Merge with latest #19092 to fix doctests after the merge of #20770 in…

… sage 7.3.beta3

src/sage/manifolds/differentiable/vectorfield.py

@@ -845,6 +845,23 @@ def add_point_plot(vector,dom,xx_list,chart_domain,chart,ambient_coords,mapping,
                 resu = set_axes_labels(resu, *labels)
         return resu
 
+    def bracket(self, other):
+        """
+        Return the Lie bracket ``[self, other]``.
+
+        EXAMPLES::
+
+            sage: M = Manifold(3, 'M', structure='smooth')
+            sage: X.<x,y,z> = M.chart()
+            sage: v = -X.frame()[0] + 2*X.frame()[1] - (x^2 - y)*X.frame()[2]
+            sage: w = (z + y) * X.frame()[1] - X.frame()[2]
+            sage: vw = v.bracket(w); vw
+            Vector field on the 3-dimensional differentiable manifold M
+            sage: vw.disp()
+            (-x^2 + y + 2) d/dy + (-y - z) d/dz
+        """
+        return other.lie_der(self)
+
 #******************************************************************************
 
 class VectorFieldParal(FiniteRankFreeModuleElement, TensorFieldParal,

src/sage/manifolds/differentiable/vectorfield_module.py

@@ -4,8 +4,9 @@
 The set of vector fields along a differentiable manifold `U` with values on
 a differentiable manifold `M` via a differentiable map `\Phi: U\rightarrow M`
 (possibly `U=M` and `\Phi=\mathrm{Id}_M`) is a module over the algebra
-`C^k(U)` of differentiable scalar fields on `U`. It is a free module iff `M` is
-parallelizable. Accordingly, two classes are devoted to vector field modules:
+`C^k(U)` of differentiable scalar fields on `U`. If `\Phi` is the identity map,
+this module is considered a Lie algebroid under the Lie bracket `[\ ,\ ]`  (cf. :wikipedia:`Lie_algebroid`). It is a free module iff `M` is parallelizable.
+Accordingly, two classes are devoted to vector field modules:
 
 - :class:`VectorFieldModule` for vector fields with values on a
   generic (in practice, not parallelizable) differentiable manifold `M`
@@ -15,6 +16,7 @@
 AUTHORS:
 
 - Eric Gourgoulhon, Michal Bejger (2014-2015): initial version
+- Travis Scrimshaw (2016): structure of Lie algebroid (:trac:`20771`)
 
 REFERENCES:
 
@@ -29,6 +31,7 @@
 #******************************************************************************
 #       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>
 #
 #  Distributed under the terms of the GNU General Public License (GPL)
 #  as published by the Free Software Foundation; either version 2 of
@@ -73,6 +76,16 @@ class VectorFieldModule(UniqueRepresentation, Parent):
     The set `\mathcal{X}(U,\Phi)` is a module over `C^k(U)`, the ring
     (algebra) of differentiable scalar fields on `U` (see
     :class:`~sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra`).
+    Furthermore, it is a Lie algebroid under the Lie bracket (cf.
+    :wikipedia:`Lie_algebroid`)
+
+    .. MATH::
+
+        [X, Y] = X \circ Y - Y \circ X
+
+    over the scalarfields if `\Phi` is the identity map. That is to say
+    the Lie bracket is antisymmetric, bilinear over the base field,
+    satisfies the Jacobi identity, and `[X, fY] = X(f) Y + f[X, Y]`.
 
     The standard case of vector fields *on* a differentiable manifold
     corresponds to `U=M` and `\Phi = \mathrm{Id}_M`; we then denote
@@ -894,6 +907,8 @@ class VectorFieldFreeModule(FiniteRankFreeModule):
     over `C^k(U)`, the ring (algebra) of differentiable scalar fields on `U`
     (see
     :class:`~sage.manifolds.differentiable.scalarfield_algebra.DiffScalarFieldAlgebra`).
+    In fact, it carries the structure of a finite-dimensional Lie algebroid
+    (cf. :wikipedia:`Lie_algebroid`).
 
     The standard case of vector fields *on* a differentiable manifold
     corresponds to `U=M` and `\Phi = \mathrm{Id}_M`; we then denote
@@ -927,8 +942,9 @@ class VectorFieldFreeModule(FiniteRankFreeModule):
         Free module X(R^2) of vector fields on the 2-dimensional differentiable
          manifold R^2
         sage: XM.category()
-        Category of finite dimensional modules over Algebra of differentiable
-         scalar fields on the 2-dimensional differentiable manifold R^2
+        Category of finite dimensional modules
+         over Algebra of differentiable scalar fields
+         on the 2-dimensional differentiable manifold R^2
         sage: XM.base_ring() is M.scalar_field_algebra()
         True
 
@@ -966,8 +982,9 @@ class VectorFieldFreeModule(FiniteRankFreeModule):
          differentiable manifold I mapped into the 2-dimensional differentiable
          manifold R^2
         sage: XIM.category()
-        Category of finite dimensional modules over Algebra of differentiable
-         scalar fields on the 1-dimensional differentiable manifold I
+        Category of finite dimensional modules
+         over Algebra of differentiable scalar fields
+         on the 1-dimensional differentiable manifold I
 
     The rank of the free module `\mathcal{X}(I,\Phi)` is the dimension
     of the manifold `\RR^2`, namely two::
@@ -1046,8 +1063,9 @@ class VectorFieldFreeModule(FiniteRankFreeModule):
         sage: isinstance(XM, FiniteRankFreeModule)
         True
         sage: XM.category()
-        Category of finite dimensional modules over Algebra of differentiable
-         scalar fields on the 1-dimensional differentiable manifold S^1
+        Category of finite dimensional modules
+         over Algebra of differentiable scalar fields
+         on the 1-dimensional differentiable manifold S^1
         sage: XM.base_ring() is M.scalar_field_algebra()
         True
 
@@ -1118,10 +1136,12 @@ def __init__(self, domain, dest_map=None):
             name += "," + self._dest_map._name + ")"
             latex_name += "," + self._dest_map._latex_name + r"\right)"
         manif = self._ambient_domain.manifold()
+        cat = Modules(domain.scalar_field_algebra()).FiniteDimensional()
         FiniteRankFreeModule.__init__(self, domain.scalar_field_algebra(),
                                manif._dim, name=name, latex_name=latex_name,
                                start_index=manif._sindex,
-                               output_formatter=DiffScalarField.coord_function)
+                               output_formatter=DiffScalarField.coord_function,
+                               category=cat)
         #
         # Special treatment when self._dest_map != identity:
         # bases of self are created from vector frames of the ambient domain