Merge #31677

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"""
@@ -3006,7 +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

@@ -559,7 +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
@@ -1042,7 +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
@@ -1134,15 +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/differentiable_submanifold.py

@@ -208,10 +208,83 @@ 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/manifold.py

@@ -717,7 +717,7 @@ def diff_degree(self):
         """
         return self._diff_degree
 
-    def open_subset(self, name, latex_name=None, coord_def={}):
+    def open_subset(self, name, latex_name=None, coord_def={}, supersets=None):
         r"""
         Create an open subset of the manifold.
 
@@ -735,6 +735,8 @@ def open_subset(self, name, latex_name=None, coord_def={}):
           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:
 
@@ -819,25 +821,39 @@ def open_subset(self, name, latex_name=None, coord_def={}):
                                       diff_degree=self._diff_degree,
                                       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)
-        #!# update vector frames and change of frames
+        if supersets is None:
+            supersets = [self]
+        for superset in supersets:
+            superset._init_open_subset(resu, coord_def=coord_def)
         return resu
 
+    def _init_open_subset(self, resu, coord_def):
+        r"""
+        Initialize ``resu`` as an open subset of ``self``.
+
+        INPUT:
+
+        - ``resu`` -- an instance of ``:class:`TopologicalManifold` or
+          a subclass.
+
+        - ``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
+
+        EXAMPLES::
+
+            sage: M = Manifold(2, 'R^2', structure='differentiable')
+            sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
+            sage: from sage.manifolds.differentiable.manifold import DifferentiableManifold
+            sage: U = DifferentiableManifold(2, 'U', field=M._field, structure=M._structure, base_manifold=M)
+            sage: M._init_open_subset(U, coord_def={c_cart: x^2+y^2<1})
+            sage: U
+            Open subset U of the 2-dimensional differentiable manifold R^2
+        """
+        super()._init_open_subset(resu, coord_def=coord_def)
+        #!# update vector frames and change of frames
+
     def diff_map(self, codomain, coord_functions=None, chart1=None,
                        chart2=None, name=None, latex_name=None):
         r"""
@@ -2736,7 +2752,7 @@ def orientation(self):
         """
         if not self._orientation:
             # try to get an orientation from super domains:
-            for sdom in self._supersets:
+            for sdom in self.open_supersets():
                 sorient = sdom._orientation
                 if sorient:
                     rst_orient = [f.restrict(self) for f in sorient]
@@ -2944,11 +2960,11 @@ def set_change_of_frame(self, frame1, frame2, change_of_frame,
                             "instance of AutomorphismFieldParal")
         fmodule.set_change_of_basis(frame1, frame2, change_of_frame,
                                     compute_inverse=compute_inverse)
-        for sdom in self._supersets:
+        for sdom in self.open_supersets():
             sdom._frame_changes[(frame1, frame2)] = change_of_frame
         if compute_inverse:
             if (frame2, frame1) not in self._frame_changes:
-                for sdom in self._supersets:
+                for sdom in self.open_supersets():
                     sdom._frame_changes[(frame2, frame1)] = change_of_frame.inverse()
 
     def vector_frame(self, *args, **kwargs):
@@ -3134,7 +3150,7 @@ def vector_frame(self, *args, **kwargs):
             # dictionary _frame_changes of self and its supersets:
             for frame_pair, chge in resu._fmodule._basis_changes.items():
                 if resu in frame_pair:
-                    for sdom in self._supersets:
+                    for sdom in self.open_supersets():
                         sdom._frame_changes[frame_pair] = chge
         return resu
 
@@ -3161,7 +3177,7 @@ def _set_covering_frame(self, frame):
         self._covering_frames.append(frame)
         self._parallelizable_parts = set([self])
         # if self contained smaller parallelizable parts, they are forgotten
-        for sd in self._supersets:
+        for sd in self.open_supersets():
             if not sd.is_manifestly_parallelizable():
                 sd._parallelizable_parts.add(self)
 

src/sage/manifolds/differentiable/pseudo_riemannian.py

@@ -672,7 +672,7 @@ def volume_form(self, contra=0):
         """
         return self.metric().volume_form(contra=contra)
 
-    def open_subset(self, name, latex_name=None, coord_def={}):
+    def open_subset(self, name, latex_name=None, coord_def={}, supersets=None):
         r"""
         Create an open subset of ``self``.
 
@@ -692,6 +692,8 @@ def open_subset(self, name, latex_name=None, coord_def={}):
           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:
 
@@ -745,22 +747,8 @@ def open_subset(self, name, latex_name=None, coord_def={}):
                                         latex_name=latex_name,
                                         metric_latex_name=self._metric_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)
-        #!# update non-coordinate vector frames and change of frames
-        #
+        if supersets is None:
+            supersets = [self]
+        for superset in supersets:
+            superset._init_open_subset(resu, coord_def=coord_def)
         return resu