diff --git a/src/sage/manifolds/differentiable/diff_map.py b/src/sage/manifolds/differentiable/diff_map.py
index 7f0c09a6b7c..4260f233e40 100644
--- a/src/sage/manifolds/differentiable/diff_map.py
+++ b/src/sage/manifolds/differentiable/diff_map.py
@@ -32,11 +32,17 @@
 #  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.manifolds.continuous_map import ContinuousMap
 from sage.parallel.decorate import parallel
 from sage.parallel.parallelism import Parallelism
 
+if TYPE_CHECKING:
+    from sage.manifolds.point import ManifoldPoint
+    from sage.tensor.modules.free_module_morphism import FiniteRankFreeModuleMorphism
 
 class DiffMap(ContinuousMap):
     r"""
@@ -515,7 +521,7 @@ def _del_derived(self):
                                           # class
         self._diff.clear()
 
-    def differential(self, point):
+    def differential(self, point: ManifoldPoint) -> FiniteRankFreeModuleMorphism:
         r"""
         Return the differential of ``self`` at a given point.
 
@@ -949,8 +955,12 @@ def pullback(self, tensor_or_codomain_subset, name=None, latex_name=None):
         tensor = tensor_or_codomain_subset
 
         from sage.manifolds.differentiable.tensorfield_paral import TensorFieldParal
-        from sage.tensor.modules.comp import (Components, CompWithSym,
-                                              CompFullySym, CompFullyAntiSym)
+        from sage.tensor.modules.comp import (
+            CompFullyAntiSym,
+            CompFullySym,
+            Components,
+            CompWithSym,
+        )
 
         def _pullback_chart(diff_map, tensor, chart1, chart2):
             r"""
@@ -1195,9 +1205,13 @@ def pushforward(self, tensor):
             Psi_*(u) = -sin(t) ∂/∂x + cos(t) ∂/∂y + ∂/∂z
 
         """
-        from sage.tensor.modules.comp import (Components, CompWithSym,
-                                              CompFullySym, CompFullyAntiSym)
         from sage.manifolds.differentiable.tensorfield_paral import TensorFieldParal
+        from sage.tensor.modules.comp import (
+            CompFullyAntiSym,
+            CompFullySym,
+            Components,
+            CompWithSym,
+        )
         vmodule = tensor.base_module()
         dest_map = vmodule.destination_map()
         dom1 = tensor.domain()
diff --git a/src/sage/manifolds/differentiable/scalarfield.py b/src/sage/manifolds/differentiable/scalarfield.py
index 9763bf46352..1b9a3c2ae2d 100644
--- a/src/sage/manifolds/differentiable/scalarfield.py
+++ b/src/sage/manifolds/differentiable/scalarfield.py
@@ -38,13 +38,15 @@
 #******************************************************************************
 
 from __future__ import annotations
-from typing import Union, TYPE_CHECKING
+
+from typing import TYPE_CHECKING, Union
+
 from sage.manifolds.scalarfield import ScalarField
 
 if TYPE_CHECKING:
     from sage.manifolds.differentiable.diff_form import DiffForm
-    from sage.manifolds.differentiable.symplectic_form import SymplecticForm
     from sage.manifolds.differentiable.metric import PseudoRiemannianMetric
+    from sage.manifolds.differentiable.symplectic_form import SymplecticForm
 
 
 class DiffScalarField(ScalarField):
@@ -713,7 +715,7 @@ def tensor_type(self):
         """
         return self._tensor_type
 
-    def differential(self):
+    def differential(self) -> DiffForm:
         r"""
         Return the differential of ``self``.
 
diff --git a/src/sage/manifolds/differentiable/tangent_space.py b/src/sage/manifolds/differentiable/tangent_space.py
index aa14cc90f70..4b90e711a79 100644
--- a/src/sage/manifolds/differentiable/tangent_space.py
+++ b/src/sage/manifolds/differentiable/tangent_space.py
@@ -24,10 +24,16 @@
 #  the License, or (at your option) any later version.
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
+from __future__ import annotations
 
+from typing import TYPE_CHECKING
+
+from sage.manifolds.differentiable.tangent_vector import TangentVector
 from sage.symbolic.ring import SR
 from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule
-from sage.manifolds.differentiable.tangent_vector import TangentVector
+
+if TYPE_CHECKING:
+    from sage.manifolds.point import ManifoldPoint
 
 class TangentSpace(FiniteRankFreeModule):
     r"""
@@ -222,7 +228,7 @@ class TangentSpace(FiniteRankFreeModule):
     """
     Element = TangentVector
 
-    def __init__(self, point, base_ring=None):
+    def __init__(self, point: ManifoldPoint, base_ring=None):
         r"""
         Construct the tangent space at a given point.
 
diff --git a/src/sage/manifolds/differentiable/tensorfield.py b/src/sage/manifolds/differentiable/tensorfield.py
index db4a2a05aa1..1390fc8076f 100644
--- a/src/sage/manifolds/differentiable/tensorfield.py
+++ b/src/sage/manifolds/differentiable/tensorfield.py
@@ -69,6 +69,7 @@
     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.manifolds.point import ManifoldPoint
     from sage.tensor.modules.comp import Components
 
 
@@ -3634,7 +3635,7 @@ def lie_derivative(self, vector):
 
     lie_der = lie_derivative
 
-    def at(self, point):
+    def at(self, point: ManifoldPoint) -> FreeModuleTensor:
         r"""
         Value of ``self`` at a point of its domain.
 
diff --git a/src/sage/tensor/modules/free_module_element.py b/src/sage/tensor/modules/free_module_element.py
index 0bde6363822..c8bec392083 100644
--- a/src/sage/tensor/modules/free_module_element.py
+++ b/src/sage/tensor/modules/free_module_element.py
@@ -29,9 +29,18 @@
 #                  http://www.gnu.org/licenses/
 #******************************************************************************
 
+from __future__ import annotations
+
+from typing import TYPE_CHECKING, Optional
+
 from sage.tensor.modules.alternating_contr_tensor import AlternatingContrTensor
 from sage.tensor.modules.comp import Components
 
+if TYPE_CHECKING:
+    from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule
+    from sage.tensor.modules.free_module_basis import FreeModuleBasis
+
+
 class FiniteRankFreeModuleElement(AlternatingContrTensor):
     r"""
     Element of a free module of finite rank over a commutative ring.
@@ -185,7 +194,13 @@ class FiniteRankFreeModuleElement(AlternatingContrTensor):
         a∧b = -2 e_0∧e_1 - 6 e_0∧e_2 + 7 e_1∧e_2
 
     """
-    def __init__(self, fmodule, name=None, latex_name=None):
+
+    def __init__(
+        self,
+        fmodule: FiniteRankFreeModule,
+        name: Optional[str] = None,
+        latex_name: Optional[str] = None,
+    ):
         r"""
         TESTS::
 
@@ -209,7 +224,7 @@ def __init__(self, fmodule, name=None, latex_name=None):
         AlternatingContrTensor.__init__(self, fmodule, 1, name=name,
                                         latex_name=latex_name)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return a string representation of ``self``.
 
@@ -227,7 +242,7 @@ def _repr_(self):
         description += "of the {}".format(self._fmodule)
         return description
 
-    def _new_comp(self, basis):
+    def _new_comp(self, basis: FreeModuleBasis) -> Components:
         r"""
         Create some (uninitialized) components of ``self`` in a given basis.
 
diff --git a/src/sage/tensor/modules/free_module_morphism.py b/src/sage/tensor/modules/free_module_morphism.py
index abb24e89203..c833d03094d 100644
--- a/src/sage/tensor/modules/free_module_morphism.py
+++ b/src/sage/tensor/modules/free_module_morphism.py
@@ -23,8 +23,15 @@
 #  the License, or (at your option) any later version.
 #                  https://www.gnu.org/licenses/
 # *****************************************************************************
-from sage.rings.integer import Integer
+from __future__ import annotations
+
+from typing import TYPE_CHECKING
+
 from sage.categories.morphism import Morphism
+from sage.rings.integer import Integer
+
+if TYPE_CHECKING:
+    from sage.tensor.modules.free_module_element import FiniteRankFreeModuleElement
 
 
 class FiniteRankFreeModuleMorphism(Morphism):
@@ -764,7 +771,9 @@ def __neg__(self):
     # Map methods
     #
 
-    def _call_(self, element):
+    def _call_(
+        self, element: FiniteRankFreeModuleElement
+    ) -> FiniteRankFreeModuleElement:
         r"""
         Action of the homomorphism ``self`` on some free module element
 
diff --git a/src/sage/tensor/modules/free_module_tensor.py b/src/sage/tensor/modules/free_module_tensor.py
index a7865299688..da85d6c455b 100644
--- a/src/sage/tensor/modules/free_module_tensor.py
+++ b/src/sage/tensor/modules/free_module_tensor.py
@@ -194,17 +194,22 @@ class being:
 # *****************************************************************************
 from __future__ import annotations
 
-from typing import TYPE_CHECKING, Dict
+from typing import TYPE_CHECKING, Dict, Optional
 
+from sage.parallel.decorate import parallel
+from sage.parallel.parallelism import Parallelism
 from sage.rings.integer import Integer
 from sage.structure.element import ModuleElementWithMutability
-from sage.tensor.modules.comp import (Components, CompWithSym, CompFullySym,
-                                      CompFullyAntiSym)
+from sage.tensor.modules.comp import (
+    CompFullyAntiSym,
+    CompFullySym,
+    Components,
+    CompWithSym,
+)
 from sage.tensor.modules.tensor_with_indices import TensorWithIndices
-from sage.parallel.decorate import parallel
-from sage.parallel.parallelism import Parallelism
 
 if TYPE_CHECKING:
+    from sage.symbolic.expression import Expression
     from sage.tensor.modules.finite_rank_free_module import FiniteRankFreeModule
     from sage.tensor.modules.free_module_basis import FreeModuleBasis
 
@@ -264,8 +269,8 @@ def __init__(
         self,
         fmodule: FiniteRankFreeModule,
         tensor_type,
-        name=None,
-        latex_name=None,
+        name: Optional[str] = None,
+        latex_name: Optional[str] = None,
         sym=None,
         antisym=None,
         parent=None,
@@ -891,7 +896,7 @@ def display_comp(self, basis=None, format_spec=None, symbol=None,
                                         only_nonzero=only_nonzero,
                                         only_nonredundant=only_nonredundant)
 
-    def set_name(self, name=None, latex_name=None):
+    def set_name(self, name: Optional[str] = None, latex_name: Optional[str] = None):
         r"""
         Set (or change) the text name and LaTeX name of ``self``.
 
@@ -2262,7 +2267,7 @@ def __truediv__(self, other):
             result._components[basis] = self._components[basis] / other
         return result
 
-    def __call__(self, *args):
+    def __call__(self, *args) -> Expression:
         r"""
         The tensor acting on linear forms and module elements as a multilinear
         map.