Differentiable Vector Bundles

Let \(K\) be a topological field. A \(C^k\)-differentiable vector bundle of rank \(n\) over the field \(K\) and over a \(C^k\)-differentiable manifold \(M\) (base space) is a \(C^k\)-differentiable manifold \(E\) (total space) together with a \(C^k\) differentiable and surjective map \(\pi: E \to M\) such that for every point \(x \in M\):

• the set \(E_x=\pi^{-1}(x)\) has the vector space structure of \(K^n\),

• there is a neighborhood \(U \subset M\) of \(x\) and a \(C^k\)-diffeomorphism \(\varphi: \pi^{-1}(x) \to U \times K^n\) such that \(v \mapsto \varphi^{-1}(y,v)\) is a linear isomorphism for any \(y \in U\).

An important case of a differentiable vector bundle over a differentiable manifold is the tensor bundle (see TensorBundle)

AUTHORS:

• Michael Jung (2019) : initial version

class sage.manifolds.differentiable.vector_bundle.DifferentiableVectorBundle(rank, name, base_space, field='real', latex_name=None, category=None, unique_tag=None)

Bases: TopologicalVectorBundle

An instance of this class represents a differentiable vector bundle \(E \to M\)

INPUT:

• rank – positive integer; rank of the vector bundle

• 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 vector space over \(K\); allowed values are

  • 'real' or an object of type RealField (e.g., RR) for a vector bundle over \(\RR\)

  • 'complex' or an object of type ComplexField (e.g., CC) for a vector bundle over \(\CC\)

  • an object in the category of topological fields (see Fields and TopologicalSpaces) for other types of topological fields

• 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() is assumed (see the category VectorBundles)

EXAMPLES:

A differentiable vector bundle of rank 2 over a 3-dimensional differentiable manifold:

sage: M = Manifold(3, 'M')
sage: E = M.vector_bundle(2, 'E', field='complex'); E
Differentiable complex vector bundle E -> M of rank 2 over the base
 space 3-dimensional differentiable manifold M
sage: E.category()
Category of smooth vector bundles over Complex Field with 53 bits of
 precision with base space 3-dimensional differentiable manifold M

At this stage, the differentiable vector bundle has the same differentiability degree as the base manifold:

sage: M.diff_degree() == E.diff_degree()
True

bundle_connection(name, latex_name=None)

Return a bundle connection on self.

OUTPUT:

• a bundle connection on self as an instance of BundleConnection

EXAMPLES:

sage: M = Manifold(3, 'M', start_index=1)
sage: X.<x,y,z> = M.chart()
sage: E = M.vector_bundle(2, 'E')
sage: e = E.local_frame('e') # standard frame for E
sage: nab = E.bundle_connection('nabla', latex_name=r'\nabla'); nab
Bundle connection nabla on the Differentiable real vector bundle
 E -> M of rank 2 over the base space 3-dimensional differentiable
 manifold M

See also

Further examples can be found in BundleConnection.

characteristic_class(*args, **kwds)

Deprecated: Use characteristic_cohomology_class() instead. See github issue #29581 for details.

characteristic_cohomology_class(*args, **kwargs)

Return a characteristic cohomology class associated with the input data.

INPUT:

• val – the input data associated with the characteristic class using the Chern-Weil homomorphism; this argument can be either a symbolic expression, a polynomial or one of the following predefined classes:

  • 'Chern' – total Chern class,

  • 'ChernChar' – Chern character,

  • 'Todd' – Todd class,

  • 'Pontryagin' – total Pontryagin class,

  • 'Hirzebruch' – Hirzebruch class,

  • 'AHat' – \(\hat{A}\) class,

  • 'Euler' – Euler class.

• base_ring – (default: QQ) base ring over which the characteristic cohomology class ring shall be defined

• name – (default: None) string representation given to the characteristic cohomology class; if None the default algebra representation or predefined name is used

• latex_name – (default: None) LaTeX name given to the characteristic class; if None the value of name is used

• class_type – (default: None) class type of the characteristic cohomology class; the following options are possible:

  • 'multiplicative' – returns a class of multiplicative type

  • 'additive' – returns a class of additive type

  • 'Pfaffian' – returns a class of Pfaffian type

  This argument must be stated if val is a polynomial or symbolic expression.

EXAMPLES:

Pontryagin class on the Minkowski space:

sage: M = Manifold(4, 'M', structure='Lorentzian', start_index=1)
sage: X.<t,x,y,z> = M.chart()
sage: g = M.metric()
sage: g[1,1] = -1
sage: g[2,2] = 1
sage: g[3,3] = 1
sage: g[4,4] = 1
sage: g.display()
g = -dt⊗dt + dx⊗dx + dy⊗dy + dz⊗dz

Let us introduce the corresponding Levi-Civita connection:

sage: nab = g.connection(); nab
Levi-Civita connection nabla_g associated with the Lorentzian
 metric g on the 4-dimensional Lorentzian manifold M
sage: nab.set_immutable()  # make nab immutable

Of course, \(\nabla_g\) is flat:

sage: nab.display()

Let us check the total Pontryagin class which must be the one element in the corresponding cohomology ring in this case:

sage: TM = M.tangent_bundle(); TM
Tangent bundle TM over the 4-dimensional Lorentzian manifold M
sage: p = TM.characteristic_cohomology_class('Pontryagin'); p
Characteristic cohomology class p(TM) of the Tangent bundle TM over
 the 4-dimensional Lorentzian manifold M
sage: p_form = p.get_form(nab); p_form.display_expansion()
p(TM, nabla_g) = 1

See also

More examples can be found in CharacteristicClass.

characteristic_cohomology_class_ring(base=Rational Field)

Return the characteristic cohomology class ring of self over a given base.

INPUT:

• base – (default: QQ) base over which the ring should be constructed; typically that would be \(\ZZ\), \(\QQ\), \(\RR\) or the symbolic ring

EXAMPLES:

sage: M = Manifold(4, 'M', start_index=1)
sage: R = M.tangent_bundle().characteristic_cohomology_class_ring()
sage: R
Algebra of characteristic cohomology classes of the Tangent bundle
 TM over the 4-dimensional differentiable manifold M
sage: p1 = R.gen(0); p1
Characteristic cohomology class (p_1)(TM) of the Tangent bundle TM
 over the 4-dimensional differentiable manifold M
sage: 1 + p1
Characteristic cohomology class (1 + p_1)(TM) of the Tangent bundle
 TM over the 4-dimensional differentiable manifold M

diff_degree()

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 \(C^k\) over its base field. The degree always corresponds to the degree of differentiability of it’s base space.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: E = M.vector_bundle(2, 'E')
sage: E.diff_degree()
+Infinity
sage: M = Manifold(2, 'M', structure='differentiable',
....:              diff_degree=3)
sage: E = M.vector_bundle(2, 'E')
sage: E.diff_degree()
3

total_space()

Return the total space of self.

Note

At this stage, the total space does not come with induced charts.

OUTPUT:

• the total space of self as an instance of DifferentiableManifold

EXAMPLES:

sage: M = Manifold(3, 'M')
sage: E = M.vector_bundle(2, 'E')
sage: E.total_space()
6-dimensional differentiable manifold E

class sage.manifolds.differentiable.vector_bundle.TensorBundle(base_space, k, l, dest_map=None)

Bases: DifferentiableVectorBundle

Tensor bundle over a differentiable manifold along a differentiable map.

An instance of this class represents the pullback tensor bundle \(\Phi^* T^{(k,l)}N\) along a differentiable map (called destination map)

\[\Phi: M \longrightarrow N\]

between two differentiable manifolds \(M\) and \(N\) over the topological field \(K\).

More precisely, \(\Phi^* T^{(k,l)}N\) consists of all pairs \((p,t) \in M \times T^{(k,l)}N\) such that \(t \in T_q^{(k,l)}N\) for \(q = \Phi(p)\), namely

\[t:\ \underbrace{T_q^*N\times\cdots\times T_q^*N}_{k\ \; \text{times}} \times \underbrace{T_q N\times\cdots\times T_q N}_{l\ \; \text{times}} \longrightarrow K\]

(\(k\) is called the contravariant and \(l\) the covariant rank of the tensor bundle).

The trivializations are directly given by charts on the codomain (called ambient domain) of \(\Phi\). In particular, let \((V, \varphi)\) be a chart of \(N\) with components \((x^1, \dots, x^n)\) such that \(q=\Phi(p) \in V\). Then, the matrix entries of \(t \in T_q^{(k,l)}N\) are given by

\[t^{a_1 \ldots a_k}_{\phantom{a_1 \ldots a_k} \, b_1 \ldots b_l} = t \left( \left.\frac{\partial}{\partial x^{a_1}}\right|_q, \dots, \left.\frac{\partial}{\partial x^{a_k}}\right|_q, \left.\mathrm{d}x^{b_1}\right|_q, \dots, \left.\mathrm{d}x^{b_l}\right|_q \right) \in K\]

and a trivialization over \(U=\Phi^{-1}(V) \subset M\) is obtained via

\[(p,t) \mapsto \left(p, t^{1 \ldots 1}_{\phantom{1 \ldots 1} \, 1 \ldots 1}, \dots, t^{n \ldots n}_{\phantom{n \ldots n} \, n \ldots n} \right) \in U \times K^{n^{(k+l)}}.\]

The standard case of a tensor bundle over a differentiable manifold corresponds to \(M=N\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(N\) (\(M\) is then an open interval of \(\RR\)).

INPUT:

• base_space – the base space (differentiable manifold) \(M\) over which the tensor bundle is defined

• k – the contravariant rank of the corresponding tensor bundle

• l – the covariant rank of the corresponding tensor bundle

• dest_map – (default: None) destination map \(\Phi:\ M \rightarrow N\) (type: DiffMap); if None, it is assumed that \(M=M\) and \(\Phi\) is the identity map of \(M\) (case of the standard tensor bundle over \(M\))

EXAMPLES:

Pullback tangent bundle of \(R^2\) along a curve \(\Phi\):

sage: M = Manifold(2, 'M')
sage: c_cart.<x,y> = M.chart()
sage: R = Manifold(1, 'R')
sage: T.<t> = R.chart()  # canonical chart on R
sage: Phi = R.diff_map(M, [cos(t), sin(t)], name='Phi') ; Phi
Differentiable map Phi from the 1-dimensional differentiable manifold R
 to the 2-dimensional differentiable manifold M
sage: Phi.display()
Phi: R → M
   t ↦ (x, y) = (cos(t), sin(t))
sage: PhiTM = R.tangent_bundle(dest_map=Phi); PhiTM
Tangent bundle Phi^*TM over the 1-dimensional differentiable manifold R
 along the Differentiable map Phi from the 1-dimensional differentiable
 manifold R to the 2-dimensional differentiable manifold M

The section module is the corresponding tensor field module:

sage: R_tensor_module = R.tensor_field_module((1,0), dest_map=Phi)
sage: R_tensor_module is PhiTM.section_module()
True

ambient_domain()

Return the codomain of the destination map.

OUTPUT:

• a DifferentiableManifold representing the codomain of the destination map

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: c_cart.<x,y> = M.chart()
sage: e_cart = c_cart.frame() # standard basis
sage: R = Manifold(1, 'R')
sage: T.<t> = R.chart()  # canonical chart on R
sage: Phi = R.diff_map(M, [cos(t), sin(t)], name='Phi') ; Phi
Differentiable map Phi from the 1-dimensional differentiable
 manifold R to the 2-dimensional differentiable manifold M
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.ambient_domain()
2-dimensional differentiable manifold M

atlas()

Return the list of charts that have been defined on the codomain of the destination map.

Note

Since an atlas of charts gives rise to an atlas of trivializations, this method directly invokes atlas() of the class TopologicalManifold.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: Y.<u,v> = M.chart()
sage: TM = M.tangent_bundle()
sage: TM.atlas()
[Chart (M, (x, y)), Chart (M, (u, v))]

change_of_frame(frame1, frame2)

Return a change of vector frames defined on the base space of self.

See also

For further details on frames on self see local_frame().

Note

Since frames on self are directly induced by vector frames on the base space, this method directly invokes change_of_frame() of the class DifferentiableManifold.

INPUT:

• frame1 – local frame 1

• frame2 – local frame 2

OUTPUT:

• a FreeModuleAutomorphism representing, at each point, the vector space automorphism \(P\) that relates frame 1, \((e_i)\) say, to frame 2, \((f_i)\) say, according to \(f_i = P(e_i)\)

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: c_uv.<u,v> = M.chart()
sage: c_xy.transition_map(c_uv, (x+y, x-y))
Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))
sage: TM = M.tangent_bundle()
sage: TM.change_of_frame(c_xy.frame(), c_uv.frame())
Field of tangent-space automorphisms on the 2-dimensional
 differentiable manifold M
sage: TM.change_of_frame(c_xy.frame(), c_uv.frame())[:]
[ 1/2  1/2]
[ 1/2 -1/2]
sage: TM.change_of_frame(c_uv.frame(), c_xy.frame())
Field of tangent-space automorphisms on the 2-dimensional
 differentiable manifold M
sage: TM.change_of_frame(c_uv.frame(), c_xy.frame())[:]
[ 1  1]
[ 1 -1]
sage: TM.change_of_frame(c_uv.frame(), c_xy.frame()) == \
....:       M.change_of_frame(c_xy.frame(), c_uv.frame()).inverse()
True

changes_of_frame()

Return the changes of vector frames defined on the base space of self with respect to the destination map.

See also

For further details on frames on self see local_frame().

OUTPUT:

• dictionary of automorphisms on the tangent bundle representing the changes of frames, the keys being the pair of frames

EXAMPLES:

Let us consider a first vector frame on a 2-dimensional differentiable manifold:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: TM = M.tangent_bundle()
sage: e = X.frame(); e
Coordinate frame (M, (∂/∂x,∂/∂y))

At this stage, the dictionary of changes of frame is empty:

sage: TM.changes_of_frame()
{}

We introduce a second frame on the manifold, relating it to frame e by a field of tangent space automorphisms:

sage: a = M.automorphism_field(name='a')
sage: a[:] = [[-y, x], [1, 2]]
sage: f = e.new_frame(a, 'f'); f
Vector frame (M, (f_0,f_1))

Then we have:

sage: TM.changes_of_frame()  # random (dictionary output)
{(Coordinate frame (M, (∂/∂x,∂/∂y)),
  Vector frame (M, (f_0,f_1))): Field of tangent-space
   automorphisms on the 2-dimensional differentiable manifold M,
 (Vector frame (M, (f_0,f_1)),
  Coordinate frame (M, (∂/∂x,∂/∂y))): Field of tangent-space
   automorphisms on the 2-dimensional differentiable manifold M}

Some checks:

sage: TM.changes_of_frame()[(e,f)] == a
True
sage: TM.changes_of_frame()[(f,e)] == a^(-1)
True

coframes()

Return the list of coframes defined on the base manifold of self with respect to the destination map.

See also

For further details on frames on self see local_frame().

OUTPUT:

• list of coframes defined on self

EXAMPLES:

Coframes on subsets of \(\RR^2\):

sage: M = Manifold(2, 'R^2')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: TM = M.tangent_bundle()
sage: TM.coframes()
[Coordinate coframe (R^2, (dx,dy))]
sage: e = TM.vector_frame('e')
sage: M.coframes()
[Coordinate coframe (R^2, (dx,dy)), Coframe (R^2, (e^0,e^1))]
sage: U = M.open_subset('U', coord_def={c_cart: x^2+y^2<1})
sage: TU = U.tangent_bundle()
sage: TU.coframes()
[Coordinate coframe (U, (dx,dy))]
sage: e.restrict(U)
Vector frame (U, (e_0,e_1))
sage: TU.coframes()
[Coordinate coframe (U, (dx,dy)), Coframe (U, (e^0,e^1))]
sage: TM.coframes()
[Coordinate coframe (R^2, (dx,dy)),
 Coframe (R^2, (e^0,e^1)),
 Coordinate coframe (U, (dx,dy)),
 Coframe (U, (e^0,e^1))]

default_frame()

Return the default vector frame defined on self.

By vector frame, it is meant a field on the manifold that provides, at each point \(p\), a vector basis of the pulled back tangent space at \(p\).

If the destination map is the identity map, the default frame is the the first one defined on the manifold, usually the coordinate frame, unless it is changed via set_default_frame().

If the destination map is non-trivial, the default frame usually must be set via set_default_frame().

OUTPUT:

• a VectorFrame representing the default vector frame

EXAMPLES:

The default vector frame is often the coordinate frame associated with the first chart defined on the manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: TM = M.tangent_bundle()
sage: TM.default_frame()
Coordinate frame (M, (∂/∂x,∂/∂y))

destination_map()

Return the destination map.

OUTPUT:

• a DifferentialMap representing the destination map

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: c_cart.<x,y> = M.chart()
sage: e_cart = c_cart.frame() # standard basis
sage: R = Manifold(1, 'R')
sage: T.<t> = R.chart()  # canonical chart on R
sage: Phi = R.diff_map(M, [cos(t), sin(t)], name='Phi') ; Phi
Differentiable map Phi from the 1-dimensional differentiable
 manifold R to the 2-dimensional differentiable manifold M
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.destination_map()
Differentiable map Phi from the 1-dimensional differentiable
 manifold R to the 2-dimensional differentiable manifold M

fiber(point)

Return the tensor bundle fiber over a point.

INPUT:

• point – ManifoldPoint; point \(p\) of the base manifold of self

OUTPUT:

• an instance of FiniteRankFreeModule representing the tensor bundle fiber over \(p\)

EXAMPLES:

sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: p = M((0,2,1), name='p'); p
Point p on the 3-dimensional differentiable manifold M
sage: TM = M.tangent_bundle(); TM
Tangent bundle TM over the 3-dimensional differentiable manifold M
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

frames()

Return the list of all vector frames defined on the base space of self with respect to the destination map.

See also

For further details on frames on self see local_frame().

OUTPUT:

• list of local frames defined on self

EXAMPLES:

Vector frames on subsets of \(\RR^2\):

sage: M = Manifold(2, 'R^2')
sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2
sage: TM = M.tangent_bundle()
sage: TM.frames()
[Coordinate frame (R^2, (∂/∂x,∂/∂y))]
sage: e = TM.vector_frame('e')
sage: TM.frames()
[Coordinate frame (R^2, (∂/∂x,∂/∂y)),
 Vector frame (R^2, (e_0,e_1))]
sage: U = M.open_subset('U', coord_def={c_cart: x^2+y^2<1})
sage: TU = U.tangent_bundle()
sage: TU.frames()
[Coordinate frame (U, (∂/∂x,∂/∂y))]
sage: TM.frames()
[Coordinate frame (R^2, (∂/∂x,∂/∂y)),
 Vector frame (R^2, (e_0,e_1)),
 Coordinate frame (U, (∂/∂x,∂/∂y))]

List of vector frames of a tensor bundle of type \((1 ,1)\) along a curve:

sage: M = Manifold(2, 'M')
sage: c_cart.<x,y> = M.chart()
sage: e_cart = c_cart.frame() # standard basis
sage: R = Manifold(1, 'R')
sage: T.<t> = R.chart()  # canonical chart on R
sage: Phi = R.diff_map(M, [cos(t), sin(t)], name='Phi') ; Phi
Differentiable map Phi from the 1-dimensional differentiable
 manifold R to the 2-dimensional differentiable manifold M
sage: Phi.display()
Phi: R → M
   t ↦ (x, y) = (cos(t), sin(t))
sage: PhiT11 = R.tensor_bundle(1, 1, dest_map=Phi); PhiT11
Tensor bundle Phi^*T^(1,1)M over the 1-dimensional differentiable
 manifold R along the Differentiable map Phi from the 1-dimensional
 differentiable manifold R to the 2-dimensional differentiable
 manifold M
sage: f = PhiT11.local_frame(); f
Vector frame (R, (∂/∂x,∂/∂y)) with values on the 2-dimensional
 differentiable manifold M
sage: PhiT11.frames()
[Vector frame (R, (∂/∂x,∂/∂y)) with values on the 2-dimensional
 differentiable manifold M]

is_manifestly_trivial()

Return True if self is known to be a trivial and False otherwise.

If False is returned, either the tensor bundle is not trivial or no vector frame has been defined on it yet.

EXAMPLES:

A just created manifold has a priori no manifestly trivial tangent bundle:

sage: M = Manifold(2, 'M')
sage: TM = M.tangent_bundle()
sage: TM.is_manifestly_trivial()
False

Defining a vector frame on it makes it trivial:

sage: e = TM.vector_frame('e')
sage: TM.is_manifestly_trivial()
True

Defining a coordinate chart on the whole manifold also makes it trivial:

sage: N = Manifold(4, 'N')
sage: X.<t,x,y,z> = N.chart()
sage: TN = N.tangent_bundle()
sage: TN.is_manifestly_trivial()
True

The situation is not so clear anymore when a destination map to a non-parallelizable manifold is stated:

sage: M = Manifold(2, 'S^2') # the 2-dimensional sphere S^2
sage: U = M.open_subset('U') # complement of the North pole
sage: c_xy.<x,y> = U.chart() # stereo coord from the North pole
sage: V = M.open_subset('V') # complement of the South pole
sage: c_uv.<u,v> = V.chart() # stereo coord from the South pole
sage: M.declare_union(U,V)   # S^2 is the union of U and V
sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2),
....:                                       y/(x^2+y^2)),
....:                                intersection_name='W',
....:                                restrictions1= x^2+y^2!=0,
....:                                restrictions2= u^2+v^2!=0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V)
sage: Phi = U.diff_map(M, {(c_xy, c_xy): [x, y]},
....:                  name='Phi') # inclusion map
sage: PhiTU = U.tangent_bundle(dest_map=Phi); PhiTU
Tangent bundle Phi^*TS^2 over the Open subset U of the
 2-dimensional differentiable manifold S^2 along the
 Differentiable map Phi from the Open subset U of the
 2-dimensional differentiable manifold S^2 to the 2-dimensional
 differentiable manifold S^2

A priori, the pullback tangent bundle is not trivial:

sage: PhiTU.is_manifestly_trivial()
False

But certainly, this bundle must be trivial since \(U\) is parallelizable. To ensure this, we need to define a local frame on \(U\) with values in \(\Phi^*TS^2\):

sage: PhiTU.local_frame('e', from_frame=c_xy.frame())
Vector frame (U, (e_0,e_1)) with values on the 2-dimensional
 differentiable manifold S^2
sage: PhiTU.is_manifestly_trivial()
True

local_frame(*args, **kwargs)

Define a vector frame on domain, possibly with values in the tangent bundle of the ambient domain.

If the basis specified by the given symbol already exists, it is simply returned. If no argument is provided the vector field module’s default frame is returned.

Notice, that a vector frame automatically induces a local frame on the tensor bundle self. More precisely, if \(e: U \to \Phi^*TN\) is a vector frame on \(U \subset M\) with values in \(\Phi^*TN\) along the destination map

\[\Phi: M \longrightarrow N\]

then the map

\[p \mapsto \Big(\underbrace{e^*(p), \dots, e^*(p)}_{k\ \; \text{times}}, \underbrace{e(p), \dots, e(p)}_{l\ \; \text{times}}\Big) \in T^{(k,l)}_q N ,\]

with \(q=\Phi(p)\), defines a basis at each point \(p \in U\) and therefore gives rise to a local frame on \(\Phi^* T^{(k,l)}N\) on the domain \(U\).

See also

VectorFrame for complete documentation.

INPUT:

• symbol – (default: None) either a string, to be used as a common base for the symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual symbols of the vector fields; can be None only if from_frame is not None (see below)

• vector_fields – tuple or list of \(n\) linearly independent vector fields on domain (\(n\) being the dimension of domain) defining the vector frame; can be omitted if the vector frame is created from scratch or if from_frame is not None

• latex_symbol – (default: None) either a string, to be used as a common base for the LaTeX symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual LaTeX symbols of the vector fields; if None, symbol is used in place of latex_symbol

• from_frame – (default: None) vector frame \(\tilde{e}\) on the codomain \(N\) of the destination map \(\Phi\); the returned frame \(e\) is then such that for all \(p \in U\), we have \(e(p) = \tilde{e}(\Phi(p))\)

• indices – (default: None; used only if symbol is a single string) tuple of strings representing the indices labelling the vector fields of the frame; if None, the indices will be generated as integers within the range declared on self

• latex_indices – (default: None) tuple of strings representing the indices for the LaTeX symbols of the vector fields; if None, indices is used instead

• symbol_dual – (default: None) same as symbol but for the dual coframe; if None, symbol must be a string and is used for the common base of the symbols of the elements of the dual coframe

• latex_symbol_dual – (default: None) same as latex_symbol but for the dual coframe

• domain – (default: None) domain on which the local frame is defined; if None is provided, the base space of self is assumed

OUTPUT:

• the vector frame corresponding to the above specifications; this is an instance of VectorFrame.

EXAMPLES:

Defining a local frame for the tangent bundle of a 3-dimensional manifold:

sage: M = Manifold(3, 'M')
sage: TM = M.tangent_bundle()
sage: e = TM.local_frame('e'); e
Vector frame (M, (e_0,e_1,e_2))
sage: e[0]
Vector field e_0 on the 3-dimensional differentiable manifold M

Specifying the domain of the vector frame:

sage: U = M.open_subset('U')
sage: f = TM.local_frame('f', domain=U); f
Vector frame (U, (f_0,f_1,f_2))
sage: f[0]
Vector field f_0 on the Open subset U of the 3-dimensional
 differentiable manifold M

See also

For more options, in particular for the choice of symbols and indices, see VectorFrame.

orientation()

Get the preferred orientation of self if available.

See orientation() for details regarding orientations on vector bundles.

The tensor bundle \(\Phi^* T^{(k,l)}N\) of a manifold is orientable if the manifold \(\Phi(M)\) is orientable. The converse does not necessarily hold true. The usual case corresponds to \(\Phi\) being the identity map, where the tensor bundle \(T^{(k,l)}M\) is orientable if and only if the manifold \(M\) is orientable.

Note

Notice that the orientation of a general tensor bundle \(\Phi^* T^{(k,l)}N\) is canonically induced by the orientation of the tensor bundle \(\Phi^* T^{(1,0)}N\) as each local frame there induces the frames on \(\Phi^* T^{(k,l)}N\) in a canonical way.

If no preferred orientation has been set before, and if the ambient space already admits a preferred orientation, the corresponding orientation is returned and henceforth fixed for the tensor bundle.

EXAMPLES:

In the trivial case, i.e. if the destination map is the identitiy and the tangent bundle is covered by one frame, the orientation is easily obtained:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: T11 = M.tensor_bundle(1, 1)
sage: T11.orientation()
[Coordinate frame (M, (∂/∂x,∂/∂y))]

The same holds true if the ambient domain admits a trivial orientation:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: R = Manifold(1, 'R')
sage: c_t.<t> = R.chart()
sage: Phi = R.diff_map(M, name='Phi')
sage: PhiT22 = R.tensor_bundle(2, 2, dest_map=Phi); PhiT22
Tensor bundle Phi^*T^(2,2)M over the 1-dimensional differentiable
 manifold R along the Differentiable map Phi from the 1-dimensional
 differentiable manifold R to the 2-dimensional differentiable
 manifold M
sage: PhiT22.local_frame()  # initialize frame
Vector frame (R, (∂/∂x,∂/∂y)) with values on the 2-dimensional
 differentiable manifold M
sage: PhiT22.orientation()
[Vector frame (R, (∂/∂x,∂/∂y)) with values on the 2-dimensional
 differentiable manifold M]
sage: PhiT22.local_frame() is PhiT22.orientation()[0]
True

In the non-trivial case, however, the orientation must be set manually by the user:

sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U'); V = M.open_subset('V')
sage: M.declare_union(U, V)
sage: c_xy.<x,y> = U.chart(); c_uv.<u,v> = V.chart()
sage: T11 = M.tensor_bundle(1, 1); T11
Tensor bundle T^(1,1)M over the 2-dimensional differentiable
 manifold M
sage: T11.orientation()
[]
sage: T11.set_orientation([c_xy.frame(), c_uv.frame()])
sage: T11.orientation()
[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))]

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))]

section(*args, **kwargs)

Return a section of self on domain, namely a tensor field on the subset domain of the base space.

Note

This method directly invokes tensor_field() of the class DifferentiableManifold.

INPUT:

• comp – (optional) either the components of the tensor field with respect to the vector frame specified by the argument frame or a dictionary of components, the keys of which are vector frames or pairs (f, c) where f is a vector frame and c the chart in which the components are expressed

• frame – (default: None; unused if comp is not given or is a dictionary) vector frame in which the components are given; if None, the default vector frame of self is assumed

• chart – (default: None; unused if comp is not given or is a dictionary) coordinate chart in which the components are expressed; if None, the default chart on the domain of frame is assumed

• domain – (default: None) domain of the section; if None, self.base_space() is assumed

• name – (default: None) name given to the tensor field

• latex_name – (default: None) LaTeX symbol to denote the tensor field; if None, the LaTeX symbol is set to name

• sym – (default: None) a symmetry or a list of symmetries among the tensor arguments: each symmetry is described by a tuple containing the positions of the involved arguments, with the convention position=0 for the first argument; for instance:

  • sym = (0,1) for a symmetry between the 1st and 2nd arguments

  • sym = [(0,2), (1,3,4)] for a symmetry between the 1st and 3rd arguments and a symmetry between the 2nd, 4th and 5th arguments

• antisym – (default: None) antisymmetry or list of antisymmetries among the arguments, with the same convention as for sym

OUTPUT:

• a TensorField (or if \(N\) is parallelizable, a TensorFieldParal) representing the defined tensor field on the domain \(U \subset M\)

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V)   # M is the union of U and V
sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart()
sage: transf = c_xy.transition_map(c_uv, (x+y, x-y),
....:                              intersection_name='W',
....:                              restrictions1= x>0,
....:                              restrictions2= u+v>0)
sage: inv = transf.inverse()
sage: W = U.intersection(V)
sage: eU = c_xy.frame() ; eV = c_uv.frame()
sage: T11M = M.tensor_bundle(1, 1); T11M
Tensor bundle T^(1,1)M over the 2-dimensional differentiable
 manifold M
sage: t = T11M.section({eU: [[1, x], [0, 2]]}, name='t'); t
Tensor field t of type (1,1) on the 2-dimensional differentiable
 manifold M
sage: t.display()
t = ∂/∂x⊗dx + x ∂/∂x⊗dy + 2 ∂/∂y⊗dy

An example of use with the arguments comp and domain:

sage: TM = M.tangent_bundle()
sage: w = TM.section([-y, x], domain=U); w
Vector field on the Open subset U of the 2-dimensional
 differentiable manifold M
sage: w.display()
-y ∂/∂x + x ∂/∂y

section_module(domain=None)

Return the section module on domain, namely the corresponding tensor field module, of self on domain.

Note

This method directly invokes tensor_field_module() of the class DifferentiableManifold.

INPUT:

• domain – (default: None) the domain of the corresponding section module; if None, the base space is assumed

OUTPUT:

• a TensorFieldModule (or if \(N\) is parallelizable, a TensorFieldFreeModule) representing the module \(\mathcal{T}^{(k,l)}(U,\Phi)\) of type-\((k,l)\) tensor fields on the domain \(U \subset M\) taking values on \(\Phi(U) \subset N\)

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: U = M.open_subset('U')
sage: TM = M.tangent_bundle()
sage: TUM = TM.section_module(domain=U); TUM
Module X(U) of vector fields on the Open subset U of the
 2-dimensional differentiable manifold M
sage: TUM is U.tensor_field_module((1,0))
True

set_change_of_frame(frame1, frame2, change_of_frame, compute_inverse=True)

Relate two vector frames by an automorphism.

This updates the internal dictionary self._frame_changes of the base space \(M\).

See also

For further details on frames on self see local_frame().

Note

Since frames on self are directly induced by vector frames on the base space, this method directly invokes set_change_of_frame() of the class DifferentiableManifold.

INPUT:

• frame1 – frame 1, denoted \((e_i)\) below

• frame2 – frame 2, denoted \((f_i)\) below

• change_of_frame – instance of class 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 automorphism is computed and the change from basis \((f_i)\) to \((e_i)\) is set to it in the internal dictionary self._frame_changes

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: e = M.vector_frame('e')
sage: f = M.vector_frame('f')
sage: a = M.automorphism_field()
sage: a[e,:] = [[1,2],[0,3]]
sage: TM = M.tangent_bundle()
sage: TM.set_change_of_frame(e, f, a)
sage: f[0].display(e)
f_0 = e_0
sage: f[1].display(e)
f_1 = 2 e_0 + 3 e_1
sage: e[0].display(f)
e_0 = f_0
sage: e[1].display(f)
e_1 = -2/3 f_0 + 1/3 f_1
sage: TM.change_of_frame(e,f)[e,:]
[1 2]
[0 3]

set_default_frame(frame)

Changing the default vector frame on self.

Note

If the destination map is the identity, the default frame of the base manifold gets changed here as well.

INPUT:

• frame – VectorFrame a vector frame defined on the base manifold

EXAMPLES:

Changing the default frame on the tangent bundle of a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: TM = M.tangent_bundle()
sage: e = TM.vector_frame('e')
sage: TM.default_frame()
Coordinate frame (M, (∂/∂x,∂/∂y))
sage: TM.set_default_frame(e)
sage: TM.default_frame()
Vector frame (M, (e_0,e_1))
sage: M.default_frame()
Vector frame (M, (e_0,e_1))

set_orientation(orientation)

Set the preferred orientation of self.

INPUT:

• orientation – a vector frame or a list of vector frames, covering the base space of self

Note

If the destination map is the identity, the preferred orientation of the base manifold gets changed here as well.

Warning

It is the user’s responsibility that the orientation set here is indeed an orientation. There is no check going on in the background. See orientation() for the definition of an orientation.

EXAMPLES:

Set an orientation on a tensor bundle:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: T11 = M.tensor_bundle(1, 1)
sage: e = T11.local_frame('e'); e
Vector frame (M, (e_0,e_1))
sage: T11.set_orientation(e)
sage: T11.orientation()
[Vector frame (M, (e_0,e_1))]

Set an orientation in the non-trivial case:

sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U'); V = M.open_subset('V')
sage: M.declare_union(U, V)
sage: c_xy.<x,y> = U.chart(); c_uv.<u,v> = V.chart()
sage: T12 = M.tensor_bundle(1, 2)
sage: e = T12.local_frame('e', domain=U)
sage: f = T12.local_frame('f', domain=V)
sage: T12.set_orientation([e, f])
sage: T12.orientation()
[Vector frame (U, (e_0,e_1)), Vector frame (V, (f_0,f_1))]

transition(chart1, chart2)

Return the change of trivializations in terms of a coordinate change between two differentiable charts defined on the codomain of the destination map.

The differentiable chart must have been defined previously, for instance by the method transition_map().

Note

Since a chart gives direct rise to a trivialization, this method is nothing but an invocation of coord_change() of the class TopologicalManifold.

INPUT:

• chart1 – chart 1

• chart2 – chart 2

OUTPUT:

• instance of CoordChange representing the transition map from chart 1 to chart 2

EXAMPLES:

Change of coordinates on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: c_uv.<u,v> = M.chart()
sage: c_xy.transition_map(c_uv, (x+y, x-y)) # defines coord. change
Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))
sage: TM = M.tangent_bundle()
sage: TM.transition(c_xy, c_uv) # returns the coord. change above
Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))

transitions()

Return the transition maps between trivialization maps in terms of coordinate changes defined via charts on the codomain of the destination map.

Note

Since a chart gives direct rise to a trivialization, this method is nothing but an invocation of coord_changes() of the class TopologicalManifold.

EXAMPLES:

Various changes of coordinates on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: c_xy.<x,y> = M.chart()
sage: c_uv.<u,v> = M.chart()
sage: xy_to_uv = c_xy.transition_map(c_uv, [x+y, x-y])
sage: TM = M.tangent_bundle()
sage: TM.transitions()
{(Chart (M, (x, y)),
  Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y))
   to Chart (M, (u, v))}
sage: uv_to_xy = xy_to_uv.inverse()
sage: TM.transitions()  # random (dictionary output)
{(Chart (M, (u, v)),
  Chart (M, (x, y))): Change of coordinates from Chart (M, (u, v))
  to Chart (M, (x, y)),
 (Chart (M, (x, y)),
  Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y))
   to Chart (M, (u, v))}
sage: c_rs.<r,s> = M.chart()
sage: uv_to_rs = c_uv.transition_map(c_rs, [-u+2*v, 3*u-v])
sage: TM.transitions()  # random (dictionary output)
{(Chart (M, (u, v)),
  Chart (M, (r, s))): Change of coordinates from Chart (M, (u, v))
   to Chart (M, (r, s)),
 (Chart (M, (u, v)),
  Chart (M, (x, y))): Change of coordinates from Chart (M, (u, v))
  to Chart (M, (x, y)),
 (Chart (M, (x, y)),
  Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y))
   to Chart (M, (u, v))}
sage: xy_to_rs = uv_to_rs * xy_to_uv
sage: TM.transitions()  # random (dictionary output)
{(Chart (M, (u, v)),
  Chart (M, (r, s))): Change of coordinates from Chart (M, (u, v))
   to Chart (M, (r, s)),
 (Chart (M, (u, v)),
  Chart (M, (x, y))): Change of coordinates from Chart (M, (u, v))
  to Chart (M, (x, y)),
 (Chart (M, (x, y)),
  Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y))
   to Chart (M, (u, v)),
 (Chart (M, (x, y)),
  Chart (M, (r, s))): Change of coordinates from Chart (M, (x, y))
   to Chart (M, (r, s))}

trivialization(coordinates='', names=None, calc_method=None)

Return a trivialization of self in terms of a chart on the codomain of the destination map.

Note

Since a chart gives direct rise to a trivialization, this method is nothing but an invocation of chart() of the class TopologicalManifold.

INPUT:

• coordinates – (default: '' (empty string)) string defining the coordinate symbols, ranges and possible periodicities, see below

• names – (default: None) unused argument, except if coordinates is not provided; it must then be a tuple containing the coordinate symbols (this is guaranteed if the shortcut operator <,> is used)

• calc_method – (default: None) string defining the calculus method to be used on this chart; must be one of

  • 'SR': Sage’s default symbolic engine (Symbolic Ring)

  • 'sympy': SymPy

  • None: the current calculus method defined on the manifold is used (cf. set_calculus_method())

The coordinates declared in the string coordinates are separated by ' ' (whitespace) and each coordinate has at most four fields, separated by a colon (':'):

1. The coordinate symbol (a letter or a few letters).

2. (optional, only for manifolds over \(\RR\)) The interval \(I\) defining the coordinate range: if not provided, the coordinate is assumed to span all \(\RR\); otherwise \(I\) must be provided in the form (a,b) (or equivalently ]a,b[) The bounds a and b can be +/-Infinity, Inf, infinity, inf or oo. For singular coordinates, non-open intervals such as [a,b] and (a,b] (or equivalently ]a,b]) are allowed. Note that the interval declaration must not contain any space character.

3. (optional) Indicator of the periodic character of the coordinate, either as period=T, where T is the period, or, for manifolds over \(\RR\) only, as the keyword periodic (the value of the period is then deduced from the interval \(I\) declared in field 2; see the example below)

4. (optional) The LaTeX spelling of the coordinate; if not provided the coordinate symbol given in the first field will be used.

The order of fields 2 to 4 does not matter and each of them can be omitted. If it contains any LaTeX expression, the string coordinates must be declared with the prefix ‘r’ (for “raw”) to allow for a proper treatment of the backslash character (see 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 trivialization.

OUTPUT:

• the created chart, as an instance of Chart or one of its subclasses, like RealDiffChart for differentiable manifolds over \(\RR\).

EXAMPLES:

Chart on a 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: TM = M.tangent_bundle()
sage: X = TM.trivialization('x y'); X
Chart (M, (x, y))
sage: X[0]
x
sage: X[1]
y
sage: X[:]
(x, y)

vector_frame(*args, **kwargs)

Define a vector frame on domain, possibly with values in the tangent bundle of the ambient domain.

If the basis specified by the given symbol already exists, it is simply returned. If no argument is provided the vector field module’s default frame is returned.

Notice, that a vector frame automatically induces a local frame on the tensor bundle self. More precisely, if \(e: U \to \Phi^*TN\) is a vector frame on \(U \subset M\) with values in \(\Phi^*TN\) along the destination map

\[\Phi: M \longrightarrow N\]

then the map

\[p \mapsto \Big(\underbrace{e^*(p), \dots, e^*(p)}_{k\ \; \text{times}}, \underbrace{e(p), \dots, e(p)}_{l\ \; \text{times}}\Big) \in T^{(k,l)}_q N ,\]

with \(q=\Phi(p)\), defines a basis at each point \(p \in U\) and therefore gives rise to a local frame on \(\Phi^* T^{(k,l)}N\) on the domain \(U\).

See also

VectorFrame for complete documentation.

INPUT:

• symbol – (default: None) either a string, to be used as a common base for the symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual symbols of the vector fields; can be None only if from_frame is not None (see below)

• vector_fields – tuple or list of \(n\) linearly independent vector fields on domain (\(n\) being the dimension of domain) defining the vector frame; can be omitted if the vector frame is created from scratch or if from_frame is not None

• latex_symbol – (default: None) either a string, to be used as a common base for the LaTeX symbols of the vector fields constituting the vector frame, or a list/tuple of strings, representing the individual LaTeX symbols of the vector fields; if None, symbol is used in place of latex_symbol

• from_frame – (default: None) vector frame \(\tilde{e}\) on the codomain \(N\) of the destination map \(\Phi\); the returned frame \(e\) is then such that for all \(p \in U\), we have \(e(p) = \tilde{e}(\Phi(p))\)

• indices – (default: None; used only if symbol is a single string) tuple of strings representing the indices labelling the vector fields of the frame; if None, the indices will be generated as integers within the range declared on self

• latex_indices – (default: None) tuple of strings representing the indices for the LaTeX symbols of the vector fields; if None, indices is used instead

• symbol_dual – (default: None) same as symbol but for the dual coframe; if None, symbol must be a string and is used for the common base of the symbols of the elements of the dual coframe

• latex_symbol_dual – (default: None) same as latex_symbol but for the dual coframe

• domain – (default: None) domain on which the local frame is defined; if None is provided, the base space of self is assumed

OUTPUT:

• the vector frame corresponding to the above specifications; this is an instance of VectorFrame.

EXAMPLES:

Defining a local frame for the tangent bundle of a 3-dimensional manifold:

sage: M = Manifold(3, 'M')
sage: TM = M.tangent_bundle()
sage: e = TM.local_frame('e'); e
Vector frame (M, (e_0,e_1,e_2))
sage: e[0]
Vector field e_0 on the 3-dimensional differentiable manifold M

Specifying the domain of the vector frame:

sage: U = M.open_subset('U')
sage: f = TM.local_frame('f', domain=U); f
Vector frame (U, (f_0,f_1,f_2))
sage: f[0]
Vector field f_0 on the Open subset U of the 3-dimensional
 differentiable manifold M

See also

For more options, in particular for the choice of symbols and indices, see VectorFrame.