ContinuousMap.preimage: New, make pullback an alias; add example

src/sage/manifolds/continuous_map.py

@@ -866,11 +866,50 @@ def image(self, subset=None, inverse=None):
                 return subset
         return ImageManifoldSubset(self, inverse=inverse, domain_subset=subset)
 
-    def pullback(self, codomain_subset, name=None, latex_name=None):
+    def preimage(self, codomain_subset, name=None, latex_name=None):
+        r"""
+        Return the preimage of ``codomain_subset`` under ``self``.
+
+        INPUT:
+
+        - ``codomain_subset`` -- an instance of :class:`ManifoldSubset`
+        - ``name`` -- string; name (symbol) given to the subset
+        - ``latex_name`` --  (default: ``None``) string; LaTeX symbol to
+          denote the subset; if none are provided, it is set to ``name``
+
+        OUTPUT:
+
+        - an instance of
+          :class:`~sage.manifolds.subsets.pullback.ManifoldSubsetPullback`
+
+        EXAMPLES::
+
+            sage: R = Manifold(1, 'R', structure='topological')  # field R
+            sage: T.<t> = R.chart()  # canonical chart on R
+            sage: R2 = Manifold(2, 'R^2', structure='topological')  # R^2
+            sage: c_xy.<x,y> = R2.chart() # Cartesian coordinates on R^2
+            sage: Phi = R.continuous_map(R2, [cos(t), sin(t)], name='Phi'); Phi
+            Continuous map Phi
+             from the 1-dimensional topological manifold R
+             to the 2-dimensional topological manifold R^2
+            sage: Q1 = R2.open_subset('Q1', coord_def={c_xy: [x>0, y>0]}); Q1
+            Open subset Q1 of the 2-dimensional topological manifold R^2
+            sage: Phi_inv_Q1 = Phi.preimage(Q1); Phi_inv_Q1
+            Subset Phi_inv_Q1 of the 1-dimensional topological manifold R
+            sage: R.point([pi/4]) in Phi_inv_Q1
+            True
+            sage: R.point([0]) in Phi_inv_Q1
+            False
+            sage: R.point([3*pi/4]) in Phi_inv_Q1
+            False
+
+        """
         from sage.manifolds.subsets.pullback import ManifoldSubsetPullback
         return ManifoldSubsetPullback(self, codomain_subset=codomain_subset,
                                       name=name, latex_name=latex_name)
 
+    pullback = preimage
+
     #
     # Monoid methods
     #

src/sage/manifolds/differentiable/diff_map.py

@@ -840,7 +840,7 @@ def jacobian_matrix(self, chart1=None, chart2=None):
         return matrix([[diff_funct[i][j].expr() for j in range(n1)]
                        for i in range(n2)])
 
-    def pullback(self, tensor):
+    def pullback(self, tensor_or_codomain_subset, name=None, latex_name=None):
         r"""
         Pullback operator associated with ``self``.
 
@@ -934,6 +934,11 @@ def pullback(self, tensor):
             (2*cos(t) + 2) dt*dt
 
         """
+        if not hasattr(tensor_or_codomain_subset, '_domain'):
+            return super().pullback(tensor_or_codomain_subset,
+                                    name=name, latex_name=latex_name)
+        tensor = tensor_or_codomain_subset
+
         from sage.manifolds.differentiable.tensorfield_paral import TensorFieldParal
         from sage.manifolds.differentiable.vectorframe import CoordFrame
         from sage.tensor.modules.comp import (Components, CompWithSym,

src/sage/manifolds/subsets/pullback.py

@@ -21,6 +21,7 @@
 from sage.rings.integer_ring import ZZ
 from sage.rings.rational_field import QQ
 from sage.manifolds.subset import ManifoldSubset
+from sage.manifolds.continuous_map import ContinuousMap
 from sage.manifolds.chart import Chart
 from sage.sets.real_set import RealSet
 from sage.geometry.polyhedron.base import is_Polyhedron
@@ -157,7 +158,7 @@ def __classcall_private__(cls, map, inverse=None, codomain_subset=None,
         if cls._is_open(codomain_subset):
 
             try:
-                coord_def = cls._coord_def(codomain_subset)
+                coord_def = cls._coord_def(map, codomain_subset)
             except NotImplementedError:
                 pass
             else:
@@ -200,7 +201,9 @@ def _is_open(codomain_subset):
         return False
 
     @staticmethod
-    def _coord_def(codomain_subset):
+    def _coord_def(map, codomain_subset):
+
+        #if isinstance(map, ContinuousMap) and isinstance(codomain_subset, Manifold):
 
         raise NotImplementedError
 
@@ -230,8 +233,8 @@ def is_open(self):
         Return if ``self`` is an open set.
 
         """
-        # Because the map is continuous, the pullback is open if and only
-        # if the codomain_subset is open.  But because other code assumes
+        # Because the map is continuous, the pullback is open if the
+        # codomain_subset is open.  But because other code assumes
         # that open subsets are instances of Manifold, we do not use this
         # fact here. Instead, the constructor is responsible for creating
         # an instance of the appropriate subclass.
@@ -319,6 +322,8 @@ def is_closed(self):
             False
 
         """
+        if self.manifold().dimension() == 0:
+            return True
         if isinstance(self._codomain_subset, ManifoldSubset):
             if self._codomain_subset.is_closed():
                 # known closed