Remove identity_map duplication

src/sage/manifolds/differentiable/manifold.py

@@ -3940,43 +3940,3 @@ 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/manifold.py

@@ -341,10 +341,12 @@
 from sage.manifolds.structure import(
                             TopologicalStructure, RealTopologicalStructure,
                             DifferentialStructure, RealDifferentialStructure)
-from typing import Union, TYPE_CHECKING, Optional
+from typing import Union, TYPE_CHECKING, Optional, overload
 if TYPE_CHECKING:
+    from sage.manifolds.manifold import TopologicalManifold
     from sage.manifolds.differentiable.manifold import DifferentiableManifold
     from sage.manifolds.continuous_map import ContinuousMap
+    from sage.manifolds.differentiable.diff_map import DiffMap
 
 
 #############################################################################
@@ -2146,8 +2148,12 @@ def homeomorphism(self, codomain, coord_functions=None, chart1=None,
         return homset(coord_functions, name=name, latex_name=latex_name,
                       is_isomorphism=True)
 
+    @overload
+    def identity_map(self: 'DifferentiableManifold') -> 'DiffMap': ...
+    @overload
+    def identity_map(self: 'TopologicalManifold') -> 'ContinuousMap': ...
     @cached_method
-    def identity_map(self) -> 'ContinuousMap':
+    def identity_map(self):
         r"""
         Identity map of ``self``.
 
@@ -2191,6 +2197,20 @@ def identity_map(self) -> 'ContinuousMap':
             sage: id(p) == p
             True
 
+        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: 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
+
         .. SEEALSO::
 
             See :class:`~sage.manifolds.continuous_map.ContinuousMap`