Add a bit of typing to manifolds

src/sage/manifolds/chart.py

@@ -2671,7 +2671,7 @@ def plot(self, chart=None, ambient_coords=None, mapping=None,
         from sage.plot.graphics import Graphics
         from sage.plot.line import line
         from sage.manifolds.continuous_map import ContinuousMap
-        from .utilities import set_axes_labels
+        from sage.manifolds.utilities import set_axes_labels
 
         # Extract the kwds options
         max_range = kwds['max_range']

src/sage/manifolds/differentiable/degenerate.py

@@ -13,6 +13,7 @@
 from sage.rings.infinity import infinity
 from sage.manifolds.structure import DegenerateStructure
 from sage.manifolds.differentiable.manifold import DifferentiableManifold
+from sage.manifolds.differentiable.metric import DegenerateMetric 
 
 ###############################################################################
 
@@ -141,7 +142,7 @@ def __init__(self, n, name, metric_name='g', signature=None, base_manifold=None,
             self._metric_latex_name = metric_latex_name
 
     def metric(self, name=None, signature=None, latex_name=None,
-               dest_map=None):
+               dest_map=None) -> DegenerateMetric:
         r"""
         Return the metric giving the null manifold structure to the
         manifold, or define a new metric tensor on the manifold.

src/sage/manifolds/differentiable/degenerate_submanifold.py

@@ -162,6 +162,7 @@
                                                       TangentTensor)
 from sage.manifolds.differentiable.differentiable_submanifold import \
     DifferentiableSubmanifold
+from sage.manifolds.differentiable.metric import DegenerateMetric
 from sage.manifolds.differentiable.vectorfield_module import VectorFieldModule
 from sage.rings.infinity import infinity
 from sage.matrix.constructor import matrix
@@ -596,14 +597,13 @@ def screen(self, name, screen, rad, latex_name=None):
         self._default_screen = self._screens[name]
         return self._screens[name]
 
-    def induced_metric(self):
+    def induced_metric(self) -> DegenerateMetric:
         r"""
         Return the pullback of the ambient metric.
 
         OUTPUT:
 
-        - induced mettric, as an instance of
-          :class:`~sage.manifolds.differentiable.metric.DegenerateMetric`
+        - induced mettric
 
         EXAMPLES:
 

src/sage/manifolds/differentiable/manifold.py

@@ -441,12 +441,17 @@
 
 from sage.categories.manifolds import Manifolds
 from sage.categories.homset import Hom
+from sage.manifolds.continuous_map import ContinuousMap
+from sage.manifolds.differentiable.diff_map import DiffMap
 from sage.rings.all import CC
 from sage.rings.real_mpfr import RR
 from sage.rings.infinity import infinity, minus_infinity
 from sage.rings.integer import Integer
 from sage.manifolds.manifold import TopologicalManifold
 from sage.manifolds.differentiable.mixed_form_algebra import MixedFormAlgebra
+from sage.manifolds.differentiable.vectorfield_module import VectorFieldModule
+from sage.manifolds.differentiable.metric import PseudoRiemannianMetric
+from sage.misc.cachefunc import cached_method
 
 ###############################################################################
 
@@ -1200,24 +1205,23 @@ def tensor_bundle(self, k, l, dest_map=None):
                                                               dest_map=dest_map)
         return self._tensor_bundles[dest_map][(k, l)]
 
-    def vector_field_module(self, dest_map=None, force_free=False):
+    def vector_field_module(self, dest_map: DiffMap=None, force_free=False) -> VectorFieldModule:
         r"""
         Return the set of vector fields defined on ``self``, possibly
         with values in another differentiable manifold, as a module over the
         algebra of scalar fields defined on the manifold.
+        That is, returns `\Gamma^\infty(\Phi^* TN)` where `\Phi:\ M \to N` is a smooth map.
 
         See :class:`~sage.manifolds.differentiable.vectorfield_module.VectorFieldModule`
         for a complete documentation.
 
         INPUT:
 
         - ``dest_map`` -- (default: ``None``) destination map, i.e. a
-          differentiable map `\Phi:\ M \rightarrow N`, where `M` is the
+          differentiable map `\Phi:\ M \to N`, where `M` is the
           current manifold and `N` a differentiable manifold;
           if ``None``, it is assumed that `N = M` and that `\Phi` is the
-          identity map (case of vector fields *on* `M`), otherwise
-          ``dest_map`` must be a
-          :class:`~sage.manifolds.differentiable.diff_map.DiffMap`
+          identity map (case of vector fields *on* `M`)
         - ``force_free`` -- (default: ``False``) if set to ``True``, force
           the construction of a *free* module (this implies that `N` is
           parallelizable)
@@ -1350,7 +1354,7 @@ def vector_field_module(self, dest_map=None, force_free=False):
                                        VectorFieldModule, VectorFieldFreeModule
         if dest_map is None:
             dest_map = self.identity_map()
-        codomain = dest_map._codomain
+        codomain = dest_map.codomain()
         if dest_map not in self._vector_field_modules:
             if codomain.is_manifestly_parallelizable() or force_free:
                 self._vector_field_modules[dest_map] = \
@@ -3703,7 +3707,7 @@ def affine_connection(self, name, latex_name=None):
                                                                AffineConnection
         return AffineConnection(self, name, latex_name)
 
-    def metric(self, name, signature=None, latex_name=None, dest_map=None):
+    def metric(self, name, signature=None, latex_name=None, dest_map=None) -> PseudoRiemannianMetric:
         r"""
         Define a pseudo-Riemannian metric on the manifold.
 
@@ -3936,3 +3940,43 @@ class :class:`~sage.manifolds.differentiable.diff_map.DiffMap`
         else:
             signat = 2 - dim
         return vmodule.metric(name, signature=signat, latex_name=latex_name)
+
+    @cached_method
+    def identity_map(self) -> DiffMap:
+        r"""
+        Identity map of ``self``.
+
+        The identity map of a differentiable manifold `M` is the trivial
+        diffeomorphism:
+
+        .. MATH::
+
+            \begin{array}{cccc}
+            \mathrm{Id}_M: & M & \longrightarrow & M \\
+                & p & \longmapsto & p
+            \end{array}
+
+        OUTPUT:
+
+        - the identity map
+
+        EXAMPLES:
+
+        Identity map of a differentiable manifold::
+
+            sage: M = Manifold(2, 'M', structure='differentiable')
+            sage: id = M.identity_map(); id
+            Identity map Id_M of the 2-dimensional differentiable manifold M
+            sage: id.display()
+            Id_M: M --> M
+            sage: isinstance(id, DiffMap)
+            True
+
+        Identity map of a topological manifold is not smooth::
+        
+            sage: M = Manifold(2, 'M', structure='topological')
+            sage: isinstance(M.identity_map(), DiffMap)
+            False
+
+        """
+        return Hom(self, self).one()

src/sage/manifolds/differentiable/metric.py

@@ -39,10 +39,13 @@
 #  the License, or (at your option) any later version.
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
+from typing import overload, TYPE_CHECKING
 
 from sage.rings.integer import Integer
 from sage.manifolds.differentiable.tensorfield import TensorField
 from sage.manifolds.differentiable.tensorfield_paral import TensorFieldParal
+if TYPE_CHECKING:
+    from sage.manifolds.differentiable.diff_form import DiffForm
 
 
 class PseudoRiemannianMetric(TensorField):
@@ -788,8 +791,7 @@ def connection(self, name=None, latex_name=None, init_coef=True):
             True
 
         """
-        from sage.manifolds.differentiable.levi_civita_connection import \
-                                                           LeviCivitaConnection
+        from sage.manifolds.differentiable.levi_civita_connection import LeviCivitaConnection
         if self._connection is None:
             if latex_name is None:
                 if name is None:
@@ -1616,6 +1618,12 @@ def sqrt_abs_det(self, frame=None):
             self._sqrt_abs_dets[frame] = resu
         return self._sqrt_abs_dets[frame]
 
+    @overload
+    def volume_form(self) -> 'DiffForm':
+        pass
+    @overload
+    def volume_form(self, contra: int) -> TensorField:
+        pass
     def volume_form(self, contra=0):
         r"""
         Volume form (Levi-Civita tensor) `\epsilon` associated with the metric.
@@ -1907,8 +1915,7 @@ def hodge_star(self, pform):
 
         """
         from sage.functions.other import factorial
-        from sage.tensor.modules.format_utilities import format_unop_txt, \
-                                                         format_unop_latex
+        from sage.tensor.modules.format_utilities import format_unop_txt, format_unop_latex
         p = pform.tensor_type()[1]
         eps = self.volume_form(p)
         if p == 0:

src/sage/manifolds/differentiable/vectorfield_module.py

@@ -38,6 +38,8 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
+from sage.manifolds.differentiable.diff_map import DiffMap
+from sage.manifolds.differentiable.manifold import DifferentiableManifold
 from sage.structure.unique_representation import UniqueRepresentation
 from sage.structure.parent import Parent
 from sage.categories.modules import Modules
@@ -59,7 +61,7 @@ class VectorFieldModule(UniqueRepresentation, Parent):
 
         \Phi:\  U \longrightarrow M,
 
-    the *vector field module* `\mathfrak{X}(U,\Phi)` is the set of
+    the *vector field module* `\mathfrak{X}(U,\Phi) = \Gamma(\Phi^* TM)` is the set of
     all vector fields of the type
 
     .. MATH::
@@ -73,6 +75,7 @@ class VectorFieldModule(UniqueRepresentation, Parent):
         \forall p \in U,\ v(p) \in T_{\Phi(p)}M,
 
     where `T_{\Phi(p)}M` is the tangent space to `M` at the point `\Phi(p)`.
+    Thus elements of `\mathfrak{X}(U,\Phi)` are vector fields along `\Phi`. 
 
     The set `\mathfrak{X}(U,\Phi)` is a module over `C^k(U)`, the ring
     (algebra) of differentiable scalar fields on `U` (see
@@ -103,9 +106,8 @@ class VectorFieldModule(UniqueRepresentation, Parent):
 
     - ``domain`` -- differentiable manifold `U` along which the
       vector fields are defined
-    - ``dest_map`` -- (default: ``None``) destination map
+    - ``dest_map`` -- (default: ``None``) smooth destination map
       `\Phi:\ U \rightarrow M`
-      (type: :class:`~sage.manifolds.differentiable.diff_map.DiffMap`);
       if ``None``, it is assumed that `U = M` and `\Phi` is the identity
       map of `M` (case of vector fields *on* `M`)
 
@@ -181,7 +183,7 @@ class VectorFieldModule(UniqueRepresentation, Parent):
     """
     Element = VectorField
 
-    def __init__(self, domain, dest_map=None):
+    def __init__(self, domain: DifferentiableManifold, dest_map: DiffMap=None):
         r"""
         Construct the module of vector fields taking values on a (a priori)
         non-parallelizable differentiable manifold.
@@ -227,7 +229,7 @@ def __init__(self, domain, dest_map=None):
             latex_name += "," + dm_latex_name
         self._name = name + ")"
         self._latex_name = latex_name + r"\right)"
-        self._ambient_domain = self._dest_map._codomain
+        self._ambient_domain = self._dest_map.codomain()
         # The member self._ring is created for efficiency (to avoid
         # calls to self.base_ring()):
         self._ring = domain.scalar_field_algebra()
@@ -1539,7 +1541,7 @@ def _repr_(self):
                            " mapped into the {}".format(self._ambient_domain)
         return description
 
-    def domain(self):
+    def domain(self) -> DifferentiableManifold:
         r"""
         Return the domain of the vector fields in ``self``.
 
@@ -1548,10 +1550,8 @@ def domain(self):
 
         OUTPUT:
 
-        - a
-          :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`
-          representing the domain of the vector fields that belong to this
-          module
+        - a differentiable manifold representing the domain of the vector fields
+          that belong to this module
 
         EXAMPLES::
 
@@ -1570,7 +1570,7 @@ def domain(self):
         """
         return self._domain
 
-    def ambient_domain(self):
+    def ambient_domain(self) -> DifferentiableManifold:
         r"""
         Return the manifold in which the vector fields of ``self``
         take their values.
@@ -1580,10 +1580,8 @@ def ambient_domain(self):
 
         OUTPUT:
 
-        - a
-          :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`
-          representing the manifold in which the vector fields of ``self``
-          take their values
+        - a differentiable manifold representing the manifold in which the 
+          vector fields of ``self`` take their values
 
         EXAMPLES::
 
@@ -1602,7 +1600,7 @@ def ambient_domain(self):
         """
         return self._ambient_domain
 
-    def destination_map(self):
+    def destination_map(self) -> DiffMap:
         r"""
         Return the differential map associated to ``self``.
 
@@ -1630,8 +1628,7 @@ def destination_map(self):
 
         OUTPUT:
 
-        - a :class:`~sage.manifolds.differentiable.diff_map.DiffMap`
-          representing the differential map `\Phi`
+        - the differentiable map `\Phi`
 
         EXAMPLES::
 

src/sage/manifolds/manifold.py

@@ -341,6 +341,10 @@
 from sage.manifolds.structure import(
                             TopologicalStructure, RealTopologicalStructure,
                             DifferentialStructure, RealDifferentialStructure)
+from typing import Union, TYPE_CHECKING, Optional
+if TYPE_CHECKING:
+    from sage.manifolds.differentiable.manifold import DifferentiableManifold
+    from sage.manifolds.continuous_map import ContinuousMap
 
 
 #############################################################################
@@ -2143,7 +2147,7 @@ def homeomorphism(self, codomain, coord_functions=None, chart1=None,
                       is_isomorphism=True)
 
     @cached_method
-    def identity_map(self):
+    def identity_map(self) -> 'ContinuousMap':
         r"""
         Identity map of ``self``.
 
@@ -2399,8 +2403,8 @@ def set_simplify_function(self, simplifying_func, method=None):
 
 _manifold_id = Integer(0)
 
-def Manifold(dim, name, latex_name=None, field='real', structure='smooth',
-             start_index=0, **extra_kwds):
+def Manifold(dim:int, name:Optional[str], latex_name:Optional[str]=None, field='real', structure='smooth',
+             start_index=0, **extra_kwds) ->  Union[TopologicalManifold, 'DifferentiableManifold']:
     r"""
     Construct a manifold of a given type over a topological field.