Some fixes in the documentation of vector bundles

Author: egourgoulhon

src/doc/en/reference/categories/index.rst

@@ -181,6 +181,7 @@ Individual Categories
    sage/categories/triangular_kac_moody_algebras
    sage/categories/unique_factorization_domains
    sage/categories/unital_algebras
+   sage/categories/vector_bundles
    sage/categories/vector_spaces
    sage/categories/weyl_groups
 

src/doc/en/reference/manifolds/vector_bundle.rst

@@ -6,6 +6,10 @@ Topological Vector Bundles
 
    sage/manifolds/vector_bundle
 
+   sage/manifolds/vector_bundle_fiber
+
+   sage/manifolds/vector_bundle_fiber_element
+
    sage/manifolds/trivialization
 
    sage/manifolds/local_frame

src/sage/manifolds/differentiable/manifold.py

@@ -1060,7 +1060,7 @@ class as the manifold.
         - ``name`` -- name given to the total space
         - ``field`` -- (default: ``'real'``) topological field giving the
           vector space structure to the fibers
-        - ``latex_name`` -- optional latex name for the total space
+        - ``latex_name`` -- optional LaTeX name for the total space
 
         OUTPUT:
 

src/sage/manifolds/differentiable/vector_bundle.py

@@ -44,7 +44,7 @@ class DifferentiableVectorBundle(TopologicalVectorBundle):
     INPUT:
 
     - ``rank`` -- positive integer; rank of the vector bundle
-    - ``name```-- string representation given to the total space
+    - ``name`` -- string representation given to the total space
     - ``base_space`` -- the base space (differentiable manifold) `M` over which
       the vector bundle is defined
     - ``field`` -- field `K` which gives the fibers the structure of a
@@ -59,11 +59,10 @@ class DifferentiableVectorBundle(TopologicalVectorBundle):
         :class:`~sage.categories.topological_spaces.TopologicalSpaces`)
         for other types of topological fields
 
-    - ``latex_name`` -- (default: ``None``) latex representation given to the
+    - ``latex_name`` -- (default: ``None``) LaTeX representation given to the
       total space
     - ``category`` -- (default: ``None``) to specify the category; if
-      ``None``, ``VectorBundles(base_space, c_field).Differentiable()`` (or
-      ``Manifolds(field).Smooth()`` if ``diff_degree`` = ``infinity``)`` is
+      ``None``, ``VectorBundles(base_space, c_field).Differentiable()`` is
       assumed (see the category
       :class:`~sage.categories.vector_bundles.VectorBundles`)
 
@@ -140,7 +139,7 @@ def diff_degree(self):
         Return the vector bundle's degree of differentiability.
 
         The degree of differentiability is the integer `k` (possibly
-        `k=\infty`) such that the vector bundle is of class `C^k`-manifold over
+        `k=\infty`) such that the vector bundle is of class `C^k` over
         its base field. The degree always corresponds to the degree of
         differentiability of it's base space.
 
@@ -390,9 +389,15 @@ def fiber(self, point):
         r"""
         Return the tensor bundle fiber at ``point``.
 
-        .. SEEALSO::
+        INPUT:
+
+        - ``point`` -- :class:`~sage.manifolds.point.ManifoldPoint`;
+          point `p` at which the fiber is defined
 
-            :meth:`~sage.manifolds.differentiable.manifold.Manifold.tangent_space`.
+        OUPUT:
+
+        - an instance of :class:`~sage.tensor.modules.finite_rank_free_module.FiniteRankFreeModule`
+          representing the tensor bundle fiber
 
         EXAMPLES::
 
@@ -405,6 +410,19 @@ def fiber(self, point):
             sage: TM.fiber(p)
             Tangent space at Point p on the 3-dimensional differentiable
              manifold M
+            sage: TM.fiber(p) is M.tangent_space(p)
+            True
+
+        ::
+
+            sage: T11M = M.tensor_bundle(1,1); T11M
+            Tensor bundle T^(1,1)M over the 3-dimensional differentiable
+             manifold M
+            sage: T11M.fiber(p)
+            Free module of type-(1,1) tensors on the Tangent space at Point p
+             on the 3-dimensional differentiable manifold M
+            sage: T11M.fiber(p) is M.tangent_space(p).tensor_module(1,1)
+            True
 
         """
         amb_point = self._dest_map(point)
@@ -417,7 +435,10 @@ def atlas(self):
 
         .. SEEALSO::
 
-            This method invokes :meth:`~sage.manifolds.manifold.Manifold.atlas`.
+            This method invokes
+            :meth:`~sage.manifolds.manifold.TopologicalManifold.atlas`
+            of class
+            :class:`~sage.manifolds.manifold.TopologicalManifold`
 
         EXAMPLES::
 
@@ -439,7 +460,7 @@ def section_module(self, domain=None):
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.differentiable.manifold.Manifold.tensor_field_module`.
+            :meth:`~sage.manifolds.differentiable.manifold.DifferentiableManifold.tensor_field_module`.
 
         INPUT:
 
@@ -483,7 +504,7 @@ def section(self, *args, **kwargs):
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.differentiable.manifold.Manifold.tensor_field`.
+            :meth:`~sage.manifolds.differentiable.manifold.DifferentiableManifold.tensor_field`.
 
         INPUT:
 
@@ -560,7 +581,7 @@ def set_change_of_frame(self, frame1, frame2, change_of_frame,
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.differentiable.manifold.Manifold.set_change_of_frame`.
+            :meth:`~sage.manifolds.differentiable.manifold.DifferentiableManifold.set_change_of_frame`.
 
         INPUT:
 
@@ -611,7 +632,8 @@ def change_of_frame(self, frame1, frame2):
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.differentiable.manifold.Manifold.change_of_frame`.
+            :meth:`~sage.manifolds.differentiable.manifold.DifferentiableManifold.change_of_frame`
+            of class :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`.
 
         INPUT:
 
@@ -659,7 +681,8 @@ def changes_of_frame(self):
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.differentiable.manifold.Manifold.changes_of_frame`.
+            :meth:`~sage.manifolds.differentiable.manifold.DifferentiableManifold.changes_of_frame`
+            of class :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`.
 
         OUTPUT:
 
@@ -717,7 +740,9 @@ def frames(self):
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.differentiable.manifold.Manifold.frames`.
+            :meth:`~sage.manifolds.differentiable.manifold.DifferentiableManifold.frames`
+            of class
+            :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`.
 
         OUTPUT:
 
@@ -764,7 +789,9 @@ def coframes(self):
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.differentiable.manifold.Manifold.coframes`.
+            :meth:`~sage.manifolds.differentiable.manifold.DifferentiableManifold.coframes`
+            of class
+            :class:`~sage.manifolds.differentiable.manifold.DifferentiableManifold`.
 
         OUTPUT:
 
@@ -816,7 +843,7 @@ def trivialization(self, coordinates='', names=None, calc_method=None):
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.manifold.Manifold.chart`.
+            :meth:`~sage.manifolds.manifold.TopologicalManifold.chart`.
 
         INPUT:
 
@@ -864,7 +891,7 @@ def trivialization(self, coordinates='', names=None, calc_method=None):
         examples below). If no interval range, no period and no LaTeX spelling
         is to be set for any coordinate, the argument ``coordinates`` can be
         omitted when the shortcut operator ``<,>`` is used to declare the
-        chart (see examples below).
+        trivialization.
 
         OUTPUT:
 
@@ -900,7 +927,7 @@ def transitions(self):
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.manifold.Manifold.coord_changes`.
+            :meth:`~sage.manifolds.manifold.TopologicalManifold.coord_changes`.
 
         EXAMPLES:
 
@@ -954,7 +981,7 @@ def transition(self, chart1, chart2):
         .. SEEALSO::
 
             This method invokes
-            :meth:`~sage.manifolds.manifold.Manifold.coord_change`.
+            :meth:`~sage.manifolds.manifold.TopologicalManifold.coord_change`.
 
         INPUT:
 
@@ -992,7 +1019,7 @@ def is_manifestly_trivial(self):
 
         EXAMPLES:
 
-        A just created manifold has no a priori manifestly trivial tensor
+        A just created manifold has a priori no manifestly trivial tangent
         bundle::
 
             sage: M = Manifold(2, 'M')

src/sage/manifolds/local_frame.py

@@ -2,8 +2,8 @@
 Local Frames
 
 The class :class:`LocalFrame` implements local frames on vector bundles
-(see :class:`sage.manifolds.vector_bundle.TopologicalVectorBundle` or
-:class:`sage.manifolds.differentiable.vector_bundle.DifferentiableVectorBundle`).
+(see :class:`~sage.manifolds.vector_bundle.TopologicalVectorBundle` or
+:class:`~sage.manifolds.differentiable.vector_bundle.DifferentiableVectorBundle`).
 
 For `k=0,1,\dots`, a *local frame* on a vector bundle `E \to M` of class `C^k`
 and rank `n` is a local section `(e_1,\dots,e_n):U \to E^n` of class `C^k`
@@ -1510,4 +1510,4 @@ def trivialization(self):
             True
 
         """
-        return self._trivialization
+        return self._trivialization

src/sage/manifolds/manifold.py

@@ -1571,7 +1571,7 @@ def vector_bundle(self, rank, name, field='real', latex_name=None):
         - ``name`` -- name given to the total space
         - ``field`` -- (default: ``'real'``) topological field giving the
           vector space structure to the fibers
-        - ``latex_name`` -- optional latex name for the total space
+        - ``latex_name`` -- optional LaTeX name for the total space
 
         OUTPUT:
 

src/sage/manifolds/section.py

@@ -62,7 +62,7 @@ class Section(ModuleElement):
 
     EXAMPLES:
 
-    A section on a non-trivial 2-rank vector bundle over a non-trivial
+    A section on a non-trivial rank 2 vector bundle over a non-trivial
     2-manifold::
 
         sage: M = Manifold(2, 'M', structure='top')
@@ -110,8 +110,9 @@ class Section(ModuleElement):
 
         sage: s.add_comp_by_continuation(fV, W, chart=c_uv)
 
-    At this stage, `s` is fully defined, having components in frames fU and fV
-    and the union of the domains of fU and fV being the whole manifold::
+    At this stage, `s` is fully defined, having components in frames ``fU``
+    and ``fV`` and the union of the domains of ``fU`` and ``fV`` being the
+    whole manifold::
 
         sage: s.display(fV)
         s = (-1/4*u^2 + 1/4*v^2 + 1/2*u + 1/2*v) (phi_V^*e_1)
@@ -576,7 +577,7 @@ def restrict(self, subdomain):
 
         EXAMPLES:
 
-        Restrictions of a section on a 2-rank vector bundle over the 2-sphere::
+        Restrictions of a section on a rank 2 vector bundle over the 2-sphere::
 
             sage: S2 = Manifold(2, 'S^2', structure='top', start_index=1)
             sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
@@ -1092,7 +1093,7 @@ def display(self, frame=None, chart=None):
 
         EXAMPLES:
 
-        Display of section on a 2-rank vector bundle over the 2-sphere::
+        Display of section on a rank 2 vector bundle over the 2-sphere::
 
             sage: S2 = Manifold(2, 'S^2', structure='top', start_index=1)
             sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
@@ -1254,7 +1255,7 @@ def at(self, point):
 
         EXAMPLES:
 
-        Vector on a 2-rank vector bundle fiber over a non-parallelizable
+        Vector on a rank 2 vector bundle fiber over a non-parallelizable
         2-dimensional manifold::
 
             sage: M = Manifold(2, 'M', structure='top')
@@ -1953,7 +1954,7 @@ def _rmul_(self, scalar):
         for dom, rst in self._restrictions.items():
             resu._restrictions[dom] = scalar.restrict(dom) * rst
         resu_name = format_mul_txt(scalar._name, '*', self._name)
-        resu_latex = format_mul_latex(scalar._latex_name, ' \cdot ',
+        resu_latex = format_mul_latex(scalar._latex_name, r' \cdot ',
                                       self._latex_name)
         resu.set_name(name=resu_name, latex_name=resu_latex)
         return resu
@@ -2010,7 +2011,7 @@ class TrivialSection(FiniteRankFreeModuleElement, Section):
 
     EXAMPLES:
 
-    A section on a trivial 3-rank vector bundle over the 3-sphere::
+    A section on a trivial rank 3 vector bundle over the 3-sphere::
 
         sage: M = Manifold(3, 'S^3', structure='top')
         sage: U = M.open_subset('U') ; V = M.open_subset('V') # complement of the North and South pole, respectively
@@ -2649,4 +2650,4 @@ def at(self, point):
             comp_resu = resu.add_comp(frame.at(point))
             for ind, val in comp._comp.items():
                 comp_resu._comp[ind] = val(point)
-        return resu
+        return resu

src/sage/manifolds/section_module.py

@@ -1,3 +1,4 @@
+# -*- coding: utf-8 -*-
 r"""
 Section Modules
 
@@ -55,6 +56,9 @@ class SectionModule(UniqueRepresentation, Parent):
 
     where `E_p` is the vector bundle fiber of `E` at the point `p`.
 
+    `C^k(U;E)` is a module over `C^k(U)`, the algebra of `C^k` scalar fields on
+    `U`.
+
     INPUT:
 
     - ``vbundle`` -- vector bundle `E` on which the sections takes its values
@@ -63,7 +67,7 @@ class SectionModule(UniqueRepresentation, Parent):
 
     EXAMPLES:
 
-    Module of sections on the Moebius bundle::
+    Module of sections on the Möbius bundle::
 
         sage: M = Manifold(1, 'S^1', structure='top', start_index=1)
         sage: U = M.open_subset('U')  # the complement of one point
@@ -85,7 +89,7 @@ class SectionModule(UniqueRepresentation, Parent):
         Module C^0(S^1;E) of sections on the 1-dimensional topological manifold
          S^1 with values in the real vector bundle E of rank 1
 
-    `C^0(S^1;E)` is a module over the algebra `C^0(M)`::
+    `C^0(S^1;E)` is a module over the algebra `C^0(S^1)`::
 
         sage: C0.category()
         Category of modules over Algebra of scalar fields on the 1-dimensional
@@ -98,7 +102,7 @@ class SectionModule(UniqueRepresentation, Parent):
         sage: isinstance(C0, FiniteRankFreeModule)
         False
 
-    since the Moebius bundle is not trivial::
+    since the Möbius bundle is not trivial::
 
         sage: E.is_manifestly_trivial()
         False
@@ -455,7 +459,7 @@ def set_default_frame(self, basis):
             sage: e is C0.default_frame()
             True
 
-        Notice, that the local frame is defined on a subset and not part of
+        Notice, that the local frame is defined on a subset and is not part of
         the section module `C^k(M;E)`::
 
             sage: C0.default_frame().domain()
@@ -524,7 +528,7 @@ class SectionFreeModule(FiniteRankFreeModule):
         sage: C0.base_ring() is M.scalar_field_algebra()
         True
 
-    The vector bundle admits a global frame and is therefore trivial:
+    The vector bundle admits a global frame and is therefore trivial::
 
         sage: E.is_manifestly_trivial()
         True
@@ -557,7 +561,7 @@ class SectionFreeModule(FiniteRankFreeModule):
         sage: C0.bases()
         [Local frame (E|_R^2, (e_0,e_1))]
 
-    The test suite is passed aswell::
+    The test suite is passed as well::
 
         sage: TestSuite(C0).run()