Merge #31764

src/doc/en/reference/manifolds/manifold.rst

@@ -21,3 +21,5 @@ Topological Manifolds
    sage/manifolds/topological_submanifold
 
    vector_bundle
+
+   sage/manifolds/family

src/sage/graphs/digraph.py

@@ -3317,12 +3317,28 @@ def layout_acyclic_dummy(self, heights=None, rankdir='up', **options):
             ...
             ValueError: `self` should be an acyclic graph
 
+        TESTS:
+
+        :trac:`31681` is fixed::
+
+            sage: H = DiGraph({0: [1], 'X': [1]}, format='dict_of_lists')
+            sage: pos = H.layout_acyclic_dummy(rankdir='up')
+            sage: pos['X'][1] == 0 and pos[0][1] == 0
+            True
+            sage: pos[1][1] == 1
+            True
         """
         if heights is None:
             if not self.is_directed_acyclic():
                 raise ValueError("`self` should be an acyclic graph")
             levels = self.level_sets()
-            levels = [sorted(z) for z in levels]
+            # Sort vertices in each level in best effort mode
+            for i in range(len(levels)):
+                try:
+                    l = sorted(levels[i])
+                    levels[i] = l
+                except:
+                    continue
             if rankdir=='down' or rankdir=='left':
                 levels.reverse()
             heights = {i: levels[i] for i in range(len(levels))}

src/sage/manifolds/chart.py

@@ -318,9 +318,9 @@ def __init__(self, domain, coordinates='', names=None, calc_method=None):
         self._restrictions = []  # to be set with method add_restrictions()
         #
         # The chart is added to the domain's atlas, as well as to all the
-        # atlases of the domain's supersets; moreover the fist defined chart
+        # atlases of the domain's supersets; moreover the first defined chart
         # is considered as the default chart
-        for sd in self._domain._supersets:
+        for sd in self._domain.open_supersets():
             # the chart is added in the top charts only if its coordinates have
             # not been used:
             for chart in sd._atlas:
@@ -346,13 +346,9 @@ def __init__(self, domain, coordinates='', names=None, calc_method=None):
         # The null and one functions of the coordinates:
         # Expression in self of the zero and one scalar fields of open sets
         # containing the domain of self:
-        for dom in self._domain._supersets:
-            if hasattr(dom, '_zero_scalar_field'):
-                # dom is an open set
-                dom._zero_scalar_field._express[self] = self.function_ring().zero()
-            if hasattr(dom, '_one_scalar_field'):
-                # dom is an open set
-                dom._one_scalar_field._express[self] = self.function_ring().one()
+        for dom in self._domain.open_supersets():
+            dom._zero_scalar_field._express[self] = self.function_ring().zero()
+            dom._one_scalar_field._express[self] = self.function_ring().one()
 
     def _init_coordinates(self, coord_list):
         r"""
@@ -932,12 +928,9 @@ def transition_map(self, other, transformations, intersection_name=None,
         The subset `W`, intersection of `U` and `V`, has been created by
         ``transition_map()``::
 
-            sage: M.list_of_subsets()
-            [1-dimensional topological manifold S^1,
-             Open subset U of the 1-dimensional topological manifold S^1,
-             Open subset V of the 1-dimensional topological manifold S^1,
-             Open subset W of the 1-dimensional topological manifold S^1]
-            sage: W = M.list_of_subsets()[3]
+            sage: F = M.subset_family(); F
+            Set {S^1, U, V, W} of open subsets of the 1-dimensional topological manifold S^1
+            sage: W = F['W']
             sage: W is U.intersection(V)
             True
             sage: M.atlas()
@@ -960,9 +953,8 @@ def transition_map(self, other, transformations, intersection_name=None,
 
         In this case, no new subset has been created since `U \cap M = U`::
 
-            sage: M.list_of_subsets()
-            [2-dimensional topological manifold R^2,
-             Open subset U of the 2-dimensional topological manifold R^2]
+            sage: M.subset_family()
+            Set {R^2, U} of open subsets of the 2-dimensional topological manifold R^2
 
         but a new chart has been created: `(U, (x, y))`::
 
@@ -3010,7 +3002,7 @@ def __init__(self, chart1, chart2, *transformations):
         # is added to the subset (and supersets) dictionary:
         if chart1._domain == chart2._domain:
             domain = chart1._domain
-            for sdom in domain._supersets:
+            for sdom in domain.open_supersets():
                 sdom._coord_changes[(chart1, chart2)] = self
 
     def _repr_(self):

src/sage/manifolds/differentiable/chart.py

@@ -357,12 +357,9 @@ def transition_map(self, other, transformations, intersection_name=None,
         The subset `W`, intersection of `U` and `V`, has been created by
         ``transition_map()``::
 
-            sage: M.list_of_subsets()
-            [1-dimensional differentiable manifold S^1,
-             Open subset U of the 1-dimensional differentiable manifold S^1,
-             Open subset V of the 1-dimensional differentiable manifold S^1,
-             Open subset W of the 1-dimensional differentiable manifold S^1]
-            sage: W = M.list_of_subsets()[3]
+            sage: F = M.subset_family(); F
+            Set {S^1, U, V, W} of open subsets of the 1-dimensional differentiable manifold S^1
+            sage: W = F['W']
             sage: W is U.intersection(V)
             True
             sage: M.atlas()
@@ -385,9 +382,8 @@ def transition_map(self, other, transformations, intersection_name=None,
 
         In this case, no new subset has been created since `U\cap M = U`::
 
-            sage: M.list_of_subsets()
-            [2-dimensional differentiable manifold R^2,
-             Open subset U of the 2-dimensional differentiable manifold R^2]
+            sage: M.subset_family()
+            Set {R^2, U} of open subsets of the 2-dimensional differentiable manifold R^2
 
         but a new chart has been created: `(U, (x, y))`::
 
@@ -563,7 +559,7 @@ def restrict(self, subset, restrictions=None):
                 sframe._restrictions[subset] = resu._frame
             # The subchart frame is not a "top frame" in the supersets
             # (including self._domain):
-            for dom in self._domain._supersets:
+            for dom in self._domain.open_supersets():
                 if resu._frame in dom._top_frames:
                     # it was added by the Chart constructor invoked in
                     # Chart.restrict above
@@ -1046,7 +1042,7 @@ def restrict(self, subset, restrictions=None):
                 sframe._restrictions[subset] = resu._frame
             # The subchart frame is not a "top frame" in the supersets
             # (including self._domain):
-            for dom in self._domain._supersets:
+            for dom in self._domain.open_supersets():
                 if resu._frame in dom._top_frames:
                     # it was added by the Chart constructor invoked in
                     # Chart.restrict above
@@ -1138,15 +1134,15 @@ def __init__(self, chart1, chart2, *transformations):
             ch_basis.add_comp(frame1)[:, chart1] = self._jacobian
             ch_basis.add_comp(frame2)[:, chart1] = self._jacobian
             vf_module._basis_changes[(frame2, frame1)] = ch_basis
-            for sdom in domain._supersets:
+            for sdom in domain.open_supersets():
                 sdom._frame_changes[(frame2, frame1)] = ch_basis
             # The inverse is computed only if it does not exist already
             # (because if it exists it may have a simpler expression than that
             #  obtained from the matrix inverse)
             if (frame1, frame2) not in vf_module._basis_changes:
                 ch_basis_inv = ch_basis.inverse()
                 vf_module._basis_changes[(frame1, frame2)] = ch_basis_inv
-                for sdom in domain._supersets:
+                for sdom in domain.open_supersets():
                     sdom._frame_changes[(frame1, frame2)] = ch_basis_inv
 
     def jacobian(self):

src/sage/manifolds/differentiable/degenerate_submanifold.py

@@ -12,7 +12,7 @@
 lightlike submanifold) and in General Relativity. In geometry of lightlike
 submanifolds, according to the dimension `r` of the radical distribution
 (see below for definition of radical distribution), degenerate submanifolds
-have been classify into 4 subgroups: `r`-lightlike submanifolds, Coisotropic
+have been classified into 4 subgroups: `r`-lightlike submanifolds, Coisotropic
 submanifolds, Isotropic submanifolds and Totally lightlike submanifolds.
 (See the book of Krishan L. Duggal and Aurel Bejancu [DS2010]_.)
 

src/sage/manifolds/differentiable/diff_form_module.py

@@ -384,16 +384,14 @@ def _an_element_(self):
 
         """
         resu = self.element_class(self._vmodule, self._degree)
-        # Non-trivial open covers of the domain:
-        open_covers = self._domain.open_covers()[1:]  # the open cover 0
-                                                      # is trivial
-        if open_covers != []:
-            oc = open_covers[0]  # the first non-trivial open cover is selected
+        for oc in self._domain.open_covers(trivial=False):
+            # the first non-trivial open cover is selected
             for dom in oc:
                 vmodule_dom = dom.vector_field_module(
                                          dest_map=self._dest_map.restrict(dom))
                 dmodule_dom = vmodule_dom.dual_exterior_power(self._degree)
                 resu.set_restriction(dmodule_dom._an_element_())
+            return resu
         return resu
 
     def _coerce_map_from_(self, other):

src/sage/manifolds/differentiable/differentiable_submanifold.py

@@ -18,6 +18,8 @@
 AUTHORS:
 
 - Florentin Jaffredo (2018): initial version
+- Eric Gourgoulhon (2018-2019): add documentation
+- Matthias Koeppe (2021): open subsets of submanifolds
 
 REFERENCES:
 
@@ -26,7 +28,9 @@
 """
 
 # *****************************************************************************
-#  Copyright (C) 2018 Florentin Jaffredo <florentin.jaffredo@polytechnique.edu>
+#  Copyright (C) 2018      Florentin Jaffredo <florentin.jaffredo@polytechnique.edu>
+#  Copyright (C) 2018-2019 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr>
+#  Copyright (C) 2021      Matthias Koeppe <mkoeppe@math.ucdavis.edu>
 #
 # This program is free software: you can redistribute it and/or modify
 # it under the terms of the GNU General Public License as published by
@@ -208,10 +212,83 @@ def _repr_(self):
              3-dimensional differentiable manifold M
 
         """
+        if self is not self._manifold:
+            return "Open subset {} of the {}".format(self._name, self._manifold)
         if self._ambient is None:
             return super(DifferentiableManifold, self).__repr__()
         if self._embedded:
             return "{}-dimensional {} submanifold {} embedded in the {}".format(
                 self._dim, self._structure.name, self._name, self._ambient)
         return "{}-dimensional {} submanifold {} immersed in the {}".format(
                 self._dim, self._structure.name, self._name, self._ambient)
+
+    def open_subset(self, name, latex_name=None, coord_def={}, supersets=None):
+        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 differentiable
+        manifold by itself.
+
+        As ``self`` is a submanifold of its ambient manifold,
+        the new open subset is also considered a submanifold of that.
+        Hence the returned object is an instance of
+        :class:`DifferentiableSubmanifold`.
+
+        INPUT:
+
+        - ``name`` -- name given to the open subset
+        - ``latex_name`` --  (default: ``None``) LaTeX symbol to denote the
+          subset; if none is 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 in the manifold's atlas and values the symbolic expressions
+          formed by the coordinates to define the subset.
+        - ``supersets`` -- (default: only ``self``) list of sets that the
+          new open subset is a subset of
+
+        OUTPUT:
+
+        - the open subset, as an instance of :class:`DifferentiableSubmanifold`
+
+        EXAMPLES::
+
+            sage: M = Manifold(3, 'M', structure="differentiable")
+            sage: N = Manifold(2, 'N', ambient=M, structure="differentiable"); N
+            2-dimensional differentiable submanifold N immersed in the
+             3-dimensional differentiable manifold M
+            sage: S = N.subset('S'); S
+            Subset S of the
+             2-dimensional differentiable submanifold N immersed in the
+              3-dimensional differentiable manifold M
+            sage: O = N.subset('O', is_open=True); O  # indirect doctest
+            Open subset O of the
+             2-dimensional differentiable submanifold N immersed in the
+              3-dimensional differentiable manifold M
+
+            sage: phi = N.diff_map(M)
+            sage: N.set_embedding(phi)
+            sage: N
+            2-dimensional differentiable submanifold N embedded in the
+             3-dimensional differentiable manifold M
+            sage: S = N.subset('S'); S
+            Subset S of the
+             2-dimensional differentiable submanifold N embedded in the
+              3-dimensional differentiable manifold M
+            sage: O = N.subset('O', is_open=True); O  # indirect doctest
+            Open subset O of the
+             2-dimensional differentiable submanifold N embedded in the
+              3-dimensional differentiable manifold M
+
+        """
+        resu = DifferentiableSubmanifold(self._dim, name, self._field,
+                                         self._structure, ambient=self._ambient,
+                                         base_manifold=self._manifold,
+                                         diff_degree=self._diff_degree,
+                                         latex_name=latex_name,
+                                         start_index=self._sindex)
+        if supersets is None:
+            supersets = [self]
+        for superset in supersets:
+            superset._init_open_subset(resu, coord_def=coord_def)
+        return resu

src/sage/manifolds/differentiable/examples/real_line.py

@@ -229,11 +229,11 @@ class OpenInterval(DifferentiableManifold):
 
     We have::
 
-        sage: I.list_of_subsets()
+        sage: list(I.subset_family())
         [Real interval (0, 1), Real interval (0, pi), Real interval (1/2, 1)]
-        sage: J.list_of_subsets()
+        sage: list(J.subset_family())
         [Real interval (0, 1), Real interval (1/2, 1)]
-        sage: K.list_of_subsets()
+        sage: list(K.subset_family())
         [Real interval (1/2, 1)]
 
     As any open subset of a manifold, open subintervals are created in a
@@ -376,9 +376,7 @@ def __init__(self, lower, upper, ambient_interval=None,
             if upper > ambient_interval.upper_bound():
                 raise ValueError("the upper bound is larger than that of "
                                  + "the containing interval")
-            self._supersets.update(ambient_interval._supersets)
-            for sd in ambient_interval._supersets:
-                sd._subsets.add(self)
+            self.declare_subset(ambient_interval)
             ambient_interval._top_subsets.add(self)
             t = ambient_interval.canonical_coordinate()
         if lower != minus_infinity:
@@ -684,7 +682,7 @@ def open_interval(self, lower, upper, name=None, latex_name=None):
             Real interval (0, pi)
             sage: J.is_subset(I)
             True
-            sage: I.list_of_subsets()
+            sage: list(I.subset_family())
             [Real interval (-4, 4), Real interval (0, pi)]
 
         ``J`` is considered as an open submanifold of ``I``::
@@ -863,7 +861,7 @@ class RealLine(OpenInterval):
         Real interval (0, 1)
         sage: I.manifold()
         Real number line R
-        sage: R.list_of_subsets()
+        sage: list(R.subset_family())
         [Real interval (0, 1), Real number line R]
 
     """

src/sage/manifolds/differentiable/examples/sphere.py

@@ -275,12 +275,9 @@ class Sphere(PseudoRiemannianSubmanifold):
         sage: stereoN, stereoS = S2.coordinate_charts('stereographic')
         sage: stereoN, stereoS
         (Chart (S^2-{NP}, (y1, y2)), Chart (S^2-{SP}, (yp1, yp2)))
-        sage: S2.open_covers()
-        [[2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3],
-         [Open subset S^2-{NP} of the 2-sphere S^2 of radius 1 smoothly
-          embedded in the Euclidean space E^3,
-          Open subset S^2-{SP} of the 2-sphere S^2 of radius 1 smoothly
-          embedded in the Euclidean space E^3]]
+        sage: list(S2.open_covers())
+        [Set {S^2} of open subsets of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3,
+         Set {S^2-{NP}, S^2-{SP}} of open subsets of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3]
 
     .. NOTE::
 
@@ -501,11 +498,10 @@ def _init_chart_domains(self):
         TESTS::
 
             sage: S2 = manifolds.Sphere(2)
-            sage: S2.open_covers()
-            [[2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3],
-             [Open subset S^2-{NP} of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3,
-              Open subset S^2-{SP} of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3]]
-            sage: S2.subsets()  # random
+            sage: list(S2.open_covers())
+            [Set {S^2} of open subsets of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3,
+             Set {S^2-{NP}, S^2-{SP}} of open subsets of the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3]
+            sage: frozenset(S2.subsets())  # random
             frozenset({Euclidean 2-sphere S^2 of radius 1,
              Open subset A of the Euclidean 2-sphere S^2 of radius 1,
              Open subset S^2-{NP,SP} of the Euclidean 2-sphere S^2 of radius 1,