sagemathgh-37829: src/sage/manifolds/differentiable: Docstring/doctes…

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URL: sagemath#37829
Reported by: Matthias Köppe
Reviewer(s): Eric Gourgoulhon

src/sage/manifolds/differentiable/affine_connection.py

@@ -149,10 +149,10 @@ class AffineConnection(SageObject):
 
     Unset components are initialized to zero::
 
-        sage: nab[:] # list of coefficients relative to the manifold's default vector frame
+        sage: nab[:]  # list of coefficients relative to the manifold's default vector frame
         [[[0, x^2, 0], [0, 0, 0], [0, 0, 0]],
-        [[0, 0, 0], [0, 0, 0], [0, 0, 0]],
-        [[0, 0, 0], [0, 0, y*z], [0, 0, 0]]]
+         [[0, 0, 0], [0, 0, 0], [0, 0, 0]],
+         [[0, 0, 0], [0, 0, y*z], [0, 0, 0]]]
 
     The treatment of connection coefficients in a given vector frame is similar
     to that of tensor components; see therefore the class
@@ -2377,7 +2377,7 @@ def curvature_form(self, i, j, frame=None):
             2-form curvature (1,1) of connection nabla w.r.t. Vector frame
              (M, (e_1,e_2,e_3)) on the 3-dimensional differentiable manifold M
             sage: nab.curvature_form(1,1,e).display(e)  # long time (if above is skipped)
-             curvature (1,1) of connection nabla w.r.t. Vector frame
+            curvature (1,1) of connection nabla w.r.t. Vector frame
              (M, (e_1,e_2,e_3)) =
               (y^3*z^4 + 2*x*y*z + (x*y^4 - x*y)*z^2) e^1∧e^2
               + (x^4*y*z^2 - x^2*y^2) e^1∧e^3 + (x^5*y*z^3 - x*z^2) e^2∧e^3

src/sage/manifolds/differentiable/bundle_connection.py

@@ -98,11 +98,11 @@ class BundleConnection(SageObject, Mutability):
         sage: nab[e, 1, 1][:] = [x+z, y-z, x*y*z]
         sage: nab.display()
         connection (1,1) of bundle connection nabla w.r.t. Local frame
-          (E|_M, (e_1,e_2)) = (x + z) dx + (y - z) dy + x*y*z dz
-         connection (1,2) of bundle connection nabla w.r.t. Local frame
-          (E|_M, (e_1,e_2)) = x*z dx + y*z dy + z^2 dz
-         connection (2,1) of bundle connection nabla w.r.t. Local frame
-          (E|_M, (e_1,e_2)) = x dx + x^2 dy + x^3 dz
+         (E|_M, (e_1,e_2)) = (x + z) dx + (y - z) dy + x*y*z dz
+        connection (1,2) of bundle connection nabla w.r.t. Local frame
+         (E|_M, (e_1,e_2)) = x*z dx + y*z dy + z^2 dz
+        connection (2,1) of bundle connection nabla w.r.t. Local frame
+         (E|_M, (e_1,e_2)) = x dx + x^2 dy + x^3 dz
 
     Notice, when we omit the frame, the default frame of the vector bundle is
     assumed (in this case ``e``)::
@@ -189,14 +189,14 @@ class BundleConnection(SageObject, Mutability):
         sage: nab.display(frame=f)
         connection (1,1) of bundle connection nabla w.r.t. Local frame
          (E|_M, (f_1,f_2)) = ((x^3 + x)*z + 2*x)/(x^2 + 1) dx + y*z dy + z^2 dz
-         connection (1,2) of bundle connection nabla w.r.t. Local frame
-          (E|_M, (f_1,f_2)) = -(x^3 + x)*z dx - (x^2 + 1)*y*z dy -
+        connection (1,2) of bundle connection nabla w.r.t. Local frame
+         (E|_M, (f_1,f_2)) = -(x^3 + x)*z dx - (x^2 + 1)*y*z dy -
           (x^2 + 1)*z^2 dz
-         connection (2,1) of bundle connection nabla w.r.t. Local frame
-          (E|_M, (f_1,f_2)) = (x*z - x)/(x^2 + 1) dx -
+        connection (2,1) of bundle connection nabla w.r.t. Local frame
+         (E|_M, (f_1,f_2)) = (x*z - x)/(x^2 + 1) dx -
           (x^2 - y*z)/(x^2 + 1) dy - (x^3 - z^2)/(x^2 + 1) dz
-         connection (2,2) of bundle connection nabla w.r.t. Local frame
-          (E|_M, (f_1,f_2)) = -x*z dx - y*z dy - z^2 dz
+        connection (2,2) of bundle connection nabla w.r.t. Local frame
+         (E|_M, (f_1,f_2)) = -x*z dx - y*z dy - z^2 dz
 
     The new connection 1-forms obey the defining formula, too::
 
@@ -214,14 +214,14 @@ class BundleConnection(SageObject, Mutability):
         ....:         print(Omega(i ,j, e).display())
         curvature (1,1) of bundle connection nabla w.r.t. Local frame
          (E|_M, (e_1,e_2)) = -(x^3 - x*y)*z dx∧dy + (-x^4*z + x*z^2) dx∧dz +
-         (-x^3*y*z + x^2*z^2) dy∧dz
-         curvature (1,2) of bundle connection nabla w.r.t. Local frame
+          (-x^3*y*z + x^2*z^2) dy∧dz
+        curvature (1,2) of bundle connection nabla w.r.t. Local frame
          (E|_M, (e_1,e_2)) = -x dx∧dz - y dy∧dz
-         curvature (2,1) of bundle connection nabla w.r.t. Local frame
+        curvature (2,1) of bundle connection nabla w.r.t. Local frame
          (E|_M, (e_1,e_2)) = 2*x dx∧dy + 3*x^2 dx∧dz
-         curvature (2,2) of bundle connection nabla w.r.t. Local frame
+        curvature (2,2) of bundle connection nabla w.r.t. Local frame
          (E|_M, (e_1,e_2)) = (x^3 - x*y)*z dx∧dy + (x^4*z - x*z^2) dx∧dz +
-         (x^3*y*z - x^2*z^2) dy∧dz
+          (x^3*y*z - x^2*z^2) dy∧dz
 
     The derived forms certainly obey the structure equations, see
     :meth:`curvature_form` for details::
@@ -483,17 +483,17 @@ def _new_forms(self, frame):
             sage: forms = nab._new_forms(e)
             sage: [forms[k] for k in sorted(forms)]
             [1-form connection (1,1) of bundle connection nabla w.r.t. Local
-             frame (E|_M, (e_1,e_2)) on the 2-dimensional differentiable
-             manifold M,
-            1-form connection (1,2) of bundle connection nabla w.r.t. Local
-             frame (E|_M, (e_1,e_2)) on the 2-dimensional differentiable
-             manifold M,
-            1-form connection (2,1) of bundle connection nabla w.r.t. Local
-             frame (E|_M, (e_1,e_2)) on the 2-dimensional differentiable
-             manifold M,
-            1-form connection (2,2) of bundle connection nabla w.r.t. Local
-            frame (E|_M, (e_1,e_2)) on the 2-dimensional differentiable
-            manifold M]
+              frame (E|_M, (e_1,e_2)) on the 2-dimensional differentiable
+              manifold M,
+             1-form connection (1,2) of bundle connection nabla w.r.t. Local
+              frame (E|_M, (e_1,e_2)) on the 2-dimensional differentiable
+              manifold M,
+             1-form connection (2,1) of bundle connection nabla w.r.t. Local
+              frame (E|_M, (e_1,e_2)) on the 2-dimensional differentiable
+              manifold M,
+             1-form connection (2,2) of bundle connection nabla w.r.t. Local
+              frame (E|_M, (e_1,e_2)) on the 2-dimensional differentiable
+              manifold M]
 
         """
         dom = frame._domain
@@ -1307,8 +1307,8 @@ def display(self, frame=None, vector_frame=None, chart=None,
             sage: nab.display()
             connection (1,1) of bundle connection nabla w.r.t. Local frame
              (E|_M, (e_1,e_2)) = x dx + y dy + z dz
-             connection (2,2) of bundle connection nabla w.r.t. Local frame
-              (E|_M, (e_1,e_2)) = x^2 dx + y^2 dy + z^2 dz
+            connection (2,2) of bundle connection nabla w.r.t. Local frame
+             (E|_M, (e_1,e_2)) = x^2 dx + y^2 dy + z^2 dz
             sage: latex(nab.display())
             \begin{array}{lcl} \omega^1_{\ \, 1} = x \mathrm{d} x +
              y \mathrm{d} y + z \mathrm{d} z \\ \omega^2_{\ \, 2} = x^{2}

src/sage/manifolds/differentiable/differentiable_submanifold.py

@@ -69,14 +69,14 @@ class DifferentiableSubmanifold(DifferentiableManifold, TopologicalSubmanifold):
     - ``field`` -- field `K` on which the sub manifold is defined; allowed
       values are
 
-        - ``'real'`` or an object of type ``RealField`` (e.g., ``RR``) for
-           a manifold over `\RR`
-        - ``'complex'`` or an object of type ``ComplexField`` (e.g., ``CC``)
-           for a manifold over `\CC`
-        - an object in the category of topological fields (see
-          :class:`~sage.categories.fields.Fields` and
-          :class:`~sage.categories.topological_spaces.TopologicalSpaces`)
-          for other types of manifolds
+      - ``'real'`` or an object of type ``RealField`` (e.g., ``RR``) for
+        a manifold over `\RR`
+      - ``'complex'`` or an object of type ``ComplexField`` (e.g., ``CC``)
+        for a manifold over `\CC`
+      - an object in the category of topological fields (see
+        :class:`~sage.categories.fields.Fields` and
+        :class:`~sage.categories.topological_spaces.TopologicalSpaces`)
+        for other types of manifolds
 
     - ``structure`` -- manifold structure (see
       :class:`~sage.manifolds.structure.TopologicalStructure` or
@@ -136,7 +136,7 @@ class DifferentiableSubmanifold(DifferentiableManifold, TopologicalSubmanifold):
         sage: phi_inv_t = M.scalar_field({CM: z-x^2-y^2})
         sage: phi_inv_t.display()
         M → ℝ
-        (x, y, z) ↦ -x^2 - y^2 + z
+           (x, y, z) ↦ -x^2 - y^2 + z
 
     `\phi` can then be declared as an embedding `N\to M`::
 

src/sage/manifolds/differentiable/multivectorfield.py

@@ -1040,8 +1040,7 @@ def wedge(self, other):
             sage: b.display()
             b = y^2 ∂/∂x∧∂/∂y + (x + z) ∂/∂x∧∂/∂z + z^2 ∂/∂y∧∂/∂z
             sage: s = a.wedge(b); s
-            3-vector field a∧b on the 3-dimensional differentiable
-             manifold M
+            3-vector field a∧b on the 3-dimensional differentiable manifold M
             sage: s.display()
             a∧b = (-x^2 + (y^3 - x - 1)*z + 2*z^2 - x) ∂/∂x∧∂/∂y∧∂/∂z
 

src/sage/manifolds/differentiable/tangent_space.py

@@ -177,28 +177,32 @@ class TangentSpace(FiniteRankFreeModule):
         [0 1]
         sage: W_Tp_xy = VectorSpace(SR, 2, inner_product_matrix=Q_Tp_xy)
         sage: Tp.bases()[0]
-        Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the 2-dimensional differentiable manifold M
-        sage: phi_Tp_xy = Tp.isomorphism_with_fixed_basis(Tp.bases()[0], codomain=W_Tp_xy); phi_Tp_xy
+        Basis (∂/∂x,∂/∂y) on the Tangent space at Point p on the
+         2-dimensional differentiable manifold M
+        sage: phi_Tp_xy = Tp.isomorphism_with_fixed_basis(Tp.bases()[0], codomain=W_Tp_xy)
+        sage: phi_Tp_xy
         Generic morphism:
-        From: Tangent space at Point p on the 2-dimensional differentiable manifold M
-        To:   Ambient quadratic space of dimension 2 over Symbolic Ring
-        Inner product matrix:
-        [1 0]
-        [0 1]
+         From: Tangent space at Point p on the 2-dimensional differentiable manifold M
+         To:   Ambient quadratic space of dimension 2 over Symbolic Ring
+               Inner product matrix:
+               [1 0]
+               [0 1]
 
         sage: Q_Tp_uv = g[c_uv.frame(),:](*p.coordinates(c_uv)); Q_Tp_uv
         [1/2   0]
         [  0 1/2]
         sage: W_Tp_uv = VectorSpace(SR, 2, inner_product_matrix=Q_Tp_uv)
         sage: Tp.bases()[1]
-        Basis (∂/∂u,∂/∂v) on the Tangent space at Point p on the 2-dimensional differentiable manifold M
-        sage: phi_Tp_uv = Tp.isomorphism_with_fixed_basis(Tp.bases()[1], codomain=W_Tp_uv); phi_Tp_uv
+        Basis (∂/∂u,∂/∂v) on the Tangent space at Point p on the
+         2-dimensional differentiable manifold M
+        sage: phi_Tp_uv = Tp.isomorphism_with_fixed_basis(Tp.bases()[1], codomain=W_Tp_uv)
+        sage: phi_Tp_uv
         Generic morphism:
-        From: Tangent space at Point p on the 2-dimensional differentiable manifold M
-        To:   Ambient quadratic space of dimension 2 over Symbolic Ring
-        Inner product matrix:
-        [1/2   0]
-        [  0 1/2]
+         From: Tangent space at Point p on the 2-dimensional differentiable manifold M
+         To:   Ambient quadratic space of dimension 2 over Symbolic Ring
+               Inner product matrix:
+               [1/2   0]
+               [  0 1/2]
 
         sage: t1, t2 = Tp.tensor((1,0)), Tp.tensor((1,0))
         sage: t1[:] = (8, 15)
@@ -211,7 +215,8 @@ class TangentSpace(FiniteRankFreeModule):
         541
 
         sage: Tp_xy_to_uv = M.change_of_frame(c_xy.frame(), c_uv.frame()).at(p); Tp_xy_to_uv
-        Automorphism of the Tangent space at Point p on the 2-dimensional differentiable manifold M
+        Automorphism of the Tangent space at Point p on the
+         2-dimensional differentiable manifold M
         sage: Tp.set_change_of_basis(Tp.bases()[0], Tp.bases()[1], Tp_xy_to_uv)
         sage: t1[Tp.bases()[1],:]
         [23, -7]

src/sage/manifolds/differentiable/vector_bundle.py

@@ -768,7 +768,7 @@ def set_change_of_frame(self, frame1, frame2, change_of_frame,
           :class:`~sage.tensor.modules.free_module_automorphism.FreeModuleAutomorphism`
           describing the automorphism `P` that relates the basis `(e_i)` to
           the basis `(f_i)` according to `f_i = P(e_i)`
-        - ``compute_inverse`` (default: True) -- if set to True, the inverse
+        - ``compute_inverse`` (default: ``True``) -- if set to ``True``, the inverse
           automorphism is computed and the change from basis `(f_i)` to `(e_i)`
           is set to it in the internal dictionary ``self._frame_changes``
 
@@ -1476,7 +1476,7 @@ def ambient_domain(self):
             sage: Phi.display()
             Phi: R → M
                t ↦ (x, y) = (cos(t), sin(t))
-                sage: PhiT11 = R.tensor_bundle(1, 1, dest_map=Phi)
+            sage: PhiT11 = R.tensor_bundle(1, 1, dest_map=Phi)
             sage: PhiT11.ambient_domain()
             2-dimensional differentiable manifold M
 
@@ -1724,24 +1724,24 @@ def orientation(self):
             []
             sage: T11.set_orientation([c_xy.frame(), c_uv.frame()])
             sage: T11.orientation()
-            [Coordinate frame (U, (∂/∂x,∂/∂y)), Coordinate frame
-             (V, (∂/∂u,∂/∂v))]
+            [Coordinate frame (U, (∂/∂x,∂/∂y)),
+             Coordinate frame (V, (∂/∂u,∂/∂v))]
 
         If the destination map is the identity, the orientation is
         automatically set for the manifold, too::
 
             sage: M.orientation()
-            [Coordinate frame (U, (∂/∂x,∂/∂y)), Coordinate frame
-             (V, (∂/∂u,∂/∂v))]
+            [Coordinate frame (U, (∂/∂x,∂/∂y)),
+             Coordinate frame (V, (∂/∂u,∂/∂v))]
 
         Conversely, if one sets an orientation on the manifold,
         the orientation on its tensor bundles is set accordingly::
 
             sage: c_tz.<t,z> = U.chart()
             sage: M.set_orientation([c_tz, c_uv])
             sage: T11.orientation()
-            [Coordinate frame (U, (∂/∂t,∂/∂z)), Coordinate frame
-             (V, (∂/∂u,∂/∂v))]
+            [Coordinate frame (U, (∂/∂t,∂/∂z)),
+             Coordinate frame (V, (∂/∂u,∂/∂v))]
 
         """
         if self._dest_map.is_identity():