def __init__(self,
vector_field_module,
degree,
name=None,
latex_name=None):
r"""
Construct a differential form.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``DiffForm`, to fit with the category framework::
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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
....: intersection_name='W', restrictions1= x>0,
....: restrictions2= u+v>0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V)
sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
sage: A = M.diff_form_module(2)
sage: XM = M.vector_field_module()
sage: a = A.element_class(XM, 2, name='a'); a
2-form a on the 2-dimensional differentiable manifold M
sage: a[e_xy,0,1] = x+y
sage: a.add_comp_by_continuation(e_uv, W, c_uv)
sage: TestSuite(a).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.diff_form``::
sage: a1 = M.diff_form(2, name='a'); a1
2-form a on the 2-dimensional differentiable manifold M
sage: type(a1) == type(a)
True
sage: a1.parent() is a.parent()
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
TensorField.__init__(
self,
vector_field_module, (0, degree),
name=name,
latex_name=latex_name,
antisym=range(degree),
parent=vector_field_module.dual_exterior_power(degree))
self._init_derived() # initialization of derived quantities
def __init__(self, tensor, embedding, screen=None):
r"""
TESTS::
sage: M = Manifold(4, 'M', structure="Lorentzian")
sage: X.<t,x,y,z> = M.chart()
sage: S = Manifold(3, 'S', ambient=M, structure='degenerate_metric')
sage: X_S.<u,v,w> = S.chart()
sage: Phi = S.diff_map(M, {(X_S, X): [u, u, v, w]},
....: name='Phi', latex_name=r'\Phi');
sage: Phi_inv = M.diff_map(S, {(X, X_S): [x,y, z]}, name='Phi_inv',
....: latex_name=r'\Phi^{-1}');
sage: S.set_immersion(Phi, inverse=Phi_inv); S.declare_embedding()
sage: g = M.metric()
sage: g[0,0], g[1,1], g[2,2], g[3,3] = -1,1,1,1
sage: v = M.vector_field(); v[1] = 1
sage: S.set_transverse(rigging=v)
sage: xi = M.vector_field(); xi[0] = 1; xi[1] = 1
sage: U = M.vector_field(); U[2] = 1; V = M.vector_field(); V[3] = 1
sage: Sc = S.screen('Sc', (U,V), xi);
sage: T1 = M.tensor_field(1,1).along(Phi); T1[0,0] = 1
sage: from sage.manifolds.differentiable.degenerate_submanifold import TangentTensor
sage: T2 = TangentTensor(T1, Phi); T2
Tensor field of type (1,1) along the degenerate hypersurface S embedded in
4-dimensional differentiable manifold M with values on the 4-dimensional
Lorentzian manifold M
"""
if not isinstance(tensor, TensorField):
raise TypeError("the second argument must be a tensor field")
self._tensor = tensor
self._tensor.set_name(tensor._name, latex_name=tensor._latex_name)
self._embedding = embedding
try:
tensor = tensor.along(embedding)
except ValueError:
pass
if isinstance(tensor, TensorFieldParal):
TensorFieldParal.__init__(self, tensor._vmodule, tensor._tensor_type, name=tensor._name,
latex_name=tensor._latex_name, sym=tensor._sym, antisym=tensor._antisym)
else:
TensorField.__init__(self, tensor._vmodule, tensor._tensor_type, name=tensor._name,
latex_name=tensor._latex_name, sym=tensor._sym, antisym=tensor._antisym)
f = tensor._domain._ambient.default_frame().along(embedding)
self[f, :] = tensor[f, :]
frame = self._domain.adapted_frame(screen)
self.display(frame)
for i in self._domain._ambient.index_generator(tensor.tensor_rank()):
for j in range(len(i)):
if i[j]==self._domain._ambient._dim-self._domain._sindex-1:
self[frame, i] = 0
def __init__(self, vector_field_module, degree, name=None, latex_name=None):
r"""
Construct a differential form.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``DiffForm`, to fit with the category framework::
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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
....: intersection_name='W', restrictions1= x>0,
....: restrictions2= u+v>0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V)
sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
sage: A = M.diff_form_module(2)
sage: XM = M.vector_field_module()
sage: a = A.element_class(XM, 2, name='a'); a
2-form a on the 2-dimensional differentiable manifold M
sage: a[e_xy,0,1] = x+y
sage: a.add_comp_by_continuation(e_uv, W, c_uv)
sage: TestSuite(a).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.diff_form``::
sage: a1 = M.diff_form(2, name='a'); a1
2-form a on the 2-dimensional differentiable manifold M
sage: type(a1) == type(a)
True
sage: a1.parent() is a.parent()
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
TensorField.__init__(self, vector_field_module, (0,degree), name=name,
latex_name=latex_name, antisym=range(degree),
parent=vector_field_module.dual_exterior_power(degree))
self._init_derived() # initialization of derived quantities
def __init__(self, vector_field_module, name=None, latex_name=None):
r"""
Construct a vector field with values on a non-parallelizable manifold.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``VectorField``, to fit with the category framework::
sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2
sage: U = M.open_subset('U') # complement of the North pole
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: V = M.open_subset('V') # complement of the South pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: M.declare_union(U,V) # S^2 is the union of U and V
sage: XM = M.vector_field_module()
sage: a = XM.element_class(XM, name='a'); a
Vector field a on the 2-dimensional differentiable manifold M
sage: a[c_xy.frame(),:] = [x, y]
sage: a[c_uv.frame(),:] = [-u, -v]
sage: TestSuite(a).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.vector_field``::
sage: a1 = M.vector_field(name='a'); a1
Vector field a on the 2-dimensional differentiable manifold M
sage: type(a1) == type(a)
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
TensorField.__init__(self,
vector_field_module, (1, 0),
name=name,
latex_name=latex_name)
# Initialization of derived quantities:
TensorField._init_derived(self)
# Initialization of list of quantities depending on self:
self._init_dependencies()
def __init__(self, vector_field_module, name=None, latex_name=None):
r"""
Construct a vector field with values on a non-parallelizable manifold.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``VectorField``, to fit with the category framework::
sage: M = Manifold(2, 'M') # the 2-dimensional sphere S^2
sage: U = M.open_subset('U') # complement of the North pole
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: V = M.open_subset('V') # complement of the South pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: M.declare_union(U,V) # S^2 is the union of U and V
sage: XM = M.vector_field_module()
sage: a = XM.element_class(XM, name='a'); a
Vector field a on the 2-dimensional differentiable manifold M
sage: a[c_xy.frame(),:] = [x, y]
sage: a[c_uv.frame(),:] = [-u, -v]
sage: TestSuite(a).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.vector_field``::
sage: a1 = M.vector_field(name='a'); a1
Vector field a on the 2-dimensional differentiable manifold M
sage: type(a1) == type(a)
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
TensorField.__init__(self, vector_field_module, (1,0), name=name,
latex_name=latex_name)
# Initialization of derived quantities:
TensorField._init_derived(self)
# Initialization of list of quantities depending on self:
self._init_dependencies()
def __init__(self, vector_field_module, name=None, latex_name=None,
is_identity=False):
r"""
Construct a field of tangent-space automorphisms on a
non-parallelizable manifold.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``AutomorphismField``, to fit with the category framework::
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: XM = M.vector_field_module()
sage: GL = XM.general_linear_group()
sage: a = GL.element_class(XM, name='a'); a
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold M
sage: a[c_xy.frame(), :] = [[1+x^2, 0], [0, 1+y^2]]
sage: a.add_comp_by_continuation(c_uv.frame(), U.intersection(V), c_uv)
sage: TestSuite(a).run(skip='_test_pickling')
Construction of the identity field::
sage: b = GL.element_class(XM, is_identity=True); b
Field of tangent-space identity maps on the 2-dimensional
differentiable manifold M
sage: TestSuite(b).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.automorphism_field``::
sage: a1 = M.automorphism_field(name='a'); a1
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold M
sage: type(a1) == type(a)
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
if is_identity:
if name is None:
name = 'Id'
if latex_name is None and name == 'Id':
latex_name = r'\mathrm{Id}'
TensorField.__init__(self, vector_field_module, (1,1), name=name,
latex_name=latex_name,
parent=vector_field_module.general_linear_group())
self._is_identity = is_identity
self._init_derived() # initialization of derived quantities
# Specific initializations for the field of identity maps:
if self._is_identity:
self._inverse = self
for dom in self._domain._subsets:
if dom.is_manifestly_parallelizable():
fmodule = dom.vector_field_module()
self._restrictions[dom] = fmodule.identity_map(name=name,
latex_name=latex_name)
def __init__(self, vector_field_module, name=None, latex_name=None,
is_identity=False):
r"""
Construct a field of tangent-space automorphisms on a
non-parallelizable manifold.
TESTS:
Construction via ``parent.element_class``, and not via a direct call
to ``AutomorphismField``, to fit with the category framework::
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: XM = M.vector_field_module()
sage: GL = XM.general_linear_group()
sage: a = GL.element_class(XM, name='a'); a
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold M
sage: a[c_xy.frame(), :] = [[1+x^2, 0], [0, 1+y^2]]
sage: a.add_comp_by_continuation(c_uv.frame(), U.intersection(V), c_uv)
sage: TestSuite(a).run(skip='_test_pickling')
Construction of the identity field::
sage: b = GL.element_class(XM, is_identity=True); b
Field of tangent-space identity maps on the 2-dimensional
differentiable manifold M
sage: TestSuite(b).run(skip='_test_pickling')
Construction with ``DifferentiableManifold.automorphism_field``::
sage: a1 = M.automorphism_field(name='a'); a1
Field of tangent-space automorphisms a on the 2-dimensional
differentiable manifold M
sage: type(a1) == type(a)
True
.. TODO::
Fix ``_test_pickling`` (in the superclass :class:`TensorField`).
"""
if is_identity:
if name is None:
name = 'Id'
if latex_name is None and name == 'Id':
latex_name = r'\mathrm{Id}'
TensorField.__init__(self, vector_field_module, (1,1), name=name,
latex_name=latex_name,
parent=vector_field_module.general_linear_group())
self._is_identity = is_identity
self._init_derived() # initialization of derived quantities
# Specific initializations for the field of identity maps:
if self._is_identity:
self._inverse = self
for dom in self._domain._subsets:
if dom.is_manifestly_parallelizable():
fmodule = dom.vector_field_module()
self._restrictions[dom] = fmodule.identity_map(name=name,
latex_name=latex_name)
