifold over the same topological field.
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See :class:`~sage.manifolds.continuous_map.ContinuousMap` for a
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complete documentation.
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INPUT:
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- ``codomain`` -- the map's codomain (must be an instance
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of :class:`TopologicalManifold`)
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- ``coord_functions`` -- (default: ``None``) if not ``None``, must be
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either
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- (i) a dictionary of
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the coordinate expressions (as lists (or tuples) of the
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coordinates of the image expressed in terms of the coordinates of
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the considered point) with the pairs of charts (chart1, chart2)
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as keys (chart1 being a chart on ``self`` and chart2 a chart on
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``codomain``)
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- (ii) a single coordinate expression in a given pair of charts, the
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latter being provided by the arguments ``chart1`` and ``chart2``
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In both cases, if the dimension of the codomain is 1, a single
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coordinate expression can be passed instead of a tuple with a single
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element
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- ``chart1`` -- (default: ``None``; used only in case (ii) above) chart
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on the current manifold defining the start coordinates involved in
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``coord_functions`` for case (ii); if none is provided, the
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coordinates are assumed to refer to the manifold's default chart
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- ``chart2`` -- (default: ``None``; used only in case (ii) above) chart
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on ``codomain`` defining the target coordinates involved in
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``coord_functions`` for case (ii); if none is provided, the
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coordinates are assumed to refer to the default chart of ``codomain``
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- ``name`` -- (default: ``None``) name given to the continuous
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map
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- ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the
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continuous map; if none is provided, the LaTeX symbol is set to
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``name``
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OUTPUT:
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- the continuous map, as an instance of
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:class:`~sage.manifolds.continuous_map.ContinuousMap`
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EXAMPLES:
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A continuous map between an open subset of `S^2` covered by regular
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spherical coordinates and `\RR^3`::
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sage: M = Manifold(2, 'S^2', structure='topological')
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sage: U = M.open_subset('U')
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sage: c_spher.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
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sage: N = Manifold(3, 'R^3', latex_name=r'\RR^3', structure='topological')
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sage: c_cart.<x,y,z> = N.chart() # Cartesian coord. on R^3
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sage: Phi = U.continuous_map(N, (sin(th)*cos(ph), sin(th)*sin(ph), cos(th)),
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....: name='Phi', latex_name=r'\Phi')
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sage: Phi
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Continuous map Phi from the Open subset U of the 2-dimensional topological manifold S^2 to the 3-dimensional topological manifold R^3
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The same definition, but with a dictionary with pairs of charts as
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keys (case (i) above)::
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sage: Phi1 = U.continuous_map(N,
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....: {(c_spher, c_cart): (sin(th)*cos(ph), sin(th)*sin(ph), cos(th))},
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....: name='Phi', latex_name=r'\Phi')
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sage: Phi1 == Phi
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True
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The continuous map acting on a point::
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sage: p = U.point((pi/2, pi)) ; p
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Point on the 2-dimensional topological manifold S^2
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sage: Phi(p)
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Point on the 3-dimensional topological manifold R^3
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sage: Phi(p).coord(c_cart)
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(-1, 0, 0)
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sage: Phi1(p) == Phi(p)
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True
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See the documentation of class
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:class:`~sage.manifolds.continuous_map.ContinuousMap` for more
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examples.
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"""
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homset = Hom(self, codomain)
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if coord_functions is None:
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coord_functions = {}
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if not isinstance(coord_functions, dict):
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# Turn coord_functions into a dictionary:
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if chart1 is None:
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chart1 = self._def_chart
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elif chart1 not in self._atlas:
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raise ValueError("{} is not a chart ".format(chart1) +
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"defined on the {}".format(self))
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if chart2 is None:
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chart2 = codomain._def_chart
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elif chart2 not in codomain._atlas:
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raise ValueError("{} is not a chart ".format(chart2) +
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" defined on the {}".format(codomain))
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coord_functions = {(chart1, chart2): coord_functions}
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return homset(coord_functions, name=name, latex_name=latex_name)
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def homeomorphism(self, codomain, coord_functions=None, chart1=None,
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chart2=None, name=None, latex_name=None):
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r"""
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Define a homeomorphism between the current manifold and another one.
1986+
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See :class:`~sage.manifolds.continuous_map.ContinuousMap` for a
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complete documentation.
1989+
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INPUT:
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- ``codomain`` -- codomain of the homeomorphism (must be an instance
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of :class:`TopologicalManifold`)
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- ``coord_functions`` -- (default: ``None``) if not ``None``, must be
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either
1996+
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- (i) a dictionary of
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the coordinate expressions (as lists (or tuples) of the
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coordinates of the image expressed in terms of the coordinates of
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the considered point) with the pairs of charts (chart1, chart2)
2001+
as keys (chart1 being a chart on ``self`` and chart2 a chart on
2002+
``codomain``)
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- (ii) a single coordinate expression in a given pair of charts, the
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latter being provided by the arguments ``chart1`` and ``chart2``
2005+
2006+
In both cases, if the dimension of the codomain is 1, a single
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coordinate expression can be passed instead of a tuple with
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a single element
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- ``chart1`` -- (default: ``None``; used only in case (ii) above) chart
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on the current manifold defining the start coordinates involved in
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``coord_functions`` for case (ii); if none is provided, the
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coordinates are assumed to refer to the manifold's default chart
2013+
- ``chart2`` -- (default: ``None``; used only in case (ii) above) chart
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on ``codomain`` defining the target coordinates involved in
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``coord_functions`` for case (ii); if none is provided, the
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coordinates are assumed to refer to the default chart of ``codomain``
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- ``name`` -- (default: ``None``) name given to the homeomorphism
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- ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the
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homeomorphism; if none is provided, the LaTeX symbol is set to
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``name``
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OUTPUT:
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2024+
- the homeomorphism, as an instance of
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:class:`~sage.manifolds.continuous_map.ContinuousMap`
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EXAMPLE:
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2029+
Homeomorphism between the open unit disk in `\RR^2` and `\RR^2`::
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2031+
sage: forget() # for doctests only
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sage: M = Manifold(2, 'M', structure='topological') # the open unit disk
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sage: c_xy.<x,y> = M.chart('x:(-1,1) y:(-1,1)') # Cartesian coord on M
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sage: c_xy.add_restrictions(x^2+y^2<1)
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sage: N = Manifold(2, 'N', structure='topological') # R^2
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sage: c_XY.<X,Y> = N.chart() # canonical coordinates on R^2
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sage: Phi = M.homeomorphism(N, [x/sqrt(1-x^2-y^2), y/sqrt(1-x^2-y^2)],
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....: name='Phi', latex_name=r'\Phi')
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sage: Phi
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Homeomorphism Phi from the 2-dimensional topological manifold M to
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the 2-dimensional topological manifold N
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sage: Phi.display()
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Phi: M --> N
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(x, y) |--> (X, Y) = (x/sqrt(-x^2 - y^2 + 1), y/sqrt(-x^2 - y^2 + 1))
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The inverse homeomorphism::
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sage: Phi^(-1)
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Homeomorphism Phi^(-1) from the 2-dimensional topological
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manifold N to the 2-dimensional topological manifold M
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sage: (Phi^(-1)).display()
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Phi^(-1): N --> M
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(X, Y) |--> (x, y) = (X/sqrt(X^2 + Y^2 + 1), Y/sqrt(X^2 + Y^2 + 1))
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See the documentation of class
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:class:`~sage.manifolds.continuous_map.ContinuousMap` for more
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examples.
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"""
2060+
homset = Hom(self, codomain)
2061+
if coord_functions is None:
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coord_functions = {}
2063+
if not isinstance(coord_functions, dict):
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# Turn coord_functions into a dictionary:
2065+
if chart1 is None:
2066+
chart1 = self._def_chart
2067+
elif chart1 not in self._atlas:
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raise ValueError("{} is not a chart ".format(chart1) +
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"defined on the {}".format(self))
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if chart2 is None:
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chart2 = codomain._def_chart
2072+
elif chart2 not in codomain._atlas:
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raise ValueError("{} is not a chart ".format(chart2) +
2074+
" defined on the {}".format(codomain))
2075+
coord_functions = {(chart1, chart2): coord_functions}
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return homset(coord_functions, name=name, latex_name=latex_name,
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is_isomorphism=True)
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def identity_map(self):
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r"""
2081+
Identity map of the manifold.
2082+
2083+
The identity map of a topological manifold `M` is the trivial
2084+
homeomorphism
2085+
2086+
.. MATH::
2087+
2088+
\begin{array}{cccc}
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\mathrm{Id}_M: & M & \longrightarrow & M \\
2090+
& p & \longmapsto & p
2091+
\end{array}
2092+
2093+
See :class:`~sage.manifolds.continuous_map.ContinuousMap` for a
2094+
complete documentation.
2095+
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OUTPUT:
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- the identity map, as an instance of
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:class:`~sage.manifolds.continuous_map.ContinuousMap`
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2101+
EXAMPLE:
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2103+
Identity map of a complex manifold::
2104+
2105+
sage: M = Manifold(2, 'M', structure='topological', field='complex')
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sage: X.<x,y> = M.chart()
2107+
sage: id = M.identity_map(); id
2108+
Identity map Id_M of the Complex 2-dimensional topological manifold M
2109+
sage: id.parent()
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Set of Morphisms from Complex 2-dimensional topological manifold M
2111+
to Complex 2-dimensional topological manifold M in Category of
2112+
manifolds over Complex Field with 53 bits of precision
2113+
sage: id.display()
2114+
Id_M: M --> M
2115+
(x, y) |--> (x, y)
2116+
2117+
The identity map acting on a point::
2118+
2119+
sage: p = M((1+I, 3-I), name='p'); p
2120+
Point p on the Complex 2-dimensional topological manifold M
2121+
sage: id(p)
2122+
Point p on the Complex 2-dimensional topological manifold M
2123+
sage: id(p) == p
2124+
True
2125+
2126+
"""
2127+
return self._identity_map
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18332129
##############################################################################
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## Constructor function
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