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Trac #30208: documentation improved
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Michael Jung committed Jul 23, 2020
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Showing 1 changed file with 106 additions and 40 deletions.
146 changes: 106 additions & 40 deletions src/sage/manifolds/differentiable/bundle_connection.py
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Original file line number Diff line number Diff line change
Expand Up @@ -30,18 +30,20 @@
- Michael Jung (2019) : initial version
"""
#******************************************************************************
# ******************************************************************************
# Copyright (C) 2019 Michael Jung <micjung@uni-potsdam.de>
#
# Distributed under the terms of the GNU General Public License (GPL)
# as published by the Free Software Foundation; either version 2 of
# the License, or (at your option) any later version.
# https://www.gnu.org/licenses/
#******************************************************************************
# ******************************************************************************

from sage.structure.sage_object import SageObject
from sage.rings.integer import Integer
from sage.manifolds.differentiable.vector_bundle import DifferentiableVectorBundle
from sage.manifolds.differentiable.vector_bundle import \
DifferentiableVectorBundle


class BundleConnection(SageObject):
r"""
Expand All @@ -59,8 +61,8 @@ class BundleConnection(SageObject):
EXAMPLES:
Define a bundle connection on a rank 2 vector bundle over some 3-dimensional
smooth manifold::
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()
Expand All @@ -70,40 +72,102 @@ class BundleConnection(SageObject):
Bundle connection nabla on the Differentiable real vector bundle E -> M
of rank 2 over the base space 3-dimensional differentiable manifold M
The bundle connection is specified by the connection 1-forms::
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
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, 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[e, 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
The other entries remain zero::
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
sage: nab[e, 2, 2].display()
connection (2,2) of bundle connection nabla w.r.t. Local frame
(E|_M, (e_1,e_2)) = 0
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[:] = 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 :meth:`set_connection_form` to specify the connection
1-forms::
sage: nab[:] = 0 # initialize with zero
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(1, 1)[:] = [x, x^2, x^3]
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::
From this, the connection 2-forms can be derived::
Notice that list assignments and :meth:`set_connection_form` delete
the connection 1-forms w.r.t. other frames for consistency reasons. To
avoid this behavior, :meth:`add_connection_form` must be used instead.
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(nab.curvature_form(i ,j).display())
....: 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 (1,2) 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 - x) dx/\dz +
(x^3*y*z - x^2*z^2 - y) dy/\dz
curvature (2,1) of bundle connection nabla w.r.t. Local frame
(E|_M, (e_1,e_2)) = 0
curvature (2,2) of bundle connection nabla w.r.t. Local frame
(E|_M, (e_1,e_2)) = 0
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
They certainly obey the structure equation::
The derived forms certainly obey the structure equations, see
:meth:`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(nab.curvature_form(i,j,e) == \
....: 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()))
....: sum(omega(k,j,e).wedge(omega(i,k,e))
....: for k in E.irange()))
sage: check # long time
[True, True, True, True]
"""

def __init__(self, vbundle, name, latex_name=None):
r"""
Construct a bundle connection.
Expand All @@ -112,7 +176,8 @@ def __init__(self, vbundle, name, latex_name=None):
sage: M = Manifold(3, 'M')
sage: E = M.vector_bundle(2, 'E')
sage: from sage.manifolds.differentiable.bundle_connection import BundleConnection
sage: from sage.manifolds.differentiable.bundle_connection \
....: import BundleConnection
sage: nab = BundleConnection(E, 'nabla', latex_name=r'\nabla')
sage: nab
Bundle connection nabla on the Differentiable real vector bundle
Expand All @@ -135,8 +200,7 @@ def __init__(self, vbundle, name, latex_name=None):
self._latex_name = self._name
else:
self._latex_name = latex_name
self._connection_forms = {} # dict. of connection coefficients, with
# the local frames as keys
self._connection_forms = {} # dict. of con. forms, with frames as keys
self._coefficients = self._connection_forms
# Initialization of derived quantities:
self._init_derived()
Expand Down Expand Up @@ -201,7 +265,7 @@ def _init_derived(self):
"""
self._curvature_forms = {} # dict. of dict. of curvature forms
# (key: local frame)
# (key: local frame)
self._hash = -1

def _del_derived(self):
Expand Down Expand Up @@ -357,7 +421,7 @@ def _new_forms(self, frame):
name += "connection " + self._name + " w.r.t. {}".format(frame)
latex_name = r"\omega^" + str(j) + r"_{\ \, " + str(i) + "}"
form = dom.diff_form(1, name=name, latex_name=latex_name)
forms_dict[(i,j)] = form
forms_dict[(i, j)] = form
return forms_dict

def connection_forms(self, frame=None):
Expand Down Expand Up @@ -400,8 +464,10 @@ def connection_forms(self, frame=None):
sage: nab[:] = 0 # initialize curvature forms
sage: nab.connection_forms() # random
{(1, 1): 1-form a on the 3-dimensional differentiable manifold M,
(1, 2): 1-form zero on the 3-dimensional differentiable manifold M,
(2, 1): 1-form zero on the 3-dimensional differentiable manifold M,
(1, 2): 1-form zero on the 3-dimensional differentiable
manifold M,
(2, 1): 1-form zero on the 3-dimensional differentiable
manifold M,
(2, 2): 1-form b on the 3-dimensional differentiable manifold M}
"""
Expand Down Expand Up @@ -462,7 +528,7 @@ def connection_form(self, i, j, frame=None):
(E|_M, (e_0,e_1)) = y^2 dx + y dy
"""
return self.connection_forms(frame)[(i,j)]
return self.connection_forms(frame)[(i, j)]

def add_connection_form(self, i, j, frame=None):
r"""
Expand Down Expand Up @@ -490,7 +556,7 @@ def add_connection_form(self, i, j, frame=None):
OUTPUT:
- connection 1-form `\omega^i_j` in the given frame, as an instance of
- connection 1-form `\omega^j_i` in the given frame, as an instance of
the class :class:`~sage.manifolds.differentiable.diff_form.DiffForm`;
if such connection 1-form did not exist previously, it is created.
See method :meth:`connection_forms` for the storage convention of the
Expand Down Expand Up @@ -544,8 +610,8 @@ def add_connection_form(self, i, j, frame=None):
raise ValueError("the {} is not".format(frame) +
" a frame on the {}".format(self._base_space))
self._connection_forms[frame] = self._new_forms(frame)
self._del_derived() # deletes the derived quantities
return self._connection_forms[frame][(i,j)]
self._del_derived() # deletes the derived quantities
return self._connection_forms[frame][(i, j)]

def set_connection_form(self, i, j, frame=None):
r"""
Expand All @@ -568,7 +634,7 @@ def set_connection_form(self, i, j, frame=None):
OUTPUT:
- connection 1-form `\omega^i_j` in the given frame, as an instance of
- connection 1-form `\omega^j_i` in the given frame, as an instance of
the class :class:`~sage.manifolds.differentiable.diff_form.DiffForm`;
if such connection 1-form did not exist previously, it is created.
See method :meth:`connection_forms` for the storage convention of the
Expand Down Expand Up @@ -686,7 +752,7 @@ def curvature_form(self, i, j, frame=None):
frame.
The *curvature 2-forms* with respect to the frame `e` are the 2-forms
`\Omega^i_j` given by the formula
`\Omega^j_i` given by the formula
.. MATH::
Expand All @@ -695,14 +761,14 @@ def curvature_form(self, i, j, frame=None):
INPUT:
- ``i``, ``j`` -- indices identifying the 2-form `\Omega^i_j`
- ``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^i_j`, as an instance of
- the 2-form `\Omega^j_i`, as an instance of
:class:`~sage.manifolds.differentiable.diff_form.DiffForm`
EXAMPLES::
Expand All @@ -729,14 +795,14 @@ def curvature_form(self, i, j, frame=None):
if frame not in self._curvature_forms:
self._curvature_forms[frame] = {}
if (i, j) not in self._curvature_forms[frame]:
name = "curvature ({},{}) of bundle connection ".format(i,j) + \
name = "curvature ({},{}) of bundle connection ".format(i, j) + \
self._name + " w.r.t. {}".format(frame)
latex_name = r"\Omega^" + str(i) + r"_{\ \, " + \
str(j) + "}"
str(j) + "}"
omega = self.connection_form
curv_form = omega(i, j, frame).exterior_derivative() + \
sum(omega(k, j, frame).wedge(omega(i, k, frame))
for k in self._vbundle.irange())
curv_form = omega(i, j, frame).exterior_derivative()
curv_form += sum(omega(k, j, frame).wedge(omega(i, k, frame))
for k in self._vbundle.irange())
curv_form.set_name(name=name, latex_name=latex_name)
self._curvature_forms[frame][(i, j)] = curv_form
return self._curvature_forms[frame][(i, j)]
Expand Down

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