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r"""

Topological Manifolds

Given a topological field `K` (in most applications, `K = \RR` or

`K = \CC`) and a non-negative integer `n`, a *topological manifold of

dimension* `n` *over K* is a topological space `M` such that

- `M` is a Hausdorff space,

- `M` is second countable,

- every point in `M` has a neighborhood homeomorphic to `K^n`.

Topological manifolds are implemented via the class

:class:`TopologicalManifold`. Open subsets of topological manifolds

are also implemented via :class:`TopologicalManifold`, since they are

topological manifolds by themselves.

In the current setting, topological manifolds are mostly described by

means of charts (see :class:`~sage.manifolds.chart.Chart`).

:class:`TopologicalManifold` serves as a base class for more specific

manifold classes.

The user interface is provided by the generic function

:func:`~sage.manifolds.manifold.Manifold`, with

with the argument ``structure`` set to ``'topological'``.

.. RUBRIC:: Example 1: the 2-sphere as a topological manifold of dimension

2 over `\RR`

One starts by declaring `S^2` as a 2-dimensional topological manifold::

sage: M = Manifold(2, 'S^2', structure='topological')

sage: M

2-dimensional topological manifold S^2

Since the base topological field has not been specified in the argument list

of ``Manifold``, `\RR` is assumed::

sage: M.base_field()

Real Field with 53 bits of precision

sage: dim(M)

Let us consider the complement of a point, the "North pole" say; this is an

open subset of `S^2`, which we call `U`::

sage: U = M.open_subset('U'); U

Open subset U of the 2-dimensional topological manifold S^2

A standard chart on `U` is provided by the stereographic projection from the

North pole to the equatorial plane::

sage: stereoN.<x,y> = U.chart(); stereoN

Chart (U, (x, y))

Thanks to the operator ``<x,y>`` on the left-hand side, the coordinates

declared in a chart (here `x` and `y`), are accessible by their names;

they are Sage's symbolic variables::

sage: y

sage: type(y)

The South pole is the point of coordinates `(x, y) = (0, 0)` in the above

chart::

sage: S = U.point((0,0), chart=stereoN, name='S'); S

Point S on the 2-dimensional topological manifold S^2

Let us call `V` the open subset that is the complement of the South pole and

let us introduce on it the chart induced by the stereographic projection from

the South pole to the equatorial plane::

sage: V = M.open_subset('V'); V

Open subset V of the 2-dimensional topological manifold S^2

sage: stereoS.<u,v> = V.chart(); stereoS

Chart (V, (u, v))

The North pole is the point of coordinates `(u, v) = (0, 0)` in this chart::

sage: N = V.point((0,0), chart=stereoS, name='N'); N

Point N on the 2-dimensional topological manifold S^2

To fully construct the manifold, we declare that it is the union of `U`

and `V`::

sage: M.declare_union(U,V)

and we provide the transition map between the charts ``stereoN`` =

`(U, (x, y))` and ``stereoS`` = `(V, (u, v))`, denoting by `W` the

intersection of `U` and `V` (`W` is the subset of `U` defined by

`x^2 + y^2 \neq 0`, as well as the subset of `V` defined by

`u^2 + v^2 \neq 0`)::

sage: stereoN_to_S = stereoN.transition_map(stereoS, [x/(x^2+y^2), y/(x^2+y^2)],

....: intersection_name='W', restrictions1= x^2+y^2!=0,

....: restrictions2= u^2+v^2!=0)

sage: stereoN_to_S

Change of coordinates from Chart (W, (x, y)) to Chart (W, (u, v))

sage: stereoN_to_S.display()

u = x/(x^2 + y^2)

v = y/(x^2 + y^2)

We give the name ``W`` to the Python variable representing `W = U \cap V`::

sage: W = U.intersection(V)

The inverse of the transition map is computed by the method

:meth:`sage.manifolds.chart.CoordChange.inverse`::

sage: stereoN_to_S.inverse()

Change of coordinates from Chart (W, (u, v)) to Chart (W, (x, y))

sage: stereoN_to_S.inverse().display()

x = u/(u^2 + v^2)

y = v/(u^2 + v^2)

At this stage, we have four open subsets on `S^2`::

sage: M.list_of_subsets()

[2-dimensional topological manifold S^2,

Open subset U of the 2-dimensional topological manifold S^2,

Open subset V of the 2-dimensional topological manifold S^2,

Open subset W of the 2-dimensional topological manifold S^2]

`W` is the open subset that is the complement of the two poles::

sage: N in W or S in W

False

The North pole lies in `V` and the South pole in `U`::

sage: N in V, N in U

(True, False)

sage: S in U, S in V

(True, False)

The manifold's (user) atlas contains four charts, two of them

being restrictions of charts to a smaller domain::

sage: M.atlas()

[Chart (U, (x, y)), Chart (V, (u, v)),

Chart (W, (x, y)), Chart (W, (u, v))]

Let us consider the point of coordinates `(1, 2)` in the chart ``stereoN``::

sage: p = M.point((1,2), chart=stereoN, name='p'); p

Point p on the 2-dimensional topological manifold S^2

sage: p.parent()

2-dimensional topological manifold S^2

sage: p in W

True

The coordinates of `p` in the chart ``stereoS`` are computed by letting

the chart act on the point::

sage: stereoS(p)

(1/5, 2/5)

Given the definition of `p`, we have of course::

sage: stereoN(p)

(1, 2)

Similarly::

sage: stereoS(N)

(0, 0)

sage: stereoN(S)

(0, 0)

A continuous map `S^2 \to \RR` (scalar field)::

sage: f = M.scalar_field({stereoN: atan(x^2+y^2), stereoS: pi/2-atan(u^2+v^2)},

....: name='f')

sage: f

Scalar field f on the 2-dimensional topological manifold S^2

sage: f.display()

f: S^2 --> R

on U: (x, y) |--> arctan(x^2 + y^2)

on V: (u, v) |--> 1/2*pi - arctan(u^2 + v^2)

sage: f(p)

arctan(5)

sage: f(N)

1/2*pi

sage: f(S)

sage: f.parent()

Algebra of scalar fields on the 2-dimensional topological manifold S^2

sage: f.parent().category()

Category of commutative algebras over Symbolic Ring

.. RUBRIC:: Example 2: the Riemann sphere as a topological manifold of

dimension 1 over `\CC`

We declare the Riemann sphere `\CC^*` as a 1-dimensional topological manifold

over `\CC`::

sage: M = Manifold(1, 'C*', structure='topological', field='complex'); M

Complex 1-dimensional topological manifold C*

We introduce a first open subset, which is actually

`\CC = \CC^*\setminus\{\infty\}` if we interpret `\CC^*` as the

Alexandroff one-point compactification of `\CC`::

sage: U = M.open_subset('U')

A natural chart on `U` is then nothing but the identity map of `\CC`, hence

we denote the associated coordinate by `z`::

sage: Z.<z> = U.chart()

The origin of the complex plane is the point of coordinate `z = 0`::

sage: O = U.point((0,), chart=Z, name='O'); O

Point O on the Complex 1-dimensional topological manifold C*

Another open subset of `\CC^*` is `V = \CC^*\setminus\{O\}`::

sage: V = M.open_subset('V')

We define a chart on `V` such that the point at infinity is the point of

coordinate `0` in this chart::

sage: W.<w> = V.chart(); W

Chart (V, (w,))

sage: inf = M.point((0,), chart=W, name='inf', latex_name=r'\infty')

sage: inf

Point inf on the Complex 1-dimensional topological manifold C*

To fully construct the Riemann sphere, we declare that it is the union

of `U` and `V`::

sage: M.declare_union(U,V)

and we provide the transition map between the two charts as `w = 1 / z`

on `A = U \cap V`::

sage: Z_to_W = Z.transition_map(W, 1/z, intersection_name='A',

....: restrictions1= z!=0, restrictions2= w!=0)

sage: Z_to_W

Change of coordinates from Chart (A, (z,)) to Chart (A, (w,))

sage: Z_to_W.display()

w = 1/z

sage: Z_to_W.inverse()

Change of coordinates from Chart (A, (w,)) to Chart (A, (z,))

sage: Z_to_W.inverse().display()

z = 1/w

Let consider the complex number `i` as a point of the Riemann sphere::

sage: i = M((I,), chart=Z, name='i'); i

Point i on the Complex 1-dimensional topological manifold C*

Its coordinates w.r.t. the charts ``Z`` and ``W`` are::

sage: Z(i)

(I,)

sage: W(i)

(-I,)

and we have::

sage: i in U

True

sage: i in V

True

The following subsets and charts have been defined::

sage: M.list_of_subsets()

[Open subset A of the Complex 1-dimensional topological manifold C*,

Complex 1-dimensional topological manifold C*,

Open subset U of the Complex 1-dimensional topological manifold C*,

Open subset V of the Complex 1-dimensional topological manifold C*]

sage: M.atlas()

[Chart (U, (z,)), Chart (V, (w,)), Chart (A, (z,)), Chart (A, (w,))]

A constant map `\CC^* \rightarrow \CC`::

sage: f = M.constant_scalar_field(3+2*I, name='f'); f

Scalar field f on the Complex 1-dimensional topological manifold C*

sage: f.display()

f: C* --> C

on U: z |--> 2*I + 3

on V: w |--> 2*I + 3

sage: f(O)

2*I + 3

sage: f(i)

2*I + 3

sage: f(inf)

2*I + 3

sage: f.parent()

Algebra of scalar fields on the Complex 1-dimensional topological

manifold C*

sage: f.parent().category()

Category of commutative algebras over Symbolic Ring

AUTHORS:

- Eric Gourgoulhon (2015): initial version

- Travis Scrimshaw (2015): structure described via

:class:`~sage.manifolds.structure.TopologicalStructure` or

:class:`~sage.manifolds.structure.RealTopologicalStructure`

REFERENCES:

- [Lee2011]_

- [Lee2013]_

- [KN1963]_

- [Huy2005]_

"""

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

# 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 __future__ import print_function

from __future__ import absolute_import

from sage.categories.fields import Fields

from sage.categories.manifolds import Manifolds

from sage.categories.homset import Hom

from sage.rings.all import CC

from sage.rings.real_mpfr import RR, RealField_class

from sage.rings.complex_field import ComplexField_class

from sage.misc.prandom import getrandbits

from sage.misc.cachefunc import cached_method

from sage.rings.integer import Integer

from sage.structure.global_options import GlobalOptions

from sage.manifolds.subset import ManifoldSubset

from sage.manifolds.structure import(

TopologicalStructure, RealTopologicalStructure,

DifferentialStructure, RealDifferentialStructure)

#############################################################################

## Global options

#############################################################################

## Class

class TopologicalManifold(ManifoldSubset):

r"""

Topological manifold over a topological field `K`.

Given a topological field `K` (in most applications, `K = \RR` or

`K = \CC`) and a non-negative integer `n`, a *topological manifold of

dimension* `n` *over K* is a topological space `M` such that

- `M` is a Hausdorff space,

- `M` is second countable, and

- every point in `M` has a neighborhood homeomorphic to `K^n`.

This is a Sage *parent* class, the corresponding *element*

class being :class:`~sage.manifolds.point.ManifoldPoint`.

INPUT:

- ``n`` -- positive integer; dimension of the manifold

- ``name`` -- string; name (symbol) given to the manifold

- ``field`` -- field `K` on which the manifold is

defined; allowed values are

- ``'real'`` or an object of type ``RealField`` (e.g., ``RR``) for

a manifold over `\RR`

- ``'complex'`` or an object of type ``ComplexField`` (e.g., ``CC``)

for a manifold over `\CC`

- an object in the category of topological fields (see

:class:`~sage.categories.fields.Fields` and

:class:`~sage.categories.topological_spaces.TopologicalSpaces`)

for other types of manifolds

- ``structure`` -- manifold structure (see

:class:`~sage.manifolds.structure.TopologicalStructure` or

:class:`~sage.manifolds.structure.RealTopologicalStructure`)

- ``ambient`` -- (default: ``None``) if not ``None``, must be a

topological manifold; the created object is then an open subset of

``ambient``

- ``latex_name`` -- (default: ``None``) string; LaTeX symbol to

denote the manifold; if none are provided, it is set to ``name``

- ``start_index`` -- (default: 0) integer; lower value of the range of

indices used for "indexed objects" on the manifold, e.g., coordinates

in a chart

- ``category`` -- (default: ``None``) to specify the category; if

``None``, ``Manifolds(field)`` is assumed (see the category

:class:`~sage.categories.manifolds.Manifolds`)

- ``unique_tag`` -- (default: ``None``) tag used to force the construction

of a new object when all the other arguments have been used previously

(without ``unique_tag``, the

:class:`~sage.structure.unique_representation.UniqueRepresentation`

behavior inherited from

:class:`~sage.manifolds.subset.ManifoldSubset`

would return the previously constructed object corresponding to these

arguments)

EXAMPLES:

A 4-dimensional topological manifold (over `\RR`)::

sage: M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='topological')

sage: M

4-dimensional topological manifold M

sage: latex(M)

\mathcal{M}

sage: type(M)

sage: M.base_field()

Real Field with 53 bits of precision

sage: dim(M)

The input parameter ``start_index`` defines the range of indices

on the manifold::

sage: M = Manifold(4, 'M', structure='topological')

sage: list(M.irange())

[0, 1, 2, 3]

sage: M = Manifold(4, 'M', structure='topological', start_index=1)

sage: list(M.irange())

[1, 2, 3, 4]

sage: list(Manifold(4, 'M', structure='topological', start_index=-2).irange())

[-2, -1, 0, 1]

A complex manifold::

sage: N = Manifold(3, 'N', structure='topological', field='complex'); N

Complex 3-dimensional topological manifold N

A manifold over `\QQ`::

sage: N = Manifold(6, 'N', structure='topological', field=QQ); N

6-dimensional topological manifold N over the Rational Field

A manifold over `\QQ_5`, the field of 5-adic numbers::

sage: N = Manifold(2, 'N', structure='topological', field=Qp(5)); N

2-dimensional topological manifold N over the 5-adic Field with capped

relative precision 20

A manifold is a Sage *parent* object, in the category of topological

manifolds over a given topological field (see

:class:`~sage.categories.manifolds.Manifolds`)::

sage: isinstance(M, Parent)

True

sage: M.category()

Category of manifolds over Real Field with 53 bits of precision

sage: from sage.categories.manifolds import Manifolds

sage: M.category() is Manifolds(RR)

True

sage: M.category() is Manifolds(M.base_field())

True

sage: M in Manifolds(RR)

True

sage: N in Manifolds(Qp(5))

True

The corresponding Sage *elements* are points::

sage: X.<t, x, y, z> = M.chart()

sage: p = M.an_element(); p

Point on the 4-dimensional topological manifold M

sage: p.parent()

4-dimensional topological manifold M

sage: M.is_parent_of(p)

True

sage: p in M

True

The manifold's points are instances of class

:class:`~sage.manifolds.point.ManifoldPoint`::

sage: isinstance(p, sage.manifolds.point.ManifoldPoint)

True

Since an open subset of a topological manifold `M` is itself a

topological manifold, open subsets of `M` are instances of the class

:class:`TopologicalManifold`::

sage: U = M.open_subset('U'); U

Open subset U of the 4-dimensional topological manifold M

sage: isinstance(U, sage.manifolds.manifold.TopologicalManifold)

True

sage: U.base_field() == M.base_field()

True

sage: dim(U) == dim(M)

True

sage: U.category()

Join of Category of subobjects of sets and Category of manifolds over

Real Field with 53 bits of precision

The manifold passes all the tests of the test suite relative to its

category::

sage: TestSuite(M).run()

.. SEEALSO::

:mod:`sage.manifolds.manifold`

"""

def __init__(self, n, name, field, structure, ambient=None,

latex_name=None, start_index=0, category=None,

unique_tag=None):

r"""

Construct a topological manifold.

TESTS::

sage: M = Manifold(3, 'M', latex_name=r'\mathbb{M}',

....: structure='topological', start_index=1)

sage: M

3-dimensional topological manifold M

sage: latex(M)

\mathbb{M}

sage: dim(M)

sage: X.<x,y,z> = M.chart()

sage: TestSuite(M).run()

Tests for open subsets::

sage: U = M.open_subset('U', coord_def={X: x>0})

sage: TestSuite(U).run()

sage: U.category() is M.category().Subobjects()

True

"""

# Initialization of the attributes _dim, _field, _field_type:

self._dim = n

if field == 'real':

self._field = RR

self._field_type = 'real'

elif field == 'complex':

self._field = CC

self._field_type = 'complex'

else:

if field not in Fields():

raise TypeError("the argument 'field' must be a field")

self._field = field

if isinstance(field, RealField_class):

self._field_type = 'real'

elif isinstance(field, ComplexField_class):

self._field_type = 'complex'

else:

self._field_type = 'neither_real_nor_complex'

# Structure and category:

self._structure = structure

if ambient is None:

ambient = self

category = Manifolds(self._field).or_subcategory(category)

category = self._structure.subcategory(category)

else:

category = ambient.category().Subobjects()

# Initialization as a manifold set:

ManifoldSubset.__init__(self, ambient, name, latex_name=latex_name,

category=category)

self._is_open = True

self._open_covers.append([self]) # list of open covers of self

if not isinstance(start_index, (int, Integer)):

raise TypeError("the starting index must be an integer")

self._sindex = start_index

self._atlas = [] # list of charts defined on subsets of self

self._top_charts = [] # list of charts defined on subsets of self

# that are not subcharts of charts on larger subsets

self._def_chart = None # default chart

self._charts_by_coord = {} # dictionary of charts whose domain is self

# (key: string formed by the coordinate

# symbols separated by a white space)

self._coord_changes = {} # dictionary of transition maps (key: pair of

# of charts)

# List of charts that individually cover self, i.e. whose

# domains are self (if non-empty, self is a coordinate domain):

self._covering_charts = []

# Algebra of scalar fields defined on self:

self._scalar_field_algebra = self._structure.scalar_field_algebra(self)

# The zero scalar field:

self._zero_scalar_field = self._scalar_field_algebra.zero()

# The unit scalar field:

self._one_scalar_field = self._scalar_field_algebra.one()

# The current calculus method on the manifold

# (to be changed by set_calculus_method)

self._calculus_method = 'SR'

def _repr_(self):

r"""

Return a string representation of the manifold.

TESTS::

sage: M = Manifold(3, 'M', structure='topological')

sage: M._repr_()

'3-dimensional topological manifold M'

sage: repr(M) # indirect doctest

'3-dimensional topological manifold M'

sage: M # indirect doctest

3-dimensional topological manifold M

sage: M = Manifold(3, 'M', structure='topological', field='complex')

sage: M._repr_()

'Complex 3-dimensional topological manifold M'

sage: M = Manifold(3, 'M', structure='topological', field=QQ)

sage: M._repr_()

'3-dimensional topological manifold M over the Rational Field'

If the manifold is actually an open subset of a larger manifold, the

string representation is different::

sage: U = M.open_subset('U')

sage: U._repr_()

'Open subset U of the 3-dimensional topological manifold M

over the Rational Field'

"""

if self is self._manifold:

if self._field_type == 'real':

return "{}-dimensional {} manifold {}".format(self._dim,

self._structure.name,

self._name)

elif self._field_type == 'complex':

if isinstance(self._structure, DifferentialStructure):

return "{}-dimensional complex manifold {}".format(

self._dim,

self._name)

else:

return "Complex {}-dimensional {} manifold {}".format(

self._dim,

self._structure.name,

self._name)

return "{}-dimensional {} manifold {} over the {}".format(

self._dim,

self._structure.name,

self._name,

self._field)

else:

return "Open subset {} of the {}".format(self._name, self._manifold)

def _an_element_(self):

r"""

Construct some point on the manifold.

EXAMPLES::

sage: M = Manifold(2, 'M', structure='topological')

sage: X.<x,y> = M.chart()

sage: p = M._an_element_(); p

Point on the 2-dimensional topological manifold M

sage: p.coord()

(0, 0)

sage: U = M.open_subset('U', coord_def={X: y>1}); U

Open subset U of the 2-dimensional topological manifold M

sage: p = U._an_element_(); p

Point on the 2-dimensional topological manifold M

sage: p in U

True

sage: p.coord()

(0, 2)

sage: V = U.open_subset('V', coord_def={X.restrict(U): x<-pi})

sage: p = V._an_element_(); p

Point on the 2-dimensional topological manifold M

sage: p in V

True

sage: p.coord()

(-pi - 1, 2)

"""

from sage.rings.infinity import Infinity

if self._def_chart is None:

return self.element_class(self)

# Attempt to construct a point in the domain of the default chart

chart = self._def_chart

if self._field_type == 'real':

coords = []

for coord_range in chart._bounds:

xmin = coord_range[0][0]

xmax = coord_range[1][0]

if xmin == -Infinity:

if xmax == Infinity:

x = 0

else:

x = xmax - 1

else:

if xmax == Infinity:

x = xmin + 1

else:

x = (xmin + xmax)/2

coords.append(x)

else:

coords = self._dim*[0]

if not chart.valid_coordinates(*coords):

# Attempt to construct a point in the domain of other charts

if self._field_type == 'real':

for ch in self._atlas:

if ch is self._def_chart:

continue # since this case has already been attempted

coords = []

for coord_range in ch._bounds:

xmin = coord_range[0][0]

xmax = coord_range[1][0]

if xmin == -Infinity:

if xmax == Infinity:

x = 0

else:

x = xmax - 1

else:

if xmax == Infinity:

x = xmin + 1

else:

x = (xmin + xmax)/2

coords.append(x)

if ch.valid_coordinates(*coords):

chart = ch

break

else:

# A generic element with specific coordinates could not be

# automatically generated, due to too complex coordinate

# conditions. An element without any coordinate set is

# returned instead:

return self.element_class(self)

else:

# Case of manifolds over a field different from R

for ch in self._atlas:

if ch is self._def_chart:

continue # since this case has already been attempted

if ch.valid_coordinates(*coords):

chart = ch

break

else:

return self.element_class(self)

# The point is constructed with check_coords=False since the check

# has just been performed above:

return self.element_class(self, coords=coords, chart=chart,

check_coords=False)

def __contains__(self, point):

r"""

Check whether a point is contained in the manifold.

EXAMPLES::

sage: M = Manifold(2, 'M', structure='topological')

sage: X.<x,y> = M.chart()

sage: p = M.point((1,2), chart=X)

sage: M.__contains__(p)

True

sage: p in M # indirect doctest

True

sage: U = M.open_subset('U', coord_def={X: x>0})

sage: U.__contains__(p)

True

sage: p in U # indirect doctest

True

sage: V = U.open_subset('V', coord_def={X.restrict(U): y<0})

sage: V.__contains__(p)

False

sage: p in V # indirect doctest

False

"""

# for efficiency, a quick test first:

if point.parent() is self:

return True

if point.parent().is_subset(self):

return True

for chart in self._atlas:

if chart in point._coordinates:

if chart.valid_coordinates( *(point._coordinates[chart]) ):

return True

for chart in point._coordinates:

for schart in chart._subcharts:

if schart in self._atlas and schart.valid_coordinates(

*(point._coordinates[chart]) ):

return True

return False

def open_subset(self, name, latex_name=None, coord_def={}):

r"""

Create an open subset of the manifold.

An open subset is a set that is (i) included in the manifold and (ii)

open with respect to the manifold's topology. It is a topological

manifold by itself. Hence the returned object is an instance of

:class:`TopologicalManifold`.

INPUT:

- ``name`` -- name given to the open subset

- ``latex_name`` -- (default: ``None``) LaTeX symbol to denote

the subset; if none are provided, it is set to ``name``

- ``coord_def`` -- (default: {}) definition of the subset in

terms of coordinates; ``coord_def`` must a be dictionary with keys

charts on the manifold and values the symbolic expressions formed

by the coordinates to define the subset

OUTPUT:

- the open subset, as an instance of :class:`TopologicalManifold`

EXAMPLES:

Creating an open subset of a 2-dimensional manifold::

sage: M = Manifold(2, 'M', structure='topological')

sage: A = M.open_subset('A'); A

Open subset A of the 2-dimensional topological manifold M

As an open subset of a topological manifold, ``A`` is itself a

topological manifold, on the same topological field and of the same

dimension as ``M``::

sage: isinstance(A, sage.manifolds.manifold.TopologicalManifold)

True

sage: A.base_field() == M.base_field()

True

sage: dim(A) == dim(M)

True

sage: A.category() is M.category().Subobjects()

True

Creating an open subset of ``A``::

sage: B = A.open_subset('B'); B

Open subset B of the 2-dimensional topological manifold M

We have then::

sage: A.subsets() # random (set output)

{Open subset B of the 2-dimensional topological manifold M,

Open subset A of the 2-dimensional topological manifold M}

sage: B.is_subset(A)

True

sage: B.is_subset(M)

True

Defining an open subset by some coordinate restrictions: the open

unit disk in `\RR^2`::

sage: M = Manifold(2, 'R^2', structure='topological')

sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2

sage: U = M.open_subset('U', coord_def={c_cart: x^2+y^2<1}); U

Open subset U of the 2-dimensional topological manifold R^2

Since the argument ``coord_def`` has been set, ``U`` is automatically

provided with a chart, which is the restriction of the Cartesian one

to ``U``::

sage: U.atlas()

[Chart (U, (x, y))]

Therefore, one can immediately check whether a point belongs

to ``U``::

sage: M.point((0,0)) in U

True

sage: M.point((1/2,1/3)) in U

True

sage: M.point((1,2)) in U

False

"""

resu = TopologicalManifold(self._dim, name, self._field,

self._structure, ambient=self._manifold,

latex_name=latex_name,

start_index=self._sindex)

resu._calculus_method = self._calculus_method

resu._supersets.update(self._supersets)

for sd in self._supersets:

sd._subsets.add(resu)

self._top_subsets.add(resu)

# Charts on the result from the coordinate definition:

for chart, restrictions in coord_def.items():

if chart not in self._atlas:

raise ValueError("the {} does not belong to ".format(chart) +

"the atlas of {}".format(self))

chart.restrict(resu, restrictions)

# Transition maps on the result inferred from those of self:

for chart1 in coord_def:

for chart2 in coord_def:

if chart2 != chart1 and (chart1, chart2) in self._coord_changes:

self._coord_changes[(chart1, chart2)].restrict(resu)

return resu

def get_chart(self, coordinates, domain=None):

r"""

Get a chart from its coordinates.

The chart must have been previously created by the method

:meth:`chart`.

INPUT:

- ``coordinates`` -- single string composed of the coordinate symbols

separated by a space

- ``domain`` -- (default: ``None``) string containing the name of the

chart's domain, which must be a subset of the current manifold; if

``None``, the current manifold is assumed

OUTPUT:

- instance of

:class:`~sage.manifolds.chart.Chart` (or of the subclass

:class:`~sage.manifolds.chart.RealChart`) representing the chart

corresponding to the above specifications

EXAMPLES::

sage: M = Manifold(2, 'M', structure='topological')

sage: X.<x,y> = M.chart()

sage: M.get_chart('x y')

Chart (M, (x, y))

sage: M.get_chart('x y') is X

True

sage: U = M.open_subset('U', coord_def={X: (y!=0,x<0)})

sage: Y.<r, ph> = U.chart(r'r:(0,+oo) ph:(0,2*pi):\phi')

sage: M.atlas()

[Chart (M, (x, y)), Chart (U, (x, y)), Chart (U, (r, ph))]

sage: M.get_chart('x y', domain='U')

Chart (U, (x, y))

sage: M.get_chart('x y', domain='U') is X.restrict(U)

True

sage: U.get_chart('r ph')

Chart (U, (r, ph))

sage: M.get_chart('r ph', domain='U')

Chart (U, (r, ph))

sage: M.get_chart('r ph', domain='U') is Y

True

"""

if domain is None:

dom = self

else:

dom = self.get_subset(domain)

try:

return dom._charts_by_coord[coordinates]

except KeyError:

raise KeyError("the coordinates '{}' ".format(coordinates) +

"do not correspond to any chart with " +

"the {} as domain".format(dom))

def dimension(self):

r"""

Return the dimension of the manifold over its base field.

EXAMPLES::

sage: M = Manifold(2, 'M', structure='topological')

sage: M.dimension()

A shortcut is ``dim()``::

sage: M.dim()

The Sage global function ``dim`` can also be used::

sage: dim(M)

"""

return self._dim

dim = dimension

def base_field(self):

r"""

Return the field on which the manifold is defined.

OUTPUT:

- a topological field

EXAMPLES::

sage: M = Manifold(3, 'M', structure='topological')

sage: M.base_field()

Real Field with 53 bits of precision

sage: M = Manifold(3, 'M', structure='topological', field='complex')

sage: M.base_field()

Complex Field with 53 bits of precision

sage: M = Manifold(3, 'M', structure='topological', field=QQ)

sage: M.base_field()

Rational Field

"""

return self._field

def base_field_type(self):

r"""

Return the type of topological field on which the manifold is defined.

OUTPUT:

- a string describing the field, with three possible values:

- ``'real'`` for the real field `\RR`

- ``'complex'`` for the complex field `\CC`

- ``'neither_real_nor_complex'`` for a field different from `\RR`

and `\CC`

EXAMPLES::

sage: M = Manifold(3, 'M', structure='topological')

sage: M.base_field_type()

'real'

sage: M = Manifold(3, 'M', structure='topological', field='complex')

sage: M.base_field_type()

'complex'

sage: M = Manifold(3, 'M', structure='topological', field=QQ)

sage: M.base_field_type()

'neither_real_nor_complex'

"""

return self._field_type

def start_index(self):

r"""

Return the first value of the index range used on the manifold.

This is the parameter ``start_index`` passed at the construction of

the manifold.

OUTPUT:

- the integer `i_0` such that all indices of indexed objects on the

manifold range from `i_0` to `i_0 + n - 1`, where `n` is the

manifold's dimension

EXAMPLES::

sage: M = Manifold(3, 'M', structure='topological')

sage: M.start_index()

sage: M = Manifold(3, 'M', structure='topological', start_index=1)

sage: M.start_index()

"""

return self._sindex

def irange(self, start=None):

r"""

Single index generator.

INPUT:

- ``start`` -- (default: ``None``) initial value `i_0` of the index;

if none are provided, the value returned by :meth:`start_index()`

is assumed

OUTPUT:

- an iterable index, starting from `i_0` and ending at

`i_0 + n - 1`, where `n` is the manifold's dimension

EXAMPLES:

Index range on a 4-dimensional manifold::

sage: M = Manifold(4, 'M', structure='topological')

sage: list(M.irange())

[0, 1, 2, 3]

sage: list(M.irange(2))

[2, 3]

Index range on a 4-dimensional manifold with starting index=1::

sage: M = Manifold(4, 'M', structure='topological', start_index=1)

sage: list(M.irange())

[1, 2, 3, 4]

sage: list(M.irange(2))

[2, 3, 4]

In general, one has always::

sage: next(M.irange()) == M.start_index()

True

"""

si = self._sindex

imax = self._dim + si

if start is None:

i = si

else:

i = start

while i < imax:

yield i

i += 1

def index_generator(self, nb_indices):

r"""

Generator of index series.

INPUT:

- ``nb_indices`` -- number of indices in a series

OUTPUT:

- an iterable index series for a generic component with the specified

number of indices

EXAMPLES:

Indices on a 2-dimensional manifold::

sage: M = Manifold(2, 'M', structure='topological', start_index=1)

sage: list(M.index_generator(2))

[(1, 1), (1, 2), (2, 1), (2, 2)]

Loops can be nested::

sage: for ind1 in M.index_generator(2):

....: print("{} : {}".format(ind1, list(M.index_generator(2))))

(1, 1) : [(1, 1), (1, 2), (2, 1), (2, 2)]

(1, 2) : [(1, 1), (1, 2), (2, 1), (2, 2)]

(2, 1) : [(1, 1), (1, 2), (2, 1), (2, 2)]

(2, 2) : [(1, 1), (1, 2), (2, 1), (2, 2)]

"""

si = self._sindex

imax = self._dim - 1 + si

ind = [si for k in range(nb_indices)]

ind_end = [si for k in range(nb_indices)]

ind_end[0] = imax+1

while ind != ind_end:

yield tuple(ind)

ret = 1

for pos in range(nb_indices-1,-1,-1):

if ind[pos] != imax:

ind[pos] += ret

ret = 0

elif ret == 1:

if pos == 0:

ind[pos] = imax + 1 # end point reached

else:

ind[pos] = si

ret = 1

def atlas(self):

r"""

Return the list of charts that have been defined on the manifold.

EXAMPLES:

Let us consider `\RR^2` as a 2-dimensional manifold::

sage: M = Manifold(2, 'R^2', structure='topological')

Immediately after the manifold creation, the atlas is empty, since no

chart has been defined yet::

sage: M.atlas()

[]

Let us introduce the chart of Cartesian coordinates::

sage: c_cart.<x,y> = M.chart()

sage: M.atlas()

[Chart (R^2, (x, y))]

The complement of the half line `\{y = 0, x \geq 0\}`::

sage: U = M.open_subset('U', coord_def={c_cart: (y!=0,x<0)})

sage: U.atlas()

[Chart (U, (x, y))]

sage: M.atlas()

[Chart (R^2, (x, y)), Chart (U, (x, y))]

Spherical (polar) coordinates on ``U``::

sage: c_spher.<r, ph> = U.chart(r'r:(0,+oo) ph:(0,2*pi):\phi')

sage: U.atlas()

[Chart (U, (x, y)), Chart (U, (r, ph))]

sage: M.atlas()

[Chart (R^2, (x, y)), Chart (U, (x, y)), Chart (U, (r, ph))]

.. SEEALSO::

:meth:`top_charts`

"""

return list(self._atlas) # Make a (shallow) copy

def top_charts(self):

r"""

Return the list of charts defined on subsets of the current manifold

that are not subcharts of charts on larger subsets.

OUTPUT:

- list of charts defined on open subsets of the manifold but not on

larger subsets

EXAMPLES:

Charts on a 2-dimensional manifold::

sage: M = Manifold(2, 'M', structure='topological')

sage: X.<x,y> = M.chart()

sage: U = M.open_subset('U', coord_def={X: x>0})

sage: Y.<u,v> = U.chart()

sage: M.top_charts()

[Chart (M, (x, y)), Chart (U, (u, v))]

Note that the (user) atlas contains one more chart: ``(U, (x,y))``,

which is not a "top" chart::

sage: M.atlas()

[Chart (M, (x, y)), Chart (U, (x, y)), Chart (U, (u, v))]

.. SEEALSO::

:meth:`atlas` for the complete list of charts defined on the

manifold.

"""

return list(self._top_charts) # Make a (shallow) copy

def default_chart(self):

r"""

Return the default chart defined on the manifold.

Unless changed via :meth:`set_default_chart`, the *default chart*

is the first one defined on a subset of the manifold (possibly itself).

OUTPUT:

- instance of :class:`~sage.manifolds.chart.Chart`

representing the default chart

EXAMPLES:

Default chart on a 2-dimensional manifold and on some subsets::

sage: M = Manifold(2, 'M', structure='topological')

sage: M.chart('x y')

Chart (M, (x, y))

sage: M.chart('u v')

Chart (M, (u, v))

sage: M.default_chart()

Chart (M, (x, y))

sage: A = M.open_subset('A')

sage: A.chart('t z')

Chart (A, (t, z))

sage: A.default_chart()

Chart (A, (t, z))

"""

return self._def_chart

def set_default_chart(self, chart):

r"""

Changing the default chart on ``self``.

INPUT:

- ``chart`` -- a chart (must be defined on some subset ``self``)

EXAMPLES:

Charts on a 2-dimensional manifold::

sage: M = Manifold(2, 'M', structure='topological')

sage: c_xy.<x,y> = M.chart()

sage: c_uv.<u,v> = M.chart()

sage: M.default_chart()

Chart (M, (x, y))

sage: M.set_default_chart(c_uv)

sage: M.default_chart()

Chart (M, (u, v))

"""

from .chart import Chart

if not isinstance(chart, Chart):

raise TypeError("{} is not a chart".format(chart))

if chart._domain is not self:

if self.is_manifestly_coordinate_domain():

raise TypeError("the chart domain must coincide with " +

"the {}".format(self))

if chart not in self._atlas:

raise ValueError("the chart must be defined on the " +

"{}".format(self))

self._def_chart = chart

def coord_change(self, chart1, chart2):

r"""

Return the change of coordinates (transition map) between two charts

defined on the manifold.

The change of coordinates must have been defined previously, for

instance by the method

:meth:`~sage.manifolds.chart.Chart.transition_map`.

INPUT:

- ``chart1`` -- chart 1

- ``chart2`` -- chart 2

OUTPUT:

- instance of :class:`~sage.manifolds.chart.CoordChange`

representing the transition map from chart 1 to chart 2

EXAMPLES:

Change of coordinates on a 2-dimensional manifold::

sage: M = Manifold(2, 'M', structure='topological')

sage: c_xy.<x,y> = M.chart()

sage: c_uv.<u,v> = M.chart()

sage: c_xy.transition_map(c_uv, (x+y, x-y)) # defines the coord. change

Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))

sage: M.coord_change(c_xy, c_uv) # returns the coord. change defined above

Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))

"""

if (chart1, chart2) not in self._coord_changes:

raise TypeError("the change of coordinates from " +

"{} to {}".format(chart1, chart2) + " has not " +

"been defined on the {}".format(self))

return self._coord_changes[(chart1, chart2)]

def coord_changes(self):

r"""

Return the changes of coordinates (transition maps) defined on

subsets of the manifold.

OUTPUT:

- dictionary of changes of coordinates, with pairs of charts as keys

EXAMPLES:

Various changes of coordinates on a 2-dimensional manifold::

sage: M = Manifold(2, 'M', structure='topological')

sage: c_xy.<x,y> = M.chart()

sage: c_uv.<u,v> = M.chart()

sage: xy_to_uv = c_xy.transition_map(c_uv, [x+y, x-y])

sage: M.coord_changes()

{(Chart (M, (x, y)),

Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))}

sage: uv_to_xy = xy_to_uv.inverse()

sage: M.coord_changes() # random (dictionary output)

{(Chart (M, (u, v)),

Chart (M, (x, y))): Change of coordinates from Chart (M, (u, v)) to Chart (M, (x, y)),

(Chart (M, (x, y)),

Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))}

sage: c_rs.<r,s> = M.chart()

sage: uv_to_rs = c_uv.transition_map(c_rs, [-u+2*v, 3*u-v])

sage: M.coord_changes() # random (dictionary output)

{(Chart (M, (u, v)),

Chart (M, (r, s))): Change of coordinates from Chart (M, (u, v)) to Chart (M, (r, s)),

(Chart (M, (u, v)),

Chart (M, (x, y))): Change of coordinates from Chart (M, (u, v)) to Chart (M, (x, y)),

(Chart (M, (x, y)),

Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))}

sage: xy_to_rs = uv_to_rs * xy_to_uv

sage: M.coord_changes() # random (dictionary output)

{(Chart (M, (u, v)),

Chart (M, (r, s))): Change of coordinates from Chart (M, (u, v)) to Chart (M, (r, s)),

(Chart (M, (u, v)),

Chart (M, (x, y))): Change of coordinates from Chart (M, (u, v)) to Chart (M, (x, y)),

(Chart (M, (x, y)),

Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v)),

(Chart (M, (x, y)),

Chart (M, (r, s))): Change of coordinates from Chart (M, (x, y)) to Chart (M, (r, s))}

"""

return self._coord_changes

def is_manifestly_coordinate_domain(self):

r"""

Return ``True`` if the manifold is known to be the domain of some

coordinate chart and ``False`` otherwise.

If ``False`` is returned, either the manifold cannot be the domain of

some coordinate chart or no such chart has been declared yet.

EXAMPLES::

sage: M = Manifold(2, 'M', structure='topological')

sage: U = M.open_subset('U')

sage: X.<x,y> = U.chart()

sage: U.is_manifestly_coordinate_domain()

True

sage: M.is_manifestly_coordinate_domain()

False

sage: Y.<u,v> = M.chart()

sage: M.is_manifestly_coordinate_domain()

True

"""

return bool(self._covering_charts)

def chart(self, coordinates='', names=None, calc_method=None):

r"""

Define a chart, the domain of which is the manifold.

A *chart* is a pair `(U, \varphi)`, where `U` is the current

manifold and `\varphi: U \rightarrow V \subset K^n`

is a homeomorphism from `U` to an open subset `V` of `K^n`, `K`

being the field on which the manifold is defined.

The components `(x^1, \ldots, x^n)` of `\varphi`, defined by

`\varphi(p) = (x^1(p), \ldots, x^n(p)) \in K^n` for any point

`p \in U`, are called the *coordinates* of the chart `(U, \varphi)`.

See :class:`~sage.manifolds.chart.Chart` for a complete

documentation.

INPUT:

- ``coordinates`` -- (default: ``''`` (empty string)) string

defining the coordinate symbols and ranges, see below

- ``names`` -- (default: ``None``) unused argument, except if

``coordinates`` is not provided; it must then be a tuple containing

the coordinate symbols (this is guaranteed if the shortcut operator

``<,>`` is used)

- ``calc_method`` -- (default: ``None``) string defining the calculus

method to be used on this chart; must be one of

- ``'SR'``: Sage's default symbolic engine (Symbolic Ring)

- ``'sympy'``: SymPy

- ``None``: the current calculus method defined on the manifold is

used (cf. :meth:`set_calculus_method`)

The coordinates declared in the string ``coordinates`` are

separated by ``' '`` (whitespace) and each coordinate has at most three

fields, separated by a colon (``':'``):

1. The coordinate symbol (a letter or a few letters).

2. (optional, only for manifolds over `\RR`) The interval `I`

defining the coordinate range: if not provided, the coordinate

is assumed to span all `\RR`; otherwise `I` must be provided

in the form ``(a,b)`` (or equivalently ``]a,b[``)

The bounds ``a`` and ``b`` can be ``+/-Infinity``, ``Inf``,

``infinity``, ``inf`` or ``oo``. For *singular* coordinates,

non-open intervals such as ``[a,b]`` and

``(a,b]`` (or equivalently ``]a,b]``) are allowed. Note that

the interval declaration must not contain any space character.

3. (optional) The LaTeX spelling of the coordinate; if not provided

the coordinate symbol given in the first field will be used.

The order of the fields 2 and 3 does not matter and each of them can

be omitted. If it contains any LaTeX expression, the string

``coordinates`` must be declared with the prefix 'r' (for "raw") to

allow for a proper treatment of the backslash character (see

examples below). If no interval range and no LaTeX spelling is to

be provided for any coordinate, the argument ``coordinates`` can be

omitted when the shortcut operator ``<,>`` is used via Sage

preparser (see examples below).

OUTPUT:

- the created chart, as an instance of

:class:`~sage.manifolds.chart.Chart` or of the subclass

:class:`~sage.manifolds.chart.RealChart` for manifolds over `\RR`.

EXAMPLES:

Chart on a 2-dimensional manifold::

sage: M = Manifold(2, 'M', structure='topological')

sage: U = M.open_subset('U')

sage: X = U.chart('x y'); X

Chart (U, (x, y))

sage: X[0]

sage: X[1]

sage: X[:]

(x, y)

The declared coordinates are not known at the global level::

sage: y

Traceback (most recent call last):

...

NameError: name 'y' is not defined

They can be recovered by the operator ``[:]`` applied to the chart::

sage: (x, y) = X[:]

sage: y

sage: type(y)

But a shorter way to proceed is to use the operator ``<,>`` in the

left-hand side of the chart declaration (there is then no need to

pass the string 'x y' to chart())::

sage: M = Manifold(2, 'M', structure='topological')

sage: U = M.open_subset('U')

sage: X.<x,y> = U.chart(); X

Chart (U, (x, y))

Indeed, the declared coordinates are then known at the global level::

sage: y

sage: (x,y) == X[:]

True

Actually the instruction ``X.<x,y> = U.chart()`` is

equivalent to the combination of the two instructions

``X = U.chart('x y')`` and ``(x,y) = X[:]``.

See the documentation of class

:class:`~sage.manifolds.chart.Chart` for more examples,

especially regarding the coordinates ranges and restrictions.

"""

if calc_method is None:

calc_method = self._calculus_method

return self._structure.chart(self, coordinates=coordinates,

names=names, calc_method=calc_method)

def is_open(self):

"""

Return if ``self`` is an open set.

In the present case (manifold or open subset of it), always

return ``True``.

TESTS::

sage: M = Manifold(2, 'M', structure='topological')

sage: M.is_open()

True

"""

return True

def scalar_field_algebra(self):

r"""

Return the algebra of scalar fields defined the manifold.

See :class:`~sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra`

for a complete documentation.

OUTPUT:

- instance of

:class:`~sage.manifolds.scalarfield_algebra.ScalarFieldAlgebra`

representing the algebra `C^0(U)` of all scalar fields defined

on `U` = ``self``

EXAMPLES:

Scalar algebra of a 3-dimensional open subset::

sage: M = Manifold(3, 'M', structure='topological')

sage: U = M.open_subset('U')

sage: CU = U.scalar_field_algebra() ; CU

Algebra of scalar fields on the Open subset U of the 3-dimensional topological manifold M

sage: CU.category()

Category of commutative algebras over Symbolic Ring

sage: CU.zero()

Scalar field zero on the Open subset U of the 3-dimensional topological manifold M

The output is cached::

sage: U.scalar_field_algebra() is CU

True

"""

return self._scalar_field_algebra

def scalar_field(self, coord_expression=None, chart=None, name=None,

latex_name=None):

r"""

Define a scalar field on the manifold.

See :class:`~sage.manifolds.scalarfield.ScalarField` (or

:class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField`

if the manifold is differentiable) for a complete documentation.

INPUT:

- ``coord_expression`` -- (default: ``None``) coordinate expression(s)

of the scalar field; this can be either

* a single coordinate expression; if the argument ``chart`` is

``'all'``, this expression is set to all the charts defined

on the open set; otherwise, the expression is set in the

specific chart provided by the argument ``chart``

* a dictionary of coordinate expressions, with the charts as keys

- ``chart`` -- (default: ``None``) chart defining the coordinates

used in ``coord_expression`` when the latter is a single

coordinate expression; if ``None``, the default chart of the

open set is assumed; if ``chart=='all'``, ``coord_expression`` is

assumed to be independent of the chart (constant scalar field)

- ``name`` -- (default: ``None``) name given to the scalar field

- ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the

scalar field; if ``None``, the LaTeX symbol is set to ``name``

If ``coord_expression`` is ``None`` or does not fully specified the

scalar field, other coordinate expressions can be added subsequently

by means of the methods

:meth:`~sage.manifolds.scalarfield.ScalarField.add_expr`,

:meth:`~sage.manifolds.scalarfield.ScalarField.add_expr_by_continuation`,

or :meth:`~sage.manifolds.scalarfield.ScalarField.set_expr`

OUTPUT:

- instance of :class:`~sage.manifolds.scalarfield.ScalarField`

(or of the subclass

:class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField`

if the manifold is differentiable) representing the defined scalar

field

EXAMPLES:

A scalar field defined by its coordinate expression in the open

set's default chart::

sage: M = Manifold(3, 'M', structure='topological')

sage: U = M.open_subset('U')

sage: c_xyz.<x,y,z> = U.chart()

sage: f = U.scalar_field(sin(x)*cos(y) + z, name='F'); f

Scalar field F on the Open subset U of the 3-dimensional topological manifold M

sage: f.display()

F: U --> R

(x, y, z) |--> cos(y)*sin(x) + z

sage: f.parent()

Algebra of scalar fields on the Open subset U of the 3-dimensional topological manifold M

sage: f in U.scalar_field_algebra()

True

Equivalent definition with the chart specified::

sage: f = U.scalar_field(sin(x)*cos(y) + z, chart=c_xyz, name='F')

sage: f.display()

F: U --> R

(x, y, z) |--> cos(y)*sin(x) + z

Equivalent definition with a dictionary of coordinate expression(s)::

sage: f = U.scalar_field({c_xyz: sin(x)*cos(y) + z}, name='F')

sage: f.display()

F: U --> R

(x, y, z) |--> cos(y)*sin(x) + z

See the documentation of class

:class:`~sage.manifolds.scalarfield.ScalarField` for more

examples.

.. SEEALSO::

:meth:`constant_scalar_field`, :meth:`zero_scalar_field`,

:meth:`one_scalar_field`

"""

if isinstance(coord_expression, dict):

# check validity of entry

for chart in coord_expression:

if not chart._domain.is_subset(self):

raise ValueError("the {} is not defined ".format(chart) +

"on some subset of the " + str(self))

alg = self.scalar_field_algebra()

return alg.element_class(alg, coord_expression=coord_expression,

name=name, latex_name=latex_name, chart=chart)

def constant_scalar_field(self, value, name=None, latex_name=None):

r"""

Define a constant scalar field on the manifold.

INPUT:

- ``value`` -- constant value of the scalar field, either a numerical

value or a symbolic expression not involving any chart coordinates

- ``name`` -- (default: ``None``) name given to the scalar field

- ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the

scalar field; if ``None``, the LaTeX symbol is set to ``name``

OUTPUT:

- instance of :class:`~sage.manifolds.scalarfield.ScalarField`

representing the scalar field whose constant value is ``value``

EXAMPLES:

A constant scalar field on the 2-sphere::

sage: M = Manifold(2, 'M', structure='topological') # the 2-dimensional sphere S^2

sage: U = M.open_subset('U') # complement of the North pole

sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole

sage: V = M.open_subset('V') # complement of the South pole

sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole

sage: M.declare_union(U,V) # S^2 is the union of U and V

sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),

....: intersection_name='W',

....: restrictions1= x^2+y^2!=0,

....: restrictions2= u^2+v^2!=0)

sage: uv_to_xy = xy_to_uv.inverse()

sage: f = M.constant_scalar_field(-1) ; f

Scalar field on the 2-dimensional topological manifold M

sage: f.display()

M --> R

on U: (x, y) |--> -1

on V: (u, v) |--> -1

We have::

sage: f.restrict(U) == U.constant_scalar_field(-1)

True

sage: M.constant_scalar_field(0) is M.zero_scalar_field()

True

.. SEEALSO::

:meth:`zero_scalar_field`, :meth:`one_scalar_field`

"""

if value == 0:

return self.zero_scalar_field()

alg = self.scalar_field_algebra()

return alg.element_class(alg, coord_expression=value, name=name,

latex_name=latex_name, chart='all')

def zero_scalar_field(self):

r"""

Return the zero scalar field defined on ``self``.

OUTPUT:

- a :class:`~sage.manifolds.scalarfield.ScalarField`

representing the constant scalar field with value `0`

EXAMPLES::

sage: M = Manifold(2, 'M', structure='topological')

sage: X.<x,y> = M.chart()

sage: f = M.zero_scalar_field() ; f

Scalar field zero on the 2-dimensional topological manifold M

sage: f.display()

zero: M --> R

(x, y) |--> 0

sage: f.parent()

Algebra of scalar fields on the 2-dimensional topological manifold M

sage: f is M.scalar_field_algebra().zero()

True

"""

return self._zero_scalar_field

def one_scalar_field(self):

r"""

Return the constant scalar field with value the unit element

of the base field of ``self``.

OUTPUT:

- a :class:`~sage.manifolds.scalarfield.ScalarField` representing

the constant scalar field with value the unit element

of the base field of ``self``

EXAMPLES::

sage: M = Manifold(2, 'M', structure='topological')

sage: X.<x,y> = M.chart()

sage: f = M.one_scalar_field(); f

Scalar field 1 on the 2-dimensional topological manifold M

sage: f.display()

1: M --> R

(x, y) |--> 1

sage: f.parent()

Algebra of scalar fields on the 2-dimensional topological manifold M

sage: f is M.scalar_field_algebra().one()

True

"""

return self._one_scalar_field

class options(GlobalOptions):

r"""

Sets and displays the options for manifolds. If no parameters

are set, then the function returns a copy of the options dictionary.

The ``options`` to manifolds can be accessed as the method

:obj:`Manifold.options`.

@OPTIONS@

EXAMPLES::

sage: M = Manifold(2, 'M', structure='topological')

sage: X.<x,y> = M.chart()

sage: g = function('g')(x, y)

For coordinate functions, the display is more "textbook" like::

sage: f = X.function(diff(g, x) + diff(g, y))

sage: f

d(g)/dx + d(g)/dy

sage: latex(f)

\frac{\partial\,g}{\partial x} + \frac{\partial\,g}{\partial y}

One can switch to Pynac notation by changing ``textbook_output``

to ``False``::

sage: Manifold.options.textbook_output=False

sage: f

diff(g(x, y), x) + diff(g(x, y), y)

sage: latex(f)

\frac{\partial}{\partial x}g\left(x, y\right)

+ \frac{\partial}{\partial y}g\left(x, y\right)

sage: Manifold.options._reset()

If there is a clear understanding that `u` and `v` are functions of

`(x,y)`, the explicit mention of the latter can be cumbersome in lengthy

tensor expressions::

sage: f = X.function(function('u')(x, y) * function('v')(x, y))

sage: f

u(x, y)*v(x, y)

We can switch it off by::

sage: M.options.omit_function_arguments=True

sage: f

u*v

sage: M.options._reset()

"""

NAME = 'manifolds'

module = 'sage.manifolds'

option_class = 'TopologicalManifold'

textbook_output = dict(default=True,

description='textbook-like output instead of the Pynac output for derivatives',

checker=lambda x: isinstance(x, bool))

omit_function_arguments = dict(default=False,

description='Determine whether the arguments of symbolic functions are printed',

checker=lambda x: isinstance(x, bool))

def _Hom_(self, other, category=None):

r"""

Construct the set of morphisms (i.e. continuous maps)

``self`` to ``other``.

INPUT:

- ``other`` -- an open subset of some topological manifold over the

same field as ``self``

- ``category`` -- (default: ``None``) not used here (to ensure

compatibility with generic hook ``_Hom_``)

OUTPUT:

- the homset `\mathrm{Hom}(U,V)`, where `U` is ``self``

and `V` is ``other``

.. SEEALSO::

For more documentation, see

:class:`~sage.manifolds.manifold_homset.TopologicalManifoldHomset`.

EXAMPLES::

sage: M = Manifold(2, 'M', structure='topological')

sage: N = Manifold(3, 'N', structure='topological')

sage: H = M._Hom_(N); H

Set of Morphisms from 2-dimensional topological manifold M to

3-dimensional topological manifold N in Category of manifolds over

Real Field with 53 bits of precision

The result is cached::

sage: H is Hom(M, N)

True

"""

return self._structure.homset(self, other)

def continuous_map(self, codomain, coord_functions=None, chart1=None,

chart2=None, name=None, latex_name=None):

r"""

Define a continuous map from ``self`` to ``codomain``.

INPUT:

- ``codomain`` -- :class:`TopologicalManifold`; the map's codomain

- ``coord_functions`` -- (default: ``None``) if not ``None``,

must be either

- (i) a dictionary of the coordinate expressions (as lists

(or tuples) of the coordinates of the image expressed in

terms of the coordinates of the considered point) with the

pairs of charts ``(chart1, chart2)`` as keys (``chart1`` being

a chart on ``self`` and ``chart2`` a chart on ``codomain``);

- (ii) a single coordinate expression in a given pair of charts, the

latter being provided by the arguments ``chart1`` and ``chart2``;

in both cases, if the dimension of the codomain is `1`, a single

coordinate expression can be passed instead of a tuple with

a single element

- ``chart1`` -- (default: ``None``; used only in case (ii) above)

chart on ``self`` defining the start coordinates involved in

``coord_functions`` for case (ii); if ``None``, the coordinates

are assumed to refer to the default chart of ``self``

- ``chart2`` -- (default: ``None``; used only in case (ii) above)

chart on ``codomain`` defining the target coordinates involved in

``coord_functions`` for case (ii); if ``None``, the coordinates

are assumed to refer to the default chart of ``codomain``

- ``name`` -- (default: ``None``) name given to the continuous map

- ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the

continuous map; if ``None``, the LaTeX symbol is set to ``name``

OUTPUT:

- the continuous map as an instance of

:class:`~sage.manifolds.continuous_map.ContinuousMap`

EXAMPLES:

A continuous map between an open subset of `S^2` covered by regular

spherical coordinates and `\RR^3`::

sage: M = Manifold(2, 'S^2', structure='topological')

sage: U = M.open_subset('U')

sage: c_spher.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')

sage: N = Manifold(3, 'R^3', latex_name=r'\RR^3', structure='topological')

sage: c_cart.<x,y,z> = N.chart() # Cartesian coord. on R^3

sage: Phi = U.continuous_map(N, (sin(th)*cos(ph), sin(th)*sin(ph), cos(th)),

....: name='Phi', latex_name=r'\Phi')

sage: Phi

Continuous map Phi from the Open subset U of the 2-dimensional topological manifold S^2 to the 3-dimensional topological manifold R^3

The same definition, but with a dictionary with pairs of charts as

keys (case (i) above)::

sage: Phi1 = U.continuous_map(N,

....: {(c_spher, c_cart): (sin(th)*cos(ph), sin(th)*sin(ph), cos(th))},

....: name='Phi', latex_name=r'\Phi')

sage: Phi1 == Phi

True

The continuous map acting on a point::

sage: p = U.point((pi/2, pi)) ; p

Point on the 2-dimensional topological manifold S^2

sage: Phi(p)

Point on the 3-dimensional topological manifold R^3

sage: Phi(p).coord(c_cart)

(-1, 0, 0)

sage: Phi1(p) == Phi(p)

True

TESTS::

sage: M = Manifold(2, 'M', structure='topological')

sage: M.continuous_map(ZZ)

Traceback (most recent call last):

...

ValueError: Integer Ring is not a manifold

over Real Field with 53 bits of precision

.. SEEALSO::

See :class:`~sage.manifolds.continuous_map.ContinuousMap`

for the complete documentation and more examples.

.. TODO::

Allow the construction of continuous maps from ``self`` to the

base field (considered as a trivial 1-dimensional manifold).

"""

if (not isinstance(codomain, TopologicalManifold)

or codomain.base_field() != self.base_field()):

raise ValueError("{} is not a manifold over {}".format(codomain, self.base_field()))

homset = Hom(self, codomain)

if coord_functions is None:

coord_functions = {}

if not isinstance(coord_functions, dict):

# Turn coord_functions into a dictionary:

if chart1 is None:

chart1 = self._def_chart

elif chart1 not in self._atlas:

raise ValueError("{} is not a chart ".format(chart1) +

"defined on the {}".format(self))

if chart2 is None:

chart2 = codomain._def_chart

elif chart2 not in codomain._atlas:

raise ValueError("{} is not a chart ".format(chart2) +

" defined on the {}".format(codomain))

coord_functions = {(chart1, chart2): coord_functions}

return homset(coord_functions, name=name, latex_name=latex_name)

def homeomorphism(self, codomain, coord_functions=None, chart1=None,

chart2=None, name=None, latex_name=None):

r"""

Define a homeomorphism between the current manifold and another one.

See :class:`~sage.manifolds.continuous_map.ContinuousMap` for a

complete documentation.

INPUT:

- ``codomain`` -- :class:`TopologicalManifold`; codomain of

the homeomorphism

- ``coord_functions`` -- (default: ``None``) if not ``None``,

must be either

- (i) a dictionary of the coordinate expressions (as lists

(or tuples) of the coordinates of the image expressed in

terms of the coordinates of the considered point) with the

pairs of charts ``(chart1, chart2)`` as keys (``chart1`` being

a chart on ``self`` and ``chart2`` a chart on ``codomain``);

- (ii) a single coordinate expression in a given pair of charts, the

latter being provided by the arguments ``chart1`` and ``chart2``;

in both cases, if the dimension of the codomain is `1`, a single

coordinate expression can be passed instead of a tuple with

a single element

- ``chart1`` -- (default: ``None``; used only in case (ii) above)

chart on ``self`` defining the start coordinates involved in

``coord_functions`` for case (ii); if ``None``, the coordinates

are assumed to refer to the default chart of ``self``

- ``chart2`` -- (default: ``None``; used only in case (ii) above)

chart on ``codomain`` defining the target coordinates involved in

``coord_functions`` for case (ii); if ``None``, the coordinates

are assumed to refer to the default chart of ``codomain``

- ``name`` -- (default: ``None``) name given to the homeomorphism

- ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the

homeomorphism; if ``None``, the LaTeX symbol is set to ``name``

OUTPUT:

- the homeomorphism, as an instance of

:class:`~sage.manifolds.continuous_map.ContinuousMap`

EXAMPLES:

Homeomorphism between the open unit disk in `\RR^2` and `\RR^2`::

sage: forget() # for doctests only

sage: M = Manifold(2, 'M', structure='topological') # the open unit disk

sage: c_xy.<x,y> = M.chart('x:(-1,1) y:(-1,1)') # Cartesian coord on M

sage: c_xy.add_restrictions(x^2+y^2<1)

sage: N = Manifold(2, 'N', structure='topological') # R^2

sage: c_XY.<X,Y> = N.chart() # canonical coordinates on R^2

sage: Phi = M.homeomorphism(N, [x/sqrt(1-x^2-y^2), y/sqrt(1-x^2-y^2)],

....: name='Phi', latex_name=r'\Phi')

sage: Phi

Homeomorphism Phi from the 2-dimensional topological manifold M to

the 2-dimensional topological manifold N

sage: Phi.display()

Phi: M --> N

(x, y) |--> (X, Y) = (x/sqrt(-x^2 - y^2 + 1), y/sqrt(-x^2 - y^2 + 1))

The inverse homeomorphism::

sage: Phi^(-1)

Homeomorphism Phi^(-1) from the 2-dimensional topological

manifold N to the 2-dimensional topological manifold M

sage: (Phi^(-1)).display()

Phi^(-1): N --> M

(X, Y) |--> (x, y) = (X/sqrt(X^2 + Y^2 + 1), Y/sqrt(X^2 + Y^2 + 1))

See the documentation of

:class:`~sage.manifolds.continuous_map.ContinuousMap` for more

examples.

"""

homset = Hom(self, codomain)

if coord_functions is None:

coord_functions = {}

if not isinstance(coord_functions, dict):

# Turn coord_functions into a dictionary:

if chart1 is None:

chart1 = self._def_chart

elif chart1 not in self._atlas:

raise ValueError("{} is not a chart ".format(chart1) +

"defined on the {}".format(self))

if chart2 is None:

chart2 = codomain._def_chart

elif chart2 not in codomain._atlas:

raise ValueError("{} is not a chart ".format(chart2) +

" defined on the {}".format(codomain))

coord_functions = {(chart1, chart2): coord_functions}

return homset(coord_functions, name=name, latex_name=latex_name,

is_isomorphism=True)

@cached_method

def identity_map(self):

r"""

Identity map of ``self``.

The identity map of a topological manifold `M` is the trivial

homeomorphism:

.. MATH::

\begin{array}{cccc}

\mathrm{Id}_M: & M & \longrightarrow & M \\

& p & \longmapsto & p

\end{array}

OUTPUT:

- the identity map as an instance of

:class:`~sage.manifolds.continuous_map.ContinuousMap`

EXAMPLES:

Identity map of a complex manifold::

sage: M = Manifold(2, 'M', structure='topological', field='complex')

sage: X.<x,y> = M.chart()

sage: id = M.identity_map(); id

Identity map Id_M of the Complex 2-dimensional topological manifold M

sage: id.parent()

Set of Morphisms from Complex 2-dimensional topological manifold M

to Complex 2-dimensional topological manifold M in Category of

manifolds over Complex Field with 53 bits of precision

sage: id.display()

Id_M: M --> M

(x, y) |--> (x, y)

The identity map acting on a point::

sage: p = M((1+I, 3-I), name='p'); p

Point p on the Complex 2-dimensional topological manifold M

sage: id(p)

Point p on the Complex 2-dimensional topological manifold M

sage: id(p) == p

True

.. SEEALSO::

See :class:`~sage.manifolds.continuous_map.ContinuousMap`

for the complete documentation.

"""

return Hom(self, self).one()

def set_calculus_method(self, method):

r"""

Set the calculus method to be used for coordinate computations on this

manifold.

The provided method is transmitted to all coordinate charts defined on

the manifold.

INPUT:

- ``method`` -- string specifying the method to be used for

coordinate computations on this manifold; one of

- ``'SR'``: Sage's default symbolic engine (Symbolic Ring)

- ``'sympy'``: SymPy

The default calculus method relies on Sage's Symbolic Ring::

sage: M = Manifold(3, 'M', structure='topological')

sage: X.<x,y,z> = M.chart()

sage: f = M.scalar_field(sin(x)*cos(y) + z^2, name='F')

sage: f.expr()

z^2 + cos(y)*sin(x)

sage: type(f.expr())

sage: parent(f.expr())

Symbolic Ring

sage: f.display()

F: M --> R

(x, y, z) |--> z^2 + cos(y)*sin(x)

Changing to SymPy::

sage: M.set_calculus_method('sympy')

sage: f.expr()

z**2 + sin(x)*cos(y)

sage: type(f.expr())

sage: parent(f.expr())

sage: f.display()

F: M --> R

(x, y, z) |--> z**2 + sin(x)*cos(y)

Changing back to the Symbolic Ring::

sage: M.set_calculus_method('SR')

sage: f.display()

F: M --> R

(x, y, z) |--> z^2 + cos(y)*sin(x)

"""

self._calculus_method = method

for chart in self._atlas :

chart.set_calculus_method(method)

##############################################################################

## Constructor function

def Manifold(dim, name, latex_name=None, field='real', structure='smooth',

start_index=0, **extra_kwds):

r"""

Construct a manifold of a given type over a topological field.

Given a topological field `K` (in most applications, `K = \RR` or

`K = \CC`) and a non-negative integer `n`, a *topological manifold of

dimension* `n` *over K* is a topological space `M` such that

- `M` is a Hausdorff space,

- `M` is second countable, and

- every point in `M` has a neighborhood homeomorphic to `K^n`.

A *real manifold* is a manifold over `\RR`. A *differentiable* (resp.

*smooth*, resp. *analytic*) *manifold* is a manifold such that all

transition maps are *differentiable* (resp. *smooth*, resp. *analytic*). A

*pseudo-Riemannian manifold* is a real differentiable manifold equipped

with a metric tensor `g` (i.e. a field of non-degenerate symmetric bilinear

forms), with the two subcases of *Riemannian manifold* (`g`

positive-definite) and *Lorentzian manifold* (`g` has signature `n-2` or

`2-n`).

INPUT:

- ``dim`` -- positive integer; dimension of the manifold

- ``name`` -- string; name (symbol) given to the manifold

- ``latex_name`` -- (default: ``None``) string; LaTeX symbol to

denote the manifold; if none are provided, it is set to ``name``

- ``field`` -- (default: ``'real'``) field `K` on which the

manifold is defined; allowed values are

- ``'real'`` or an object of type ``RealField`` (e.g. ``RR``) for a

manifold over `\RR`

- ``'complex'`` or an object of type ``ComplexField`` (e.g. ``CC``)

for a manifold over `\CC`

- an object in the category of topological fields (see

:class:`~sage.categories.fields.Fields` and

:class:`~sage.categories.topological_spaces.TopologicalSpaces`)

for other types of manifolds

- ``structure`` -- (default: ``'smooth'``) to specify the structure or

type of manifold; allowed values are

- ``'topological'`` or ``'top'`` for a topological manifold

- ``'differentiable'`` or ``'diff'`` for a differentiable manifold

- ``'smooth'`` for a smooth manifold

- ``'analytic'`` for an analytic manifold

- ``'pseudo-Riemannian'`` for a real differentiable manifold equipped

with a pseudo-Riemannian metric; the signature is specified via the

keyword argument ``signature`` (see below)

- ``'Riemannian'`` for a real differentiable manifold equipped with a

Riemannian (i.e. positive definite) metric

- ``'Lorentzian'`` for a real differentiable manifold equipped with a

Lorentzian metric; the signature convention is specified by the

keyword argument ``signature='positive'`` (default) or ``'negative'``

- ``start_index`` -- (default: 0) integer; lower value of the range of

indices used for "indexed objects" on the manifold, e.g. coordinates

in a chart

- ``extra_kwds`` -- keywords meaningful only for some specific types

of manifolds:

- ``diff_degree`` -- (only for differentiable manifolds; default:

``infinity``): the degree of differentiability

- ``metric_name`` -- (only for pseudo-Riemannian manifolds; default:

``'g'``) string; name (symbol) given to the metric

- ``metric_latex_name`` -- (only for pseudo-Riemannian manifolds;

default: ``None``) string; LaTeX symbol to denote the metric; if none

is provided, the symbol is set to ``metric_name``

- ``signature`` -- (only for pseudo-Riemannian manifolds; default:

``None``) signature `S` of the metric as a single integer:

`S = n_+ - n_-`, where `n_+` (resp. `n_-`) is the number of positive

terms (resp. negative terms) in any diagonal writing of the

metric components; if ``signature`` is not provided, `S` is set to the

manifold's dimension (Riemannian signature); for Lorentzian manifolds

the values ``signature='positive'`` (default) or

``signature='negative'`` are allowed to indicate the chosen signature

convention.

OUTPUT:

- a manifold of the specified type, as an instance of

:class:`~sage.manifolds.manifold.TopologicalManifold` or one of its

subclasses

:class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`

:class:`~sage.manifolds.differentiable.pseudo_riemannian.PseudoRiemannianManifold`

EXAMPLES:

A 3-dimensional real topological manifold::

sage: M = Manifold(3, 'M', structure='topological'); M

3-dimensional topological manifold M

Given the default value of the parameter ``field``, the above is

equivalent to::

sage: M = Manifold(3, 'M', structure='topological', field='real'); M

3-dimensional topological manifold M

A complex topological manifold::

sage: M = Manifold(3, 'M', structure='topological', field='complex'); M

Complex 3-dimensional topological manifold M

A topological manifold over `\QQ`::

sage: M = Manifold(3, 'M', structure='topological', field=QQ); M

3-dimensional topological manifold M over the Rational Field

A 3-dimensional real differentiable manifold of class `C^4`::

sage: M = Manifold(3, 'M', field='real', structure='differentiable',

....: diff_degree=4); M

3-dimensional differentiable manifold M

Since the default value of the parameter ``field`` is ``'real'``, the above

is equivalent to::

sage: M = Manifold(3, 'M', structure='differentiable', diff_degree=4)

sage: M

3-dimensional differentiable manifold M

sage: M.base_field_type()

'real'

A 3-dimensional real smooth manifold::

sage: M = Manifold(3, 'M', structure='differentiable', diff_degree=+oo)

sage: M

3-dimensional differentiable manifold M

Instead of ``structure='differentiable', diff_degree=+oo``, it suffices to

use ``structure='smooth'`` to get the same result::

sage: M = Manifold(3, 'M', structure='smooth'); M

3-dimensional differentiable manifold M

sage: M.diff_degree()

+Infinity

Actually, since ``'smooth'`` is the default value of the parameter

``structure``, the creation of a real smooth manifold can be shortened to::

sage: M = Manifold(3, 'M'); M

3-dimensional differentiable manifold M

sage: M.diff_degree()

+Infinity

For a complex smooth manifold, we have to set the parameter ``field``::

sage: M = Manifold(3, 'M', field='complex'); M

3-dimensional complex manifold M

sage: M.diff_degree()

+Infinity

See the documentation of classes

:class:`~sage.manifolds.manifold.TopologicalManifold`,

:class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`

and

:class:`~sage.manifolds.differentiable.pseudo_riemannian.PseudoRiemannianManifold`

for more detailed examples.

.. RUBRIC:: Uniqueness of manifold objects

Suppose we construct a manifold named `M`::

sage: M = Manifold(2, 'M', structure='topological')

sage: X.<x,y> = M.chart()

At some point, we change our mind and would like to restart with a new

manifold, using the same name `M` and keeping the previous manifold for

reference::

sage: M_old = M # for reference

sage: M = Manifold(2, 'M', structure='topological')

This results in a brand new object::

sage: M.atlas()

[]

The object ``M_old`` is intact::

sage: M_old.atlas()

[Chart (M, (x, y))]

Both objects have the same display::

sage: M

2-dimensional topological manifold M

sage: M_old

2-dimensional topological manifold M

but they are different::

sage: M != M_old

True

Let us introduce a chart on ``M``, using the same coordinate symbols

as for ``M_old``::

sage: X.<x,y> = M.chart()

The charts are displayed in the same way::

sage: M.atlas()

[Chart (M, (x, y))]

sage: M_old.atlas()

[Chart (M, (x, y))]

but they are actually different::

sage: M.atlas()[0] != M_old.atlas()[0]

True

Moreover, the two manifolds ``M`` and ``M_old`` are still considered

distinct::

sage: M != M_old

True

This reflects the fact that the equality of manifold objects holds only

for identical objects, i.e. one has ``M1 == M2`` if, and only if,

``M1 is M2``. Actually, the manifold classes inherit from

:class:`~sage.misc.fast_methods.WithEqualityById`::

sage: isinstance(M, sage.misc.fast_methods.WithEqualityById)

True

"""

from time import time

from sage.rings.infinity import infinity

from sage.manifolds.differentiable.manifold import DifferentiableManifold

from sage.manifolds.differentiable.pseudo_riemannian import PseudoRiemannianManifold

# Some sanity checks

if not isinstance(dim, (int, Integer)):

raise TypeError("the manifold dimension must be an integer")

if dim < 1:

raise ValueError("the manifold dimension must be strictly positive")

if structure in ['topological', 'top']:

if field == 'real' or isinstance(field, RealField_class):

structure = RealTopologicalStructure()

else:

structure = TopologicalStructure()

return TopologicalManifold(dim, name, field, structure,

latex_name=latex_name,

start_index=start_index,

unique_tag=getrandbits(128)*time())

elif structure in ['differentiable', 'diff', 'smooth']:

if 'diff_degree' in extra_kwds:

diff_degree = extra_kwds['diff_degree']

if structure == 'smooth' and diff_degree != infinity:

raise ValueError("diff_degree = {} is ".format(diff_degree) +

"not compatible with a smooth structure")

else:

diff_degree = infinity

if field == 'real' or isinstance(field, RealField_class):

structure = RealDifferentialStructure()

else:

structure = DifferentialStructure()

return DifferentiableManifold(dim, name, field, structure,

diff_degree=diff_degree,

latex_name=latex_name,

start_index=start_index,

unique_tag=getrandbits(128)*time())

elif structure in ['pseudo-Riemannian', 'Riemannian', 'Lorentzian']:

if 'diff_degree' in extra_kwds:

diff_degree = extra_kwds['diff_degree']

else:

diff_degree = infinity

if 'metric_name' in extra_kwds:

metric_name = extra_kwds['metric_name']

else:

metric_name = 'g'

if 'metric_latex_name' in extra_kwds:

metric_latex_name = extra_kwds['metric_latex_name']

else:

metric_latex_name = None

if structure == 'pseudo-Riemannian':

if 'signature' in extra_kwds:

signature = extra_kwds['signature']

else:

signature = None

elif structure == 'Riemannian':

signature = dim

elif structure == 'Lorentzian':

if 'signature' in extra_kwds:

signat = extra_kwds['signature']

if signat == 'positive' or signat == dim - 2:

signature = dim - 2

elif signat == 'negative' or signat == 2 - dim:

signature = 2 - dim

else:

raise ValueError("signature {} not ".format(signat) +

"compatible with a Lorentzian " +

"manifold of dimension {}".format(dim))

else:

signature = dim - 2 # default value for a Lorentzian manifold

return PseudoRiemannianManifold(dim, name, metric_name=metric_name,

signature=signature,

diff_degree=diff_degree,

latex_name=latex_name,

metric_latex_name=metric_latex_name,

start_index=start_index,

unique_tag=getrandbits(128)*time())

raise NotImplementedError("manifolds of type {} are ".format(structure) +

"not implemented")

Manifold.options = TopologicalManifold.options

# Deprecations from trac:18555. July 2016

from sage.misc.superseded import deprecated_function_alias

Manifold.global_options=deprecated_function_alias(18555, TopologicalManifold.options)

ManifoldOptions = deprecated_function_alias(18555, TopologicalManifold.options)

TopologicalManifold.global_options=deprecated_function_alias(18555, TopologicalManifold.options)

Coverage for local/lib/python2.7/site-packages/sage/manifolds/manifold.py : 84%

335 statements 280 run 55 missing 0 excluded