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 bundlename
– string representation given to the total spacebase_space
– the base space (differentiable manifold) \(M\) over which the vector bundle is definedfield
– field \(K\) which gives the fibers the structure of a vector space over \(K\); allowed values are'real'
or an object of typeRealField
(e.g.,RR
) for a vector bundle over \(\RR\)'complex'
or an object of typeComplexField
(e.g.,CC
) for a vector bundle over \(\CC\)an object in the category of topological fields (see
Fields
andTopologicalSpaces
) for other types of topological fields
latex_name
– (default:None
) LaTeX representation given to the total spacecategory
– (default:None
) to specify the category; ifNone
,VectorBundles(base_space, c_field).Differentiable()
is assumed (see the categoryVectorBundles
)
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 ofBundleConnection
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 trac ticket #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 definedname
– (default:None
) string representation given to the characteristic cohomology class; ifNone
the default algebra representation or predefined name is usedlatex_name
– (default:None
) LaTeX name given to the characteristic class; ifNone
the value ofname
is usedclass_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 ofDifferentiableManifold
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
\[t:\ \underbrace{T_q^*N\times\cdots\times T_q^*N}_{k\ \; \mbox{times}} \times \underbrace{T_q N\times\cdots\times T_q N}_{l\ \; \mbox{times}} \longrightarrow K\](\(k\) 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 definedk
– the contravariant rank of the corresponding tensor bundlel
– the covariant rank of the corresponding tensor bundledest_map
– (default:None
) destination map \(\Phi:\ M \rightarrow N\) (type:DiffMap
); ifNone
, 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 classTopologicalManifold
.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
seelocal_frame()
.Note
Since frames on
self
are directly induced by vector frames on the base space, this method directly invokeschange_of_frame()
of the classDifferentiableManifold
.INPUT:
frame1
– local frame 1frame2
– 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
seelocal_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
seelocal_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 ofself
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
seelocal_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
ifself
is known to be a trivial andFalse
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
\[p \mapsto \Big(\underbrace{e^*(p), \dots, e^*(p)}_{k\ \; \mbox{times}}, \underbrace{e(p), \dots, e(p)}_{l\ \; \mbox{times}}\Big) \in T^{(k,l)}_q N ,\]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 beNone
only iffrom_frame
is notNone
(see below)vector_fields
– tuple or list of \(n\) linearly independent vector fields ondomain
(\(n\) being the dimension ofdomain
) defining the vector frame; can be omitted if the vector frame is created from scratch or iffrom_frame
is notNone
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; ifNone
,symbol
is used in place oflatex_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 ifsymbol
is a single string) tuple of strings representing the indices labelling the vector fields of the frame; ifNone
, the indices will be generated as integers within the range declared onself
latex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the vector fields; ifNone
,indices
is used insteadsymbol_dual
– (default:None
) same assymbol
but for the dual coframe; ifNone
,symbol
must be a string and is used for the common base of the symbols of the elements of the dual coframelatex_symbol_dual
– (default:None
) same aslatex_symbol
but for the dual coframedomain
– (default:None
) domain on which the local frame is defined; ifNone
is provided, the base space ofself
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
ondomain
, namely a tensor field on the subsetdomain
of the base space.Note
This method directly invokes
tensor_field()
of the classDifferentiableManifold
.INPUT:
comp
– (optional) either the components of the tensor field with respect to the vector frame specified by the argumentframe
or a dictionary of components, the keys of which are vector frames or pairs(f, c)
wheref
is a vector frame andc
the chart in which the components are expressedframe
– (default:None
; unused ifcomp
is not given or is a dictionary) vector frame in which the components are given; ifNone
, the default vector frame ofself
is assumedchart
– (default:None
; unused ifcomp
is not given or is a dictionary) coordinate chart in which the components are expressed; ifNone
, the default chart on the domain offrame
is assumeddomain
– (default:None
) domain of the section; ifNone
,self.base_space()
is assumedname
– (default:None
) name given to the tensor fieldlatex_name
– (default:None
) LaTeX symbol to denote the tensor field; ifNone
, the LaTeX symbol is set toname
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 conventionposition=0
for the first argument; for instance:sym = (0,1)
for a symmetry between the 1st and 2nd argumentssym = [(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 forsym
OUTPUT:
a
TensorField
(or if \(N\) is parallelizable, aTensorFieldParal
) 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
anddomain
: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, ofself
ondomain
.Note
This method directly invokes
tensor_field_module()
of the classDifferentiableManifold
.INPUT:
domain
– (default:None
) the domain of the corresponding section module; ifNone
, the base space is assumed
OUTPUT:
a
TensorFieldModule
(or if \(N\) is parallelizable, aTensorFieldFreeModule
) 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
seelocal_frame()
.Note
Since frames on
self
are directly induced by vector frames on the base space, this method directly invokesset_change_of_frame()
of the classDifferentiableManifold
.INPUT:
frame1
– frame 1, denoted \((e_i)\) belowframe2
– frame 2, denoted \((f_i)\) belowchange_of_frame
– instance of classFreeModuleAutomorphism
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 dictionaryself._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 ofself
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 classTopologicalManifold
.INPUT:
chart1
– chart 1chart2
– 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 classTopologicalManifold
.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 classTopologicalManifold
.INPUT:
coordinates
– (default:''
(empty string)) string defining the coordinate symbols, ranges and possible periodicities, see belownames
– (default:None
) unused argument, except ifcoordinates
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'
: SymPyNone
: 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 (':'
):The coordinate symbol (a letter or a few letters).
(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 boundsa
andb
can be+/-Infinity
,Inf
,infinity
,inf
oroo
. 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.(optional) Indicator of the periodic character of the coordinate, either as
period=T
, whereT
is the period, or, for manifolds over \(\RR\) only, as the keywordperiodic
(the value of the period is then deduced from the interval \(I\) declared in field 2; see the example below)(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 argumentcoordinates
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, likeRealDiffChart
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
\[p \mapsto \Big(\underbrace{e^*(p), \dots, e^*(p)}_{k\ \; \mbox{times}}, \underbrace{e(p), \dots, e(p)}_{l\ \; \mbox{times}}\Big) \in T^{(k,l)}_q N ,\]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 beNone
only iffrom_frame
is notNone
(see below)vector_fields
– tuple or list of \(n\) linearly independent vector fields ondomain
(\(n\) being the dimension ofdomain
) defining the vector frame; can be omitted if the vector frame is created from scratch or iffrom_frame
is notNone
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; ifNone
,symbol
is used in place oflatex_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 ifsymbol
is a single string) tuple of strings representing the indices labelling the vector fields of the frame; ifNone
, the indices will be generated as integers within the range declared onself
latex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the vector fields; ifNone
,indices
is used insteadsymbol_dual
– (default:None
) same assymbol
but for the dual coframe; ifNone
,symbol
must be a string and is used for the common base of the symbols of the elements of the dual coframelatex_symbol_dual
– (default:None
) same aslatex_symbol
but for the dual coframedomain
– (default:None
) domain on which the local frame is defined; ifNone
is provided, the base space ofself
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
.