diff --git a/src/sage/manifolds/chart.py b/src/sage/manifolds/chart.py
index b35fe15476c..e5f7ad47a46 100644
--- a/src/sage/manifolds/chart.py
+++ b/src/sage/manifolds/chart.py
@@ -3037,7 +3037,7 @@ 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']
diff --git a/src/sage/manifolds/differentiable/degenerate.py b/src/sage/manifolds/differentiable/degenerate.py
index aa3a301255b..57828ad8dee 100644
--- a/src/sage/manifolds/differentiable/degenerate.py
+++ b/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 
 
 ###############################################################################
 
@@ -143,7 +144,7 @@ def __init__(self, n, name, metric_name=None, signature=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.
diff --git a/src/sage/manifolds/differentiable/degenerate_submanifold.py b/src/sage/manifolds/differentiable/degenerate_submanifold.py
index 03c2f769b12..7ae03974d69 100644
--- a/src/sage/manifolds/differentiable/degenerate_submanifold.py
+++ b/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
@@ -590,7 +591,7 @@ 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.
 
diff --git a/src/sage/manifolds/differentiable/diff_form.py b/src/sage/manifolds/differentiable/diff_form.py
index 0aa3469770d..6b045a91c57 100644
--- a/src/sage/manifolds/differentiable/diff_form.py
+++ b/src/sage/manifolds/differentiable/diff_form.py
@@ -45,13 +45,14 @@
 # *****************************************************************************
 
 from __future__ import annotations
-from typing import Union, TYPE_CHECKING
+from typing import Optional, Union, TYPE_CHECKING
 from sage.misc.cachefunc import cached_method
 from sage.tensor.modules.free_module_alt_form import FreeModuleAltForm
 from sage.manifolds.differentiable.tensorfield import TensorField
 from sage.manifolds.differentiable.tensorfield_paral import TensorFieldParal
 
 if TYPE_CHECKING:
+    from sage.manifolds.differentiable.vectorfield_module import VectorFieldModule
     from sage.manifolds.differentiable.metric import PseudoRiemannianMetric
     from sage.manifolds.differentiable.symplectic_form import SymplecticForm
 
@@ -384,7 +385,7 @@ def _del_derived(self):
         self.exterior_derivative.clear_cache()
 
     @cached_method
-    def exterior_derivative(self):
+    def exterior_derivative(self) -> DiffForm:
         r"""
         Compute the exterior derivative of ``self``.
 
@@ -590,7 +591,7 @@ def wedge(self, other: DiffForm) -> DiffForm:
                                           other_r._restrictions[dom])
         return resu
 
-    def degree(self):
+    def degree(self) -> int:
         r"""
         Return the degree of ``self``.
 
@@ -615,7 +616,9 @@ def degree(self):
 
     def hodge_dual(
         self,
-        nondegenerate_tensor: Union[PseudoRiemannianMetric, SymplecticForm, None] = None,
+        nondegenerate_tensor: Union[
+            PseudoRiemannianMetric, SymplecticForm, None
+        ] = None,
     ) -> DiffForm:
         r"""
         Compute the Hodge dual of the differential form with respect to some non-degenerate
@@ -1228,8 +1231,8 @@ class DiffFormParal(FreeModuleAltForm, TensorFieldParal, DiffForm):
         no symmetry;  no antisymmetry
 
     """
-    def __init__(self, vector_field_module, degree, name=None,
-                 latex_name=None):
+    def __init__(self, vector_field_module: VectorFieldModule, degree: int, name: Optional[str] = None,
+                 latex_name: Optional[str] = None):
         r"""
         Construct a differential form.
 
@@ -1382,7 +1385,7 @@ def __call__(self, *args):
         return TensorFieldParal.__call__(self, *args)
 
     @cached_method
-    def exterior_derivative(self):
+    def exterior_derivative(self) -> DiffFormParal:
         r"""
         Compute the exterior derivative of ``self``.
 
diff --git a/src/sage/manifolds/differentiable/manifold.py b/src/sage/manifolds/differentiable/manifold.py
index f83cbdfe096..359edbfa205 100644
--- a/src/sage/manifolds/differentiable/manifold.py
+++ b/src/sage/manifolds/differentiable/manifold.py
@@ -443,12 +443,15 @@
 from typing import Optional, TYPE_CHECKING
 from sage.categories.manifolds import Manifolds
 from sage.categories.homset import Hom
+from sage.manifolds.differentiable.diff_map import DiffMap
 from sage.rings.cc 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
 
 if TYPE_CHECKING:
     from sage.manifolds.differentiable.vectorfield_module import VectorFieldModule
@@ -3945,7 +3948,9 @@ 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: str, signature: Optional[int] = None, 
+               latex_name: Optional[str] = None,
+               dest_map: Optional[DiffMap] = None) -> PseudoRiemannianMetric:
         r"""
         Define a pseudo-Riemannian metric on the manifold.
 
diff --git a/src/sage/manifolds/differentiable/metric.py b/src/sage/manifolds/differentiable/metric.py
index feff8a0899b..686af0d3505 100644
--- a/src/sage/manifolds/differentiable/metric.py
+++ b/src/sage/manifolds/differentiable/metric.py
@@ -41,16 +41,18 @@
 #  the License, or (at your option) any later version.
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
-
 from __future__ import annotations
-from typing import TYPE_CHECKING
-from sage.rings.integer import Integer
+
+from typing import TYPE_CHECKING, overload
+
 from sage.manifolds.differentiable.tensorfield import TensorField
 from sage.manifolds.differentiable.tensorfield_paral import TensorFieldParal
+from sage.rings.integer import Integer
 
 if TYPE_CHECKING:
     from sage.manifolds.differentiable.diff_form import DiffForm
 
+
 class PseudoRiemannianMetric(TensorField):
     r"""
     Pseudo-Riemannian metric with values on an open subset of a
@@ -1624,7 +1626,11 @@ def sqrt_abs_det(self, frame=None):
             self._sqrt_abs_dets[frame] = resu
         return self._sqrt_abs_dets[frame]
 
-    def volume_form(self, contra: int = 0) -> TensorField:
+    @overload
+    def volume_form(self) -> DiffForm: ...
+    @overload
+    def volume_form(self, contra: int) -> TensorField: ...
+    def volume_form(self, contra=0):
         r"""
         Volume form (Levi-Civita tensor) `\epsilon` associated with the metric.
 
diff --git a/src/sage/manifolds/differentiable/tensorfield.py b/src/sage/manifolds/differentiable/tensorfield.py
index 784ae9a6c41..808368cdb6c 100644
--- a/src/sage/manifolds/differentiable/tensorfield.py
+++ b/src/sage/manifolds/differentiable/tensorfield.py
@@ -53,20 +53,23 @@
 # *****************************************************************************
 
 from __future__ import annotations
+
+from typing import TYPE_CHECKING, Optional, Tuple, Union
+
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.structure.element import ModuleElementWithMutability
 from sage.tensor.modules.free_module_tensor import FreeModuleTensor
 from sage.tensor.modules.tensor_with_indices import TensorWithIndices
 
-from typing import Optional, TYPE_CHECKING, Tuple, Union
-
 if TYPE_CHECKING:
-    from sage.manifolds.differentiable.manifold import DifferentiableManifold
     from sage.manifolds.differentiable.diff_map import DiffMap
+    from sage.manifolds.differentiable.manifold import DifferentiableManifold
     from sage.manifolds.differentiable.metric import PseudoRiemannianMetric
-    from sage.manifolds.differentiable.symplectic_form import SymplecticForm
     from sage.manifolds.differentiable.poisson_tensor import PoissonTensorField
+    from sage.manifolds.differentiable.symplectic_form import SymplecticForm
+    from sage.manifolds.differentiable.vectorfield_module import VectorFieldModule
+    from sage.tensor.modules.comp import Components
 
 
 TensorType = Tuple[int, int]
@@ -402,8 +405,17 @@ class TensorField(ModuleElementWithMutability):
         ValueError: the name of an immutable element cannot be changed
 
     """
-    def __init__(self, vector_field_module, tensor_type, name=None,
-                 latex_name=None, sym=None, antisym=None, parent=None):
+
+    def __init__(
+        self,
+        vector_field_module: VectorFieldModule,
+        tensor_type: TensorType,
+        name: Optional[str] = None,
+        latex_name: Optional[str] = None,
+        sym=None,
+        antisym=None,
+        parent=None,
+    ):
         r"""
         Construct a tensor field.
 
@@ -840,7 +852,7 @@ def domain(self) -> DifferentiableManifold:
         """
         return self._domain
 
-    def base_module(self):
+    def base_module(self) -> VectorFieldModule:
         r"""
         Return the vector field module on which ``self`` acts as a tensor.
 
@@ -867,7 +879,7 @@ def base_module(self):
         """
         return self._vmodule
 
-    def tensor_type(self) -> Tuple[int, int]:
+    def tensor_type(self) -> TensorType:
         r"""
         Return the tensor type of ``self``.
 
@@ -968,7 +980,7 @@ def set_immutable(self):
             rst.set_immutable()
         super().set_immutable()
 
-    def set_restriction(self, rst):
+    def set_restriction(self, rst: TensorField):
         r"""
         Define a restriction of ``self`` to some subdomain.
 
@@ -1035,7 +1047,9 @@ def set_restriction(self, rst):
                                                        latex_name=self._latex_name)
         self._is_zero = False  # a priori
 
-    def restrict(self, subdomain: DifferentiableManifold, dest_map: Optional[DiffMap] = None) -> TensorField:
+    def restrict(
+        self, subdomain: DifferentiableManifold, dest_map: Optional[DiffMap] = None
+    ) -> TensorField:
         r"""
         Return the restriction of ``self`` to some subdomain.
 
@@ -1385,7 +1399,7 @@ def _add_comp_unsafe(self, basis=None):
         rst = self.restrict(basis._domain, dest_map=basis._dest_map)
         return rst._add_comp_unsafe(basis)
 
-    def add_comp(self, basis=None):
+    def add_comp(self, basis=None) -> Components:
         r"""
         Return the components of ``self`` in a given vector frame
         for assignment.
@@ -3718,7 +3732,13 @@ def at(self, point):
             if point in dom:
                 return rst.at(point)
 
-    def up(self, non_degenerate_form: Union['PseudoRiemannianMetric', 'SymplecticForm', 'PoissonTensorField'], pos: Optional[int] = None) -> 'TensorField':
+    def up(
+        self,
+        non_degenerate_form: Union[
+            "PseudoRiemannianMetric", "SymplecticForm", "PoissonTensorField"
+        ],
+        pos: Optional[int] = None,
+    ) -> "TensorField":
         r"""
         Compute a dual of the tensor field by raising some index with the
         given tensor field (usually, a pseudo-Riemannian metric, a symplectic form or a Poisson tensor).
@@ -3857,14 +3877,15 @@ def up(self, non_degenerate_form: Union['PseudoRiemannianMetric', 'SymplecticFor
                 k = result._tensor_type[0]
                 result = result.up(non_degenerate_form, k)
             return result
-        
-        if pos<n_con or pos>self._tensor_rank-1:
+
+        if pos < n_con or pos > self._tensor_rank - 1:
             print("pos = {}".format(pos))
             raise ValueError("position out of range")
 
         from sage.manifolds.differentiable.metric import PseudoRiemannianMetric
         from sage.manifolds.differentiable.symplectic_form import SymplecticForm
         from sage.manifolds.differentiable.poisson_tensor import PoissonTensorField
+
         if isinstance(non_degenerate_form, PseudoRiemannianMetric):
             return self.contract(pos, non_degenerate_form.inverse(), 1)
         elif isinstance(non_degenerate_form, SymplecticForm):
@@ -3874,7 +3895,11 @@ def up(self, non_degenerate_form: Union['PseudoRiemannianMetric', 'SymplecticFor
         else:
             raise ValueError("The non-degenerate form has to be a metric, a symplectic form or a Poisson tensor field")
 
-    def down(self, non_degenerate_form: Union[PseudoRiemannianMetric, SymplecticForm], pos: Optional[int] = None) -> TensorField:
+    def down(
+        self,
+        non_degenerate_form: Union[PseudoRiemannianMetric, SymplecticForm],
+        pos: Optional[int] = None,
+    ) -> TensorField:
         r"""
         Compute a dual of the tensor field by lowering some index with a
         given non-degenerate form (pseudo-Riemannian metric or symplectic form).
@@ -4009,7 +4034,7 @@ def down(self, non_degenerate_form: Union[PseudoRiemannianMetric, SymplecticForm
             result = self
             for p in range(n_con):
                 k = result._tensor_type[0]
-                result = result.down(non_degenerate_form, k-1)
+                result = result.down(non_degenerate_form, k - 1)
             return result
         if not isinstance(pos, (int, Integer)):
             raise TypeError("the argument 'pos' must be an integer")
diff --git a/src/sage/manifolds/differentiable/tensorfield_paral.py b/src/sage/manifolds/differentiable/tensorfield_paral.py
index 6bcd68becb6..373f491cfd3 100644
--- a/src/sage/manifolds/differentiable/tensorfield_paral.py
+++ b/src/sage/manifolds/differentiable/tensorfield_paral.py
@@ -303,12 +303,19 @@
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
 
-from sage.tensor.modules.free_module_tensor import FreeModuleTensor
+from __future__ import annotations
+
+from typing import TYPE_CHECKING
+
 from sage.manifolds.chart import Chart
 from sage.manifolds.differentiable.tensorfield import TensorField
 from sage.parallel.decorate import parallel
 from sage.parallel.parallelism import Parallelism
 from sage.symbolic.ring import SR
+from sage.tensor.modules.free_module_tensor import FreeModuleTensor
+
+if TYPE_CHECKING:
+    from sage.tensor.modules.comp import Components
 
 class TensorFieldParal(FreeModuleTensor, TensorField):
     r"""
@@ -1122,7 +1129,7 @@ class :class:`~sage.tensor.modules.comp.Components`; if such
         # The add_comp operation is performed on the subdomain:
         return rst.add_comp(basis=basis)
 
-    def comp(self, basis=None, from_basis=None):
+    def comp(self, basis=None, from_basis=None) -> Components:
         r"""
         Return the components in a given vector frame.
 
diff --git a/src/sage/manifolds/differentiable/vectorfield_module.py b/src/sage/manifolds/differentiable/vectorfield_module.py
index 8c888136430..0b9dedc3b1c 100644
--- a/src/sage/manifolds/differentiable/vectorfield_module.py
+++ b/src/sage/manifolds/differentiable/vectorfield_module.py
@@ -27,7 +27,7 @@
 
 """
 
-#******************************************************************************
+# ******************************************************************************
 #       Copyright (C) 2015 Eric Gourgoulhon <eric.gourgoulhon@obspm.fr>
 #       Copyright (C) 2015 Michal Bejger <bejger@camk.edu.pl>
 #       Copyright (C) 2016 Travis Scrimshaw <tscrimsh@umn.edu>
@@ -36,16 +36,23 @@
 #  as published by the Free Software Foundation; either version 2 of
 #  the License, or (at your option) any later version.
 #                  http://www.gnu.org/licenses/
-#******************************************************************************
+# ******************************************************************************
+
+from __future__ import annotations
+
+from typing import TYPE_CHECKING, Optional
 
-from typing import Optional
-from sage.structure.unique_representation import UniqueRepresentation
-from sage.structure.parent import Parent
 from sage.categories.modules import Modules
+from sage.manifolds.differentiable.diff_map import DiffMap
+from sage.manifolds.differentiable.vectorfield import VectorField, VectorFieldParal
 from sage.misc.cachefunc import cached_method
 from sage.rings.integer import Integer
+from sage.structure.parent import Parent
+from sage.structure.unique_representation import UniqueRepresentation
 from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule
-from sage.manifolds.differentiable.vectorfield import VectorField, VectorFieldParal
+
+if TYPE_CHECKING:
+    from sage.manifolds.differentiable.manifold import DifferentiableManifold
 
 
 class VectorFieldModule(UniqueRepresentation, Parent):
@@ -182,7 +189,7 @@ class VectorFieldModule(UniqueRepresentation, Parent):
     """
     Element = VectorField
 
-    def __init__(self, domain, dest_map=None):
+    def __init__(self, domain: DifferentiableManifold, dest_map: Optional[DiffMap] = None):
         r"""
         Construct the module of vector fields taking values on a (a priori)
         non-parallelizable differentiable manifold.
@@ -387,7 +394,7 @@ def _latex_(self):
         else:
             return self._latex_name
 
-    def domain(self):
+    def domain(self) -> DifferentiableManifold:
         r"""
         Return the domain of the vector fields in this module.
 
@@ -416,7 +423,7 @@ 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 this module take
         their values.
@@ -1050,7 +1057,7 @@ def zero(self):
         zero.set_immutable()
         return zero
 
-    def metric(self, name, signature=None, latex_name=None):
+    def metric(self, name: str, signature: Optional[int] = None, latex_name: Optional[str] = None):
         r"""
         Construct a metric (symmetric bilinear
         form) on the current vector field module.
@@ -1610,7 +1617,7 @@ 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``.
 
@@ -1641,7 +1648,7 @@ 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.
@@ -1673,7 +1680,7 @@ 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``.
 
diff --git a/src/sage/manifolds/manifold.py b/src/sage/manifolds/manifold.py
index 4c3d350b044..58e34843230 100644
--- a/src/sage/manifolds/manifold.py
+++ b/src/sage/manifolds/manifold.py
@@ -325,20 +325,32 @@
 #                  https://www.gnu.org/licenses/
 # ****************************************************************************
 
+from __future__ import annotations
+
+from typing import TYPE_CHECKING, Optional, Union, overload
+
+import sage.rings.abc
 from sage.categories.fields import Fields
-from sage.categories.manifolds import Manifolds
 from sage.categories.homset import Hom
-import sage.rings.abc
-from sage.rings.cc import CC
-from sage.rings.real_mpfr import RR
-from sage.misc.prandom import getrandbits
+from sage.categories.manifolds import Manifolds
+from sage.manifolds.structure import (
+    DifferentialStructure,
+    RealDifferentialStructure,
+    RealTopologicalStructure,
+    TopologicalStructure,
+)
+from sage.manifolds.subset import ManifoldSubset
 from sage.misc.cachefunc import cached_method
+from sage.misc.prandom import getrandbits
+from sage.rings.cc import CC
 from sage.rings.integer import Integer
+from sage.rings.real_mpfr import RR
 from sage.structure.global_options import GlobalOptions
-from sage.manifolds.subset import ManifoldSubset
-from sage.manifolds.structure import (
-    TopologicalStructure, RealTopologicalStructure,
-    DifferentialStructure, RealDifferentialStructure)
+
+if TYPE_CHECKING:
+    from sage.manifolds.continuous_map import ContinuousMap
+    from sage.manifolds.differentiable.diff_map import DiffMap
+    from sage.manifolds.differentiable.manifold import DifferentiableManifold
 
 
 #############################################################################
@@ -2444,6 +2456,10 @@ 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: TopologicalManifold) -> ContinuousMap: ...
+    @overload
+    def identity_map(self: DifferentiableManifold) -> DiffMap: ...
     @cached_method
     def identity_map(self):
         r"""
@@ -2701,8 +2717,16 @@ def set_simplify_function(self, simplifying_func, method=None):
 
 _manifold_id = Integer(0)
 
-def Manifold(dim, name, latex_name=None, field='real', structure=None,
-             start_index=0, **extra_kwds):
+
+def Manifold(
+    dim: int,
+    name: Optional[str],
+    latex_name: Optional[str] = None,
+    field: str = "real",
+    structure: Optional[str] = None,
+    start_index: int = 0,
+    **extra_kwds,
+) -> Union[TopologicalManifold, DifferentiableManifold]:
     r"""
     Construct a manifold of a given type over a topological field.
 
diff --git a/src/sage/manifolds/topological_submanifold.py b/src/sage/manifolds/topological_submanifold.py
index 62aa78fef3f..91f6f0d6062 100644
--- a/src/sage/manifolds/topological_submanifold.py
+++ b/src/sage/manifolds/topological_submanifold.py
@@ -513,8 +513,9 @@ def declare_embedding(self):
                              "before calling declare_embedding()")
         self._embedded = True
 
-    def set_embedding(self, phi, inverse=None, var=None,
-                      t_inverse=None):
+    def set_embedding(
+        self, phi: ContinuousMap, inverse=None, var=None, t_inverse=None
+    ):
         r"""
         Register the embedding of an embedded submanifold.
 
@@ -812,7 +813,7 @@ def plot(self, param, u, v, chart1=None, chart2=None, **kwargs):
 
         return ParametricSurface((fx, fy, fz), (u, v), **kwargs)
 
-    def ambient(self):
+    def ambient(self) -> TopologicalManifold:
         r"""
         Return the manifold in which ``self`` is immersed or embedded.
 
@@ -825,7 +826,7 @@ def ambient(self):
         """
         return self._ambient
 
-    def immersion(self):
+    def immersion(self) -> ContinuousMap:
         r"""
         Return the immersion of ``self`` into the ambient manifold.
 
@@ -849,9 +850,10 @@ def immersion(self):
         """
         if not self._immersed:
             raise ValueError("the submanifold is not immersed")
+        assert self._immersion
         return self._immersion
 
-    def embedding(self):
+    def embedding(self) -> ContinuousMap:
         r"""
         Return the embedding of ``self`` into the ambient manifold.
 
@@ -875,6 +877,7 @@ def embedding(self):
         """
         if not self._embedded:
             raise ValueError("the submanifold is not embedded")
+        assert self._immersion
         return self._immersion
 
     def as_subset(self):
diff --git a/src/sage/tensor/modules/free_module_tensor.py b/src/sage/tensor/modules/free_module_tensor.py
index f5c2da72c7c..487d0d83e0e 100644
--- a/src/sage/tensor/modules/free_module_tensor.py
+++ b/src/sage/tensor/modules/free_module_tensor.py
@@ -192,6 +192,7 @@ class being:
 #  the License, or (at your option) any later version.
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
+from typing import Dict
 
 from sage.rings.integer import Integer
 from sage.structure.element import ModuleElementWithMutability
@@ -200,6 +201,7 @@ class being:
 from sage.tensor.modules.tensor_with_indices import TensorWithIndices
 from sage.parallel.decorate import parallel
 from sage.parallel.parallelism import Parallelism
+from sage.tensor.modules.free_module_basis import FreeModuleBasis
 
 class FreeModuleTensor(ModuleElementWithMutability):
     r"""
@@ -287,7 +289,7 @@ def __init__(self, fmodule, tensor_type, name=None, latex_name=None,
             self._latex_name = self._name
         else:
             self._latex_name = latex_name
-        self._components = {}  # dict. of the sets of components on various
+        self._components: Dict[FreeModuleBasis, Components] = {}  # dict. of the sets of components on various
                               # bases, with the bases as keys (initially empty)
 
         # Treatment of symmetry declarations: