Bundle Connections

Let \(E \to M\) be a smooth vector bundle of rank \(n\) over a smooth manifold \(M\) and over a non-discrete topological field \(K\) (typically \(K=\RR\) or \(K=\CC\)). A bundle connection on this vector bundle is a \(K\)-linear map

\[\nabla : C^\infty(M;E) \to C^\infty(M;E \otimes T^*M)\]

such that the Leibniz rule applies for each scalar field \(f \in C^\infty(M)\) and section \(s \in C^\infty(M;E)\):

\[\nabla(f \, s) = f \cdot \nabla s + s \otimes \mathrm{d}f .\]

If \(e\) is a local frame on \(E\), we have

\[\nabla e_i = \sum^n_{j=1} e_j \otimes \omega^j_i ,\]

and the corresponding \(n \times n\)-matrix \((\omega^j_i)_{i,j}\) consisting of one forms is called connection matrix of \(\nabla\) with respect to \(e\).

AUTHORS:

  • Michael Jung (2019) : initial version

class sage.manifolds.differentiable.bundle_connection.BundleConnection(vbundle, name, latex_name=None)

Bases: sage.structure.sage_object.SageObject

An instance of this class represents a bundle connection \(\nabla\) on a smooth vector bundle \(E \to M\).

INPUT:

  • vbundle – the vector bundle on which the connection is defined (must be an instance of class DifferentiableVectorBundle)

  • name – name given to the bundle connection

  • latex_name – (default: None) LaTeX symbol to denote the bundle connection; if None, it is set to name

EXAMPLES:

Define a bundle connection on a rank 2 vector bundle over some 3-dimensional smooth manifold:

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'); nab
Bundle connection nabla on the Differentiable real vector bundle E -> M
 of rank 2 over the base space 3-dimensional differentiable manifold M

First, let us initialize all connection 1-forms w.r.t. the frame e to zero:

sage: nab[e, :] = [[0, 0], [0, 0]]

This line can be shortened by the following:

sage: nab[e, :] = 0  # initialize to zero

The connection 1-forms are now initialized being differential 1-forms:

sage: nab[e, 1, 1].parent()
Free module Omega^1(M) of 1-forms on the 3-dimensional differentiable
 manifold M
sage: nab[e, 1, 1].display()
connection (1,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_1,e_2)) = 0

Now, we want to specify some non-zero entries:

sage: nab[e, 1, 2][:] = [x*z, y*z, z^2]
sage: nab[e, 2, 1][:] = [x, x^2, x^3]
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

Notice, when we omit the frame, the default frame of the vector bundle is assumed (in this case e):

sage: nab[2, 2].display()
connection (2,2) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_1,e_2)) = 0

The same holds for the assignment:

sage: nab[1, 2] = 0
sage: nab[e, 1, 2].display()
connection (1,2) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_1,e_2)) = 0

We can also use set_connection_form() to specify the connection 1-forms:

sage: nab[:] = 0  # re-initialize to zero
sage: nab.set_connection_form(1, 2)[:] = [x*z, y*z, z^2]
sage: nab.set_connection_form(2, 1)[:] = [x, x^2, x^3]
sage: nab[1, 2].display()
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
sage: nab[2, 1].display()
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

Note

Notice that list assignments and set_connection_form() delete the connection 1-forms w.r.t. other frames for consistency reasons. To avoid this behavior, add_connection_form() must be used instead.

By definition, a bundle connection acts on vector fields and sections:

sage: v = M.vector_field((x^2,y^2,z^2), name='v'); v.display()
v = x^2 d/dx + y^2 d/dy + z^2 d/dz
sage: s = E.section((x-y^2, -z), name='s'); s.display()
s = (-y^2 + x) e_1 - z e_2
sage: nab_vs = nab(v, s); nab_vs
Section nabla_v(s) on the 3-dimensional differentiable manifold M with
 values in the real vector bundle E of rank 2
sage: nab_vs.display()
nabla_v(s) = (-x^3*z^3 - 2*y^3 + x^2 - (x^2*y^2 + x^3)*z) e_1 +
 (-(y^2 - x)*z^4 - (x^3*y^2 + y^5 - x^4 - x*y^3)*z - z^2) e_2

The bundle connection action certainly obeys the defining formula for the connection 1-forms:

sage: vframe = X.frame()
sage: all(nab(vframe[k], e[i]) == sum(nab[e, i, j](vframe[k])*e[j]
....:                                 for j in E.irange())
....:     for i in E.irange() for k in M.irange())
True

The connection 1-forms are computed automatically for different frames:

sage: f = E.local_frame('f', ((1+x^2)*e[1], e[1]-e[2]))
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 -
  (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 -
  (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

The new connection 1-forms obey the defining formula, too:

sage: all(nab(vframe[k], f[i]) == sum(nab[f, i, j](vframe[k])*f[j]
....:                                 for j in E.irange())
....:     for i in E.irange() for k in M.irange())
True

After the connection has been specified, the curvature 2-forms can be derived:

sage: Omega = nab.curvature_form
sage: for i in E.irange():
....:     for j in E.irange():
....:         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
 (E|_M, (e_1,e_2)) = -x dx/\dz - y dy/\dz
 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
 (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

The derived forms certainly obey the structure equations, see curvature_form() for details:

sage: omega = nab.connection_form
sage: check = []
sage: for i in E.irange():  # long time
....:     for j in E.irange():
....:         check.append(Omega(i,j,e) == \
....:                       omega(i,j,e).exterior_derivative() + \
....:         sum(omega(k,j,e).wedge(omega(i,k,e))
....:             for k in E.irange()))
sage: check  # long time
[True, True, True, True]
add_connection_form(i, j, form=None, frame=None)

Return the connection form \(\omega^j_i\) in a given frame for assignment.

See method connection_forms() for details about the definition of the connection forms.

To delete the connection forms in other frames, use the method set_connection_form() instead.

INPUT:

  • i, j – indices identifying the 1-form \(\omega^j_i\)

  • frame – (default: None) local frame in which the connection 1-form is defined; if None, the default frame of the vector bundle is assumed.

Warning

If the connection has already forms in other frames, it is the user’s responsibility to make sure that the 1-forms to be added are consistent with them.

OUTPUT:

  • connection 1-form \(\omega^j_i\) in the given frame, as an instance of the class DiffForm; if such connection 1-form did not exist previously, it is created. See method connection_forms() for the storage convention of the connection 1-forms.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = 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')
sage: nab.add_connection_form(0, 1, frame=e)[:] = [x^2, x]
sage: nab[e, 0, 1].display()
connection (0,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_0,e_1)) = x^2 dx + x dy

Since e is the vector bundle’s default local frame, its mention may be omitted:

sage: nab.add_connection_form(1, 0)[:] = [y^2, y]
sage: nab[1, 0].display()
connection (1,0) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_0,e_1)) = y^2 dx + y dy

Adding connection 1-forms w.r.t. to another local frame:

sage: f = E.local_frame('f')
sage: nab.add_connection_form(1, 1, frame=f)[:] = [x, y]
sage: nab[f, 1, 1].display()
connection (1,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (f_0,f_1)) = x dx + y dy

The forms w.r.t. the frame e have been kept:

sage: nab[e, 0, 1].display()
connection (0,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_0,e_1)) = x^2 dx + x dy

To delete them, use the method set_connection_form() instead.

connection_form(i, j, frame=None)

Return the connection 1-form corresponding to the given index and local frame.

See also

Consult connection_forms() for detailed information.

INPUT:

  • i, j – indices identifying the 1-form \(\omega^j_i\)

  • frame – (default: None) local frame relative to which the connection 1-forms are defined; if None, the default frame of the vector bundle’s corresponding section module is assumed.

OUTPUT:

  • the 1-form \(\omega^j_i\), as an instance of DiffForm

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = 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')
sage: nab.set_connection_form(0, 1)[:] = [x^2, x]
sage: nab.set_connection_form(1, 0)[:] = [y^2, y]
sage: nab.connection_form(0, 1).display()
connection (0,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_0,e_1)) = x^2 dx + x dy
sage: nab.connection_form(1, 0).display()
connection (1,0) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_0,e_1)) = y^2 dx + y dy
connection_forms(frame=None)

Return the connection forms relative to the given frame.

If \(e\) is a local frame on \(E\), we have

\[\nabla e_i = \sum^n_{j=1} e_j \otimes \omega^j_i ,\]

and the corresponding \(n \times n\)-matrix \((\omega^j_i)_{i,j}\) consisting of one forms is called connection matrix of \(\nabla\) with respect to \(e\).

If the connection coefficients are not known already, they are computed from the above formula.

INPUT:

  • frame – (default: None) local frame relative to which the connection forms are required; if none is provided, the vector bundle’s default frame is assumed

OUTPUT:

  • connection forms relative to the frame frame, as a dictionary with tuples \((i, j)\) as key and one forms as instances of diff_form as value representing the matrix entries.

EXAMPLES:

Connection forms of a bundle connection on a rank 2 vector bundle over a 3-dimensional manifold:

sage: M = Manifold(3, 'M', start_index=1)
sage: c_xyz.<x,y,z> = M.chart()
sage: E = M.vector_bundle(2, 'E')
sage: e = E.local_frame('e')
sage: nab = E.bundle_connection('nabla', r'\nabla')
sage: nab[:] = 0  # initialize curvature forms
sage: forms = nab.connection_forms()
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 3-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 3-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 3-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 3-dimensional differentiable
 manifold M]
curvature_form(i, j, frame=None)

Return the curvature 2-form corresponding to the given index and local frame.

The curvature 2-forms with respect to the frame \(e\) are the 2-forms \(\Omega^j_i\) given by the formula

\[\Omega^j_i = \mathrm{d} \omega^j_i + \sum^n_{k=1} \omega^j_k \wedge \omega^k_i\]

INPUT:

  • i, j – indices identifying the 2-form \(\Omega^j_i\)

  • frame – (default: None) local frame relative to which the curvature 2-forms are defined; if None, the default frame of the vector bundle is assumed.

OUTPUT:

  • the 2-form \(\Omega^j_i\), as an instance of DiffForm

EXAMPLES:

sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart()
sage: E = M.vector_bundle(1, 'E')
sage: nab = E.bundle_connection('nabla', latex_name=r'\nabla')
sage: e = E.local_frame('e')
sage: nab.set_connection_form(1, 1)[:] = [x^2, x]
sage: curv = nab.curvature_form(1, 1); curv
2-form curvature (1,1) of bundle connection nabla w.r.t. Local
 frame (E|_M, (e_1)) on the 2-dimensional differentiable manifold M
sage: curv.display()
curvature (1,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_1)) = dx/\dy
del_other_forms(frame=None)

Delete all the connection forms but those corresponding to frame.

INPUT:

  • frame – (default: None) local frame, the connection forms w.r.t. which are to be kept; if None, the default frame of the vector bundle is assumed.

EXAMPLES:

We first create two sets of connection forms:

sage: M = Manifold(2, 'M', start_index=1)
sage: X.<x,y> = M.chart()
sage: E = M.vector_bundle(2, 'E')
sage: nab = E.bundle_connection('nabla', latex_name=r'\nabla')
sage: e = E.local_frame('e')
sage: nab.set_connection_form(1, 1, frame=e)[:] = [x^2, x]
sage: f = E.local_frame('f')
sage: nab.add_connection_form(1, 1, frame=f)[:] = [y^2, y]
sage: nab[e, 1, 1].display()
connection (1,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_1,e_2)) = x^2 dx + x dy
sage: nab[f, 1, 1].display()
connection (1,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (f_1,f_2)) = y^2 dx + y dy

Let us delete the connection forms w.r.t. all frames except for frame e:

sage: nab.del_other_forms(e)
sage: nab[e, 1, 1].display()
connection (1,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_1,e_2)) = x^2 dx + x dy

The connection forms w.r.t. frame e have indeed been deleted:

sage: nab[f, :]
Traceback (most recent call last):
...
ValueError: no basis could be found for computing the components in
 the Local frame (E|_M, (e_1,e_2))
display(frame=None, vector_frame=None, chart=None, only_nonzero=True)

Display all the connection 1-forms w.r.t. to a given local frame, one per line.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

INPUT:

  • frame – (default: None) local frame of the vector bundle relative to which the connection 1-forms are defined; if None, the default frame of the bundle is used

  • vector_frame – (default: None) vector frame of the manifold relative to which the connection 1-forms should be displayed; if None, the default frame of the local frame’s domain is used

  • chart – (default: None) chart specifying the coordinate expression of the connection 1-forms; if None, the default chart of the domain of frame is used

  • only_nonzero – (default: True) boolean; if True, only nonzero connection coefficients are displayed

EXAMPLES:

Set connection 1-forms:

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
sage: nab[:] = 0
sage: nab[1, 1][:] = [x, y, z]
sage: nab[2, 2][:] = [x^2, y^2, z^2]

By default, only the nonzero connection coefficients are displayed:

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
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}
 \mathrm{d} x + y^{2} \mathrm{d} y + z^{2} \mathrm{d} z \end{array}

By default, the displayed connection 1-forms are those w.r.t. the default frame of the vector bundle. The aforementioned is therefore equivalent to:

sage: nab.display(frame=E.default_frame())
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

Moreover, the connection 1-forms are displayed w.r.t. the default vector frame on the local frame’s domain, i.e.:

sage: domain = e.domain()
sage: nab.display(vector_frame=domain.default_frame())
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

By default, the parameter only_nonzero is set to True. Otherwise, the connection 1-forms being zero are shown as well:

sage: nab.display(only_nonzero=False)
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 (1,2) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_1,e_2)) = 0
connection (2,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_1,e_2)) = 0
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
set_connection_form(i, j, form=None, frame=None)

Return the connection form \(\omega^j_i\) in a given frame for assignment.

See method connection_forms() for details about the definition of the connection forms.

The connection forms with respect to other frames are deleted, in order to avoid any inconsistency. To keep them, use the method add_connection_form() instead.

INPUT:

  • i, j – indices identifying the 1-form \(\omega^j_i\)

  • frame – (default: None) local frame in which the connection 1-form is defined; if None, the default frame of the vector bundle is assumed.

OUTPUT:

  • connection 1-form \(\omega^j_i\) in the given frame, as an instance of the class DiffForm; if such connection 1-form did not exist previously, it is created. See method connection_forms() for the storage convention of the connection 1-forms.

EXAMPLES:

Setting the connection forms of a bundle connection w.r.t. some local frame:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = 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')
sage: nab.set_connection_form(0, 1)[:] = [x^2, x]
sage: nab[0, 1].display()
connection (0,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_0,e_1)) = x^2 dx + x dy

Since e is the vector bundle’s default local frame, its mention may be omitted:

sage: nab.set_connection_form(1, 0)[:] = [y^2, y]
sage: nab[1, 0].display()
connection (1,0) of bundle connection nabla w.r.t. Local frame
 (E|_M, (e_0,e_1)) = y^2 dx + y dy

Setting connection 1-forms w.r.t. to another local frame:

sage: f = E.local_frame('f')
sage: nab.set_connection_form(1, 1, frame=f)[:] = [x, y]
sage: nab[f, 1, 1].display()
connection (1,1) of bundle connection nabla w.r.t. Local frame
 (E|_M, (f_0,f_1)) = x dx + y dy

The forms w.r.t. the frame e have been deleted:

sage: nab[e, 0, 1].display()
Traceback (most recent call last):
...
ValueError: no basis could be found for computing the components in
 the Local frame (E|_M, (f_0,f_1))

To keep them, use the method add_connection_form() instead.

vector_bundle()

Return the vector bundle on which the bundle connection is defined.

OUTPUT:

EXAMPLES:

sage: M = Manifold(3, 'M', start_index=1)
sage: c_xyz.<x,y,z> = M.chart()
sage: E = M.vector_bundle(2, 'E')
sage: nab = E.bundle_connection('nabla', r'\nabla')
sage: nab.vector_bundle()
Differentiable real vector bundle E -> M of rank 2 over the base
 space 3-dimensional differentiable manifold M