Trivializations#

The class Trivialization implements trivializations on vector bundles. The corresponding transition maps between two trivializations are represented by TransitionMap.

AUTHORS:

Michael Jung (2019) : initial version

class sage.manifolds.trivialization.TransitionMap(triv1, triv2, transf, compute_inverse=True)#

Bases: SageObject

Transition map between two trivializations.

Given a vector bundle \(\pi : E \to M\) of class \(C^k\) and rank \(n\) over the field \(K\), and two trivializations \(\varphi_U : \pi^{-1}(U) \to U \times K^n\) and \(\varphi_V : \pi^{-1}(V) \to V \times K^n\), the transition map from \(\varphi_U\) to \(\varphi_V\) is given by the composition

\[\varphi_V \circ \varphi_U^{-1} : U \cap V \times K^n \to U \cap V \times K^n .\]

This composition is of the form

\[(p, v) \mapsto (p, g(p)v),\]

where \(p \mapsto g(p)\) is a \(C^k\) family of invertible \(n\times n\) matrices.

INPUT:

triv1 – trivialization 1
triv2 – trivialization 2
transf – the transformation between both trivializations in form of a matrix of scalar fields (ScalarField) or coordinate functions (ChartFunction), or a bundle automorphism (FreeModuleAutomorphism)
compute_inverse – (default: True) determines whether the inverse shall be computed or not

EXAMPLES:

Transition map of two trivializations on a real rank 2 vector bundle of the 2-sphere:

sage: S2 = Manifold(2, 'S^2', structure='top')
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E')
sage: phi_U = E.trivialization('phi_U', domain=U)
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: phi_U_to_phi_V = phi_U.transition_map(phi_V, [[0,1],[1,0]])
sage: phi_U_to_phi_V
Transition map from Trivialization (phi_U, E|_U) to Trivialization
 (phi_V, E|_V)

automorphism()#

Return the automorphism connecting both trivializations.

The family of matrices \(p \mapsto g(p)\) given by the transition map induce a bundle automorphism

\[\varphi_U^{-1} \circ \varphi_V : \pi^{-1}(U \cap V) \to \pi^{-1}(U \cap V)\]

correlating the local frames induced by the trivializations in the following way:

\[(\varphi_U^{-1} \circ \varphi_V) (\varphi_V^*e_i) = \varphi_U^*e_i .\]

Then, for each point \(p \in M\), the matrix \(g(p)\) is the representation of the induced automorphism on the fiber \(E_p=\pi^{-1}(p)\) in the basis \(\left( (\varphi_V^*e_i)(p)\right)_{i=1,\dots,n}\).

EXAMPLES:

sage: S2 = Manifold(2, 'S^2', structure='top')
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E')
sage: phi_U = E.trivialization('phi_U', latex_name=r'\varphi_U',
....:                          domain=U); phi_U
Trivialization (phi_U, E|_U)
sage: phi_V = E.trivialization('phi_V', latex_name=r'\varphi_V',
....:                          domain=V); phi_V
Trivialization (phi_V, E|_V)
sage: phi_U_to_phi_V = phi_U.transition_map(phi_V, [[0,1],[1,0]])
sage: aut = phi_U_to_phi_V.automorphism(); aut
Automorphism phi_U^(-1)*phi_V of the Free module C^0(W;E) of
 sections on the Open subset W of the 2-dimensional topological
 manifold S^2 with values in the real vector bundle E of rank 2
sage: aut.display(phi_U.frame().restrict(W))
phi_U^(-1)*phi_V = (phi_U^*e_1)⊗(phi_U^*e^2) +
 (phi_U^*e_2)⊗(phi_U^*e^1)

det()#

Return the determinant of self.

OUTPUT:

An instance of ScalarField.

EXAMPLES:

sage: S2 = Manifold(2, 'S^2', structure='top')
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E')
sage: phi_U = E.trivialization('phi_U', latex_name=r'\varphi_U',
....:                          domain=U); phi_U
Trivialization (phi_U, E|_U)
sage: phi_V = E.trivialization('phi_V', latex_name=r'\varphi_V',
....:                          domain=V); phi_V
Trivialization (phi_V, E|_V)
sage: phi_U_to_phi_V = phi_U.transition_map(phi_V, [[0,1],[1,0]])
sage: det = phi_U_to_phi_V.det(); det
Scalar field det(phi_U^(-1)*phi_V) on the Open subset W of the
 2-dimensional topological manifold S^2
sage: det.display()
det(phi_U^(-1)*phi_V): W → ℝ
     (x, y) ↦ -1
     (u, v) ↦ -1

inverse()#

Return the inverse transition map.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='top')
sage: X.<x,y> = M.chart()
sage: U = M.open_subset('U'); V = M.open_subset('V')
sage: XU = X.restrict(U); XV = X.restrict(U)
sage: W = U.intersection(V)
sage: XW = X.restrict(W)
sage: E = M.vector_bundle(2, 'E')
sage: phi_U = E.trivialization('phi_U', domain=U)
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: phi_U_to_phi_V = phi_U.transition_map(phi_V, [[1,1],[-1,1]],
....:                                       compute_inverse=False)
sage: phi_V_to_phi_U = phi_U_to_phi_V.inverse(); phi_V_to_phi_U
Transition map from Trivialization (phi_V, E|_V) to Trivialization (phi_U, E|_U)
sage: phi_V_to_phi_U.automorphism() == phi_U_to_phi_V.automorphism().inverse()
True

matrix()#

Return the matrix representation the transition map.

EXAMPLES:

Local trivializations on a real rank 2 vector bundle over the 2-sphere:

sage: S2 = Manifold(2, 'S^2', structure='top')
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E')
sage: phi_U = E.trivialization('phi_U', latex_name=r'\varphi_U',
....:                          domain=U); phi_U
Trivialization (phi_U, E|_U)
sage: phi_V = E.trivialization('phi_V', latex_name=r'\varphi_V',
....:                          domain=V); phi_V
Trivialization (phi_V, E|_V)

The input is coerced into a bundle automorphism. From there, the matrix can be recovered:

sage: phi_U_to_phi_V = phi_U.transition_map(phi_V, [[0,1],[1,0]])
sage: matrix = phi_U_to_phi_V.matrix(); matrix
[Scalar field zero on the Open subset W of the 2-dimensional
 topological manifold S^2      Scalar field 1 on the Open subset
 W of the 2-dimensional topological manifold S^2]
 [     Scalar field 1 on the Open subset W of the 2-dimensional
 topological manifold S^2 Scalar field zero on the Open subset W of
 the 2-dimensional topological manifold S^2]

Let us check the matrix components:

sage: matrix[0,0].display()
zero: W → ℝ
    (x, y) ↦ 0
    (u, v) ↦ 0
sage: matrix[0,1].display()
1: W → ℝ
 (x, y) ↦ 1
 (u, v) ↦ 1
sage: matrix[1,0].display()
1: W → ℝ
 (x, y) ↦ 1
 (u, v) ↦ 1
sage: matrix[1,1].display()
zero: W → ℝ
    (x, y) ↦ 0
    (u, v) ↦ 0

class sage.manifolds.trivialization.Trivialization(vector_bundle, name, domain, latex_name=None)#

Bases: UniqueRepresentation, SageObject

A local trivialization of a given vector bundle.

Let \(\pi:E \to M\) be a vector bundle of rank \(n\) and class \(C^k\) over the field \(K\) (see TopologicalVectorBundle or DifferentiableVectorBundle). A local trivialization over an open subset \(U \subset M\) is a \(C^k\)-diffeomorphism \(\varphi: \pi^{-1}(U) \to U \times K^n\) such that \(\pi \circ \varphi^{-1}=\mathrm{pr}_1\) and \(v \mapsto \varphi^{-1}(q,v)\) is a linear isomorphism for any \(q \in U\).

Note

Notice that frames and trivializations are equivalent concepts (for further details see LocalFrame). However, in order to facilitate applications and being consistent with the implementations of charts, trivializations are introduced separately.

EXAMPLES:

Local trivializations on a real rank 2 vector bundle over the 2-sphere:

sage: S2 = Manifold(2, 'S^2', structure='top')
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E')
sage: phi_U = E.trivialization('phi_U', latex_name=r'\varphi_U',
....:                          domain=U); phi_U
Trivialization (phi_U, E|_U)
sage: phi_V = E.trivialization('phi_V', latex_name=r'\varphi_V',
....:                          domain=V); phi_V
Trivialization (phi_V, E|_V)
sage: phi_U_to_phi_V = phi_U.transition_map(phi_V, [[0,1],[1,0]]); phi_U_to_phi_V
Transition map from Trivialization (phi_U, E|_U) to Trivialization
 (phi_V, E|_V)

The LaTeX output gives the following:

sage: latex(phi_U)
\varphi_U : E |_{U} \to U \times \Bold{R}^2
sage: latex(phi_V)
\varphi_V : E |_{V} \to V \times \Bold{R}^2

The trivializations are part of the vector bundle atlas:

sage: E.atlas()
[Trivialization (phi_U, E|_U), Trivialization (phi_V, E|_V)]

Each trivialization induces a local trivialization frame:

sage: fU = phi_U.frame(); fU
Trivialization frame (E|_U, ((phi_U^*e_1),(phi_U^*e_2)))
sage: fV = phi_V.frame(); fV
Trivialization frame (E|_V, ((phi_V^*e_1),(phi_V^*e_2)))

and the transition map connects these two frames via a bundle automorphism:

sage: aut = phi_U_to_phi_V.automorphism(); aut
Automorphism phi_U^(-1)*phi_V of the Free module C^0(W;E) of sections on
 the Open subset W of the 2-dimensional topological manifold S^2 with
 values in the real vector bundle E of rank 2
sage: aut.display(fU.restrict(W))
phi_U^(-1)*phi_V = (phi_U^*e_1)⊗(phi_U^*e^2) + (phi_U^*e_2)⊗(phi_U^*e^1)
sage: aut.display(fV.restrict(W))
phi_U^(-1)*phi_V = (phi_V^*e_1)⊗(phi_V^*e^2) + (phi_V^*e_2)⊗(phi_V^*e^1)

The automorphisms are listed in the frame changes of the vector bundle:

sage: E.changes_of_frame() # random
{(Local frame (E|_W, ((phi_U^*e_1),(phi_U^*e_2))),
 Local frame (E|_W, ((phi_V^*e_1),(phi_V^*e_2)))): Automorphism
 phi_U^(-1)*phi_V^(-1) of the Free module C^0(W;E) of sections on the
 Open subset W of the 2-dimensional topological manifold S^2 with values
 in the real vector bundle E of rank 2,
 (Local frame (E|_W, ((phi_V^*e_1),(phi_V^*e_2))),
 Local frame (E|_W, ((phi_U^*e_1),(phi_U^*e_2)))): Automorphism
 phi_U^(-1)*phi_V of the Free module C^0(W;E) of sections on the Open
 subset W of the 2-dimensional topological manifold S^2 with values in
 the real vector bundle E of rank 2}

Let us check the components of fU with respect to the frame fV:

sage: fU[0].comp(fV.restrict(W))[:]
[0, 1]
sage: fU[1].comp(fV.restrict(W))[:]
[1, 0]

base_space()#

Return the manifold on which the trivialization is defined.

EXAMPLES:

sage: M = Manifold(2, 'M', structure='top')
sage: U = M.open_subset('U')
sage: E = M.vector_bundle(2, 'E')
sage: phi = E.trivialization('phi', domain=U)
sage: phi.base_space()
2-dimensional topological manifold M

coframe()#

Return the standard coframe induced by self.