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)[source]¶
- Bases: - TopologicalVectorBundle- An instance of this class represents a differentiable vector bundle \(E \to M\) - INPUT: - rank– positive integer; rank of the vector bundle
- name– string representation given to the total space
- base_space– the base space (differentiable manifold) \(M\) over which the vector bundle is defined
- field– field \(K\) which gives the fibers the structure of a vector space over \(K\); allowed values are- 'real'or an object of type- RealField(e.g.,- RR) for a vector bundle over \(\RR\)
- 'complex'or an object of type- ComplexField(e.g.,- CC) for a vector bundle over \(\CC\)
- an object in the category of topological fields (see - Fieldsand- TopologicalSpaces) for other types of topological fields
 
- latex_name– (default:- None) LaTeX representation given to the total space
- category– (default:- None) to specify the category; if- None,- VectorBundles(base_space, c_field).Differentiable()is assumed (see the category- VectorBundles)
 - 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 - >>> from sage.all import * >>> M = Manifold(Integer(3), 'M') >>> E = M.vector_bundle(Integer(2), 'E', field='complex'); E Differentiable complex vector bundle E -> M of rank 2 over the base space 3-dimensional differentiable manifold M >>> 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 - >>> from sage.all import * >>> M.diff_degree() == E.diff_degree() True - bundle_connection(name, latex_name=None)[source]¶
- Return a bundle connection on - self.- OUTPUT: - a bundle connection on - selfas an instance of- BundleConnection
 - 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 - >>> from sage.all import * >>> M = Manifold(Integer(3), 'M', start_index=Integer(1)) >>> X = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = X._first_ngens(3) >>> E = M.vector_bundle(Integer(2), 'E') >>> e = E.local_frame('e') # standard frame for E >>> 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)[source]¶
- Deprecated: Use - characteristic_cohomology_class()instead. See Issue #29581 for details.
 - characteristic_cohomology_class(*args, **kwargs)[source]¶
- 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 defined
- name– (default:- None) string representation given to the characteristic cohomology class; if- Nonethe default algebra representation or predefined name is used
- latex_name– (default:- None) LaTeX name given to the characteristic class; if- Nonethe value of- nameis used
- class_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 - valis 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 - >>> from sage.all import * >>> M = Manifold(Integer(4), 'M', structure='Lorentzian', start_index=Integer(1)) >>> X = M.chart(names=('t', 'x', 'y', 'z',)); (t, x, y, z,) = X._first_ngens(4) >>> g = M.metric() >>> g[Integer(1),Integer(1)] = -Integer(1) >>> g[Integer(2),Integer(2)] = Integer(1) >>> g[Integer(3),Integer(3)] = Integer(1) >>> g[Integer(4),Integer(4)] = Integer(1) >>> 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 - >>> from sage.all import * >>> nab = g.connection(); nab Levi-Civita connection nabla_g associated with the Lorentzian metric g on the 4-dimensional Lorentzian manifold M >>> nab.set_immutable() # make nab immutable - Of course, \(\nabla_g\) is flat: - sage: nab.display() - >>> from sage.all import * >>> 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 - >>> from sage.all import * >>> TM = M.tangent_bundle(); TM Tangent bundle TM over the 4-dimensional Lorentzian manifold M >>> p = TM.characteristic_cohomology_class('Pontryagin'); p Characteristic cohomology class p(TM) of the Tangent bundle TM over the 4-dimensional Lorentzian manifold M >>> 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)[source]¶
- Return the characteristic cohomology class ring of - selfover 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 - >>> from sage.all import * >>> M = Manifold(Integer(4), 'M', start_index=Integer(1)) >>> R = M.tangent_bundle().characteristic_cohomology_class_ring() >>> R Algebra of characteristic cohomology classes of the Tangent bundle TM over the 4-dimensional differentiable manifold M >>> p1 = R.gen(Integer(0)); p1 Characteristic cohomology class (p_1)(TM) of the Tangent bundle TM over the 4-dimensional differentiable manifold M >>> Integer(1) + p1 Characteristic cohomology class (1 + p_1)(TM) of the Tangent bundle TM over the 4-dimensional differentiable manifold M 
 - diff_degree()[source]¶
- 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> E = M.vector_bundle(Integer(2), 'E') >>> E.diff_degree() +Infinity >>> M = Manifold(Integer(2), 'M', structure='differentiable', ... diff_degree=Integer(3)) >>> E = M.vector_bundle(Integer(2), 'E') >>> E.diff_degree() 3 
 - total_space()[source]¶
- Return the total space of - self.- Note - At this stage, the total space does not come with induced charts. - OUTPUT: - the total space of - selfas an instance of- DifferentiableManifold
 - EXAMPLES: - sage: M = Manifold(3, 'M') sage: E = M.vector_bundle(2, 'E') sage: E.total_space() 6-dimensional differentiable manifold E - >>> from sage.all import * >>> M = Manifold(Integer(3), 'M') >>> E = M.vector_bundle(Integer(2), 'E') >>> E.total_space() 6-dimensional differentiable manifold E 
 
- class sage.manifolds.differentiable.vector_bundle.TensorBundle(base_space, k, l, dest_map=None)[source]¶
- 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 \[t:\ \underbrace{T_q^*N\times\cdots\times T_q^*N}_{k\ \; \text{times}} \times \underbrace{T_q N\times\cdots\times T_q N}_{l\ \; \text{times}} \longrightarrow K\]- (\(k\) 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 defined
- k– the contravariant rank of the corresponding tensor bundle
- l– the covariant rank of the corresponding tensor bundle
- dest_map– (default:- None) destination map \(\Phi:\ M \rightarrow N\) (type:- DiffMap); if- None, 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2) >>> R = Manifold(Integer(1), 'R') >>> T = R.chart(names=('t',)); (t,) = T._first_ngens(1)# canonical chart on R >>> 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 >>> Phi.display() Phi: R → M t ↦ (x, y) = (cos(t), sin(t)) >>> 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 - >>> from sage.all import * >>> R_tensor_module = R.tensor_field_module((Integer(1),Integer(0)), dest_map=Phi) >>> R_tensor_module is PhiTM.section_module() True - ambient_domain()[source]¶
- Return the codomain of the destination map. - OUTPUT: - a - DifferentiableManifoldrepresenting 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2) >>> e_cart = c_cart.frame() # standard basis >>> R = Manifold(Integer(1), 'R') >>> T = R.chart(names=('t',)); (t,) = T._first_ngens(1)# canonical chart on R >>> 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 >>> Phi.display() Phi: R → M t ↦ (x, y) = (cos(t), sin(t)) >>> PhiT11 = R.tensor_bundle(Integer(1), Integer(1), dest_map=Phi) >>> PhiT11.ambient_domain() 2-dimensional differentiable manifold M 
 - atlas()[source]¶
- 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 class- TopologicalManifold.- 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))] - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2) >>> Y = M.chart(names=('u', 'v',)); (u, v,) = Y._first_ngens(2) >>> TM = M.tangent_bundle() >>> TM.atlas() [Chart (M, (x, y)), Chart (M, (u, v))] 
 - change_of_frame(frame1, frame2)[source]¶
- Return a change of vector frames defined on the base space of - self.- See also - For further details on frames on - selfsee- local_frame().- Note - Since frames on - selfare directly induced by vector frames on the base space, this method directly invokes- change_of_frame()of the class- DifferentiableManifold.- INPUT: - frame1– local frame 1
- frame2– local frame 2
 - OUTPUT: - a - FreeModuleAutomorphismrepresenting, 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2) >>> c_uv = M.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2) >>> c_xy.transition_map(c_uv, (x+y, x-y)) Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v)) >>> TM = M.tangent_bundle() >>> TM.change_of_frame(c_xy.frame(), c_uv.frame()) Field of tangent-space automorphisms on the 2-dimensional differentiable manifold M >>> TM.change_of_frame(c_xy.frame(), c_uv.frame())[:] [ 1/2 1/2] [ 1/2 -1/2] >>> TM.change_of_frame(c_uv.frame(), c_xy.frame()) Field of tangent-space automorphisms on the 2-dimensional differentiable manifold M >>> TM.change_of_frame(c_uv.frame(), c_xy.frame())[:] [ 1 1] [ 1 -1] >>> 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()[source]¶
- Return the changes of vector frames defined on the base space of - selfwith respect to the destination map.- See also - For further details on frames on - selfsee- local_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)) - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2) >>> TM = M.tangent_bundle() >>> 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() {} - >>> from sage.all import * >>> TM.changes_of_frame() {} - We introduce a second frame on the manifold, relating it to frame - eby 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)) - >>> from sage.all import * >>> a = M.automorphism_field(name='a') >>> a[:] = [[-y, x], [Integer(1), Integer(2)]] >>> 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} - >>> from sage.all import * >>> 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 - >>> from sage.all import * >>> TM.changes_of_frame()[(e,f)] == a True >>> TM.changes_of_frame()[(f,e)] == a**(-Integer(1)) True 
 - coframes()[source]¶
- Return the list of coframes defined on the base manifold of - selfwith respect to the destination map.- See also - For further details on frames on - selfsee- local_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))] - >>> from sage.all import * >>> M = Manifold(Integer(2), 'R^2') >>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2)# Cartesian coordinates on R^2 >>> TM = M.tangent_bundle() >>> TM.coframes() [Coordinate coframe (R^2, (dx,dy))] >>> e = TM.vector_frame('e') >>> M.coframes() [Coordinate coframe (R^2, (dx,dy)), Coframe (R^2, (e^0,e^1))] >>> U = M.open_subset('U', coord_def={c_cart: x**Integer(2)+y**Integer(2)<Integer(1)}) >>> TU = U.tangent_bundle() >>> TU.coframes() [Coordinate coframe (U, (dx,dy))] >>> e.restrict(U) Vector frame (U, (e_0,e_1)) >>> TU.coframes() [Coordinate coframe (U, (dx,dy)), Coframe (U, (e^0,e^1))] >>> 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()[source]¶
- 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 - VectorFramerepresenting 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)) - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2) >>> TM = M.tangent_bundle() >>> TM.default_frame() Coordinate frame (M, (∂/∂x,∂/∂y)) 
 - destination_map()[source]¶
- Return the destination map. - OUTPUT: - a - DifferentialMaprepresenting 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2) >>> e_cart = c_cart.frame() # standard basis >>> R = Manifold(Integer(1), 'R') >>> T = R.chart(names=('t',)); (t,) = T._first_ngens(1)# canonical chart on R >>> 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 >>> Phi.display() Phi: R → M t ↦ (x, y) = (cos(t), sin(t)) >>> PhiT11 = R.tensor_bundle(Integer(1), Integer(1), dest_map=Phi) >>> PhiT11.destination_map() Differentiable map Phi from the 1-dimensional differentiable manifold R to the 2-dimensional differentiable manifold M 
 - fiber(point)[source]¶
- Return the tensor bundle fiber over a point. - INPUT: - point–- ManifoldPoint; point \(p\) of the base manifold of- self
 - OUTPUT: - an instance of - FiniteRankFreeModulerepresenting 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 - >>> from sage.all import * >>> M = Manifold(Integer(3), 'M') >>> X = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = X._first_ngens(3) >>> p = M((Integer(0),Integer(2),Integer(1)), name='p'); p Point p on the 3-dimensional differentiable manifold M >>> TM = M.tangent_bundle(); TM Tangent bundle TM over the 3-dimensional differentiable manifold M >>> TM.fiber(p) Tangent space at Point p on the 3-dimensional differentiable manifold M >>> 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 - >>> from sage.all import * >>> T11M = M.tensor_bundle(Integer(1),Integer(1)); T11M Tensor bundle T^(1,1)M over the 3-dimensional differentiable manifold M >>> T11M.fiber(p) Free module of type-(1,1) tensors on the Tangent space at Point p on the 3-dimensional differentiable manifold M >>> T11M.fiber(p) is M.tangent_space(p).tensor_module(Integer(1),Integer(1)) True 
 - frames()[source]¶
- Return the list of all vector frames defined on the base space of - selfwith respect to the destination map.- See also - For further details on frames on - selfsee- local_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))] - >>> from sage.all import * >>> M = Manifold(Integer(2), 'R^2') >>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2)# Cartesian coordinates on R^2 >>> TM = M.tangent_bundle() >>> TM.frames() [Coordinate frame (R^2, (∂/∂x,∂/∂y))] >>> e = TM.vector_frame('e') >>> TM.frames() [Coordinate frame (R^2, (∂/∂x,∂/∂y)), Vector frame (R^2, (e_0,e_1))] >>> U = M.open_subset('U', coord_def={c_cart: x**Integer(2)+y**Integer(2)<Integer(1)}) >>> TU = U.tangent_bundle() >>> TU.frames() [Coordinate frame (U, (∂/∂x,∂/∂y))] >>> 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] - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2) >>> e_cart = c_cart.frame() # standard basis >>> R = Manifold(Integer(1), 'R') >>> T = R.chart(names=('t',)); (t,) = T._first_ngens(1)# canonical chart on R >>> 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 >>> Phi.display() Phi: R → M t ↦ (x, y) = (cos(t), sin(t)) >>> PhiT11 = R.tensor_bundle(Integer(1), Integer(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 >>> f = PhiT11.local_frame(); f Vector frame (R, (∂/∂x,∂/∂y)) with values on the 2-dimensional differentiable manifold M >>> PhiT11.frames() [Vector frame (R, (∂/∂x,∂/∂y)) with values on the 2-dimensional differentiable manifold M] 
 - is_manifestly_trivial()[source]¶
- Return - Trueif- selfis known to be a trivial and- Falseotherwise.- If - Falseis 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> TM = M.tangent_bundle() >>> 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 - >>> from sage.all import * >>> e = TM.vector_frame('e') >>> 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 - >>> from sage.all import * >>> N = Manifold(Integer(4), 'N') >>> X = N.chart(names=('t', 'x', 'y', 'z',)); (t, x, y, z,) = X._first_ngens(4) >>> TN = N.tangent_bundle() >>> 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'S^2') # the 2-dimensional sphere S^2 >>> U = M.open_subset('U') # complement of the North pole >>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2)# stereo coord from the North pole >>> V = M.open_subset('V') # complement of the South pole >>> c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2)# stereo coord from the South pole >>> M.declare_union(U,V) # S^2 is the union of U and V >>> xy_to_uv = c_xy.transition_map(c_uv, (x/(x**Integer(2)+y**Integer(2)), ... y/(x**Integer(2)+y**Integer(2))), ... intersection_name='W', ... restrictions1= x**Integer(2)+y**Integer(2)!=Integer(0), ... restrictions2= u**Integer(2)+v**Integer(2)!=Integer(0)) >>> uv_to_xy = xy_to_uv.inverse() >>> W = U.intersection(V) >>> Phi = U.diff_map(M, {(c_xy, c_xy): [x, y]}, ... name='Phi') # inclusion map >>> 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 - >>> from sage.all import * >>> 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 - >>> from sage.all import * >>> 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 >>> PhiTU.is_manifestly_trivial() True 
 - local_frame(*args, **kwargs)[source]¶
- 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 \[p \mapsto \Big(\underbrace{e^*(p), \dots, e^*(p)}_{k\ \; \text{times}}, \underbrace{e(p), \dots, e(p)}_{l\ \; \text{times}}\Big) \in T^{(k,l)}_q N ,\]- 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 - VectorFramefor 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 be- Noneonly if- from_frameis not- None(see below)
- vector_fields– tuple or list of \(n\) linearly independent vector fields on- domain(\(n\) being the dimension of- domain) defining the vector frame; can be omitted if the vector frame is created from scratch or if- from_frameis not- None
- 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; if- None,- symbolis used in place of- latex_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 if- symbolis a single string) tuple of strings representing the indices labelling the vector fields of the frame; if- None, the indices will be generated as integers within the range declared on- self
- latex_indices– (default:- None) tuple of strings representing the indices for the LaTeX symbols of the vector fields; if- None,- indicesis used instead
- symbol_dual– (default:- None) same as- symbolbut for the dual coframe; if- None,- symbolmust be a string and is used for the common base of the symbols of the elements of the dual coframe
- latex_symbol_dual– (default:- None) same as- latex_symbolbut for the dual coframe
- domain– (default:- None) domain on which the local frame is defined; if- Noneis provided, the base space of- selfis 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 - >>> from sage.all import * >>> M = Manifold(Integer(3), 'M') >>> TM = M.tangent_bundle() >>> e = TM.local_frame('e'); e Vector frame (M, (e_0,e_1,e_2)) >>> e[Integer(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 - >>> from sage.all import * >>> U = M.open_subset('U') >>> f = TM.local_frame('f', domain=U); f Vector frame (U, (f_0,f_1,f_2)) >>> f[Integer(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()[source]¶
- Get the preferred orientation of - selfif 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 identity 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))] - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2) >>> T11 = M.tensor_bundle(Integer(1), Integer(1)) >>> 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2) >>> R = Manifold(Integer(1), 'R') >>> c_t = R.chart(names=('t',)); (t,) = c_t._first_ngens(1) >>> Phi = R.diff_map(M, name='Phi') >>> PhiT22 = R.tensor_bundle(Integer(2), Integer(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 >>> PhiT22.local_frame() # initialize frame Vector frame (R, (∂/∂x,∂/∂y)) with values on the 2-dimensional differentiable manifold M >>> PhiT22.orientation() [Vector frame (R, (∂/∂x,∂/∂y)) with values on the 2-dimensional differentiable manifold M] >>> PhiT22.local_frame() is PhiT22.orientation()[Integer(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))] - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> U = M.open_subset('U'); V = M.open_subset('V') >>> M.declare_union(U, V) >>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2); c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2) >>> T11 = M.tensor_bundle(Integer(1), Integer(1)); T11 Tensor bundle T^(1,1)M over the 2-dimensional differentiable manifold M >>> T11.orientation() [] >>> T11.set_orientation([c_xy.frame(), c_uv.frame()]) >>> 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))] - >>> from sage.all import * >>> 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))] - >>> from sage.all import * >>> c_tz = U.chart(names=('t', 'z',)); (t, z,) = c_tz._first_ngens(2) >>> M.set_orientation([c_tz, c_uv]) >>> T11.orientation() [Coordinate frame (U, (∂/∂t,∂/∂z)), Coordinate frame (V, (∂/∂u,∂/∂v))] 
 - section(*args, **kwargs)[source]¶
- Return a section of - selfon- domain, namely a tensor field on the subset- domainof the base space.- Note - This method directly invokes - tensor_field()of the class- DifferentiableManifold.- INPUT: - comp– (optional) either the components of the tensor field with respect to the vector frame specified by the argument- frameor a dictionary of components, the keys of which are vector frames or pairs- (f, c)where- fis a vector frame and- cthe chart in which the components are expressed
- frame– (default:- None; unused if- compis not given or is a dictionary) vector frame in which the components are given; if- None, the default vector frame of- selfis assumed
- chart– (default:- None; unused if- compis not given or is a dictionary) coordinate chart in which the components are expressed; if- None, the default chart on the domain of- frameis assumed
- domain– (default:- None) domain of the section; if- None,- self.base_space()is assumed
- name– (default:- None) name given to the tensor field
- latex_name– (default:- None) LaTeX symbol to denote the tensor field; if- None, the LaTeX symbol is set to- name
- 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 convention- position=0for the first argument; for instance:- sym = (0,1)for a symmetry between the 1st and 2nd arguments
- sym = [(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 for- sym
 - OUTPUT: - a - TensorField(or if \(N\) is parallelizable, a- TensorFieldParal) 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> U = M.open_subset('U') ; V = M.open_subset('V') >>> M.declare_union(U,V) # M is the union of U and V >>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2); c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2) >>> transf = c_xy.transition_map(c_uv, (x+y, x-y), ... intersection_name='W', ... restrictions1= x>Integer(0), ... restrictions2= u+v>Integer(0)) >>> inv = transf.inverse() >>> W = U.intersection(V) >>> eU = c_xy.frame() ; eV = c_uv.frame() >>> T11M = M.tensor_bundle(Integer(1), Integer(1)); T11M Tensor bundle T^(1,1)M over the 2-dimensional differentiable manifold M >>> t = T11M.section({eU: [[Integer(1), x], [Integer(0), Integer(2)]]}, name='t'); t Tensor field t of type (1,1) on the 2-dimensional differentiable manifold M >>> t.display() t = ∂/∂x⊗dx + x ∂/∂x⊗dy + 2 ∂/∂y⊗dy - An example of use with the arguments - compand- domain:- 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 - >>> from sage.all import * >>> TM = M.tangent_bundle() >>> w = TM.section([-y, x], domain=U); w Vector field on the Open subset U of the 2-dimensional differentiable manifold M >>> w.display() -y ∂/∂x + x ∂/∂y 
 - section_module(domain=None)[source]¶
- Return the section module on - domain, namely the corresponding tensor field module, of- selfon- domain.- Note - This method directly invokes - tensor_field_module()of the class- DifferentiableManifold.- INPUT: - domain– (default:- None) the domain of the corresponding section module; if- None, the base space is assumed
 - OUTPUT: - a - TensorFieldModule(or if \(N\) is parallelizable, a- TensorFieldFreeModule) 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 - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2) >>> U = M.open_subset('U') >>> TM = M.tangent_bundle() >>> 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 >>> TUM is U.tensor_field_module((Integer(1),Integer(0))) True 
 - set_change_of_frame(frame1, frame2, change_of_frame, compute_inverse=True)[source]¶
- Relate two vector frames by an automorphism. - This updates the internal dictionary - self._frame_changesof the base space \(M\).- See also - For further details on frames on - selfsee- local_frame().- Note - Since frames on - selfare directly induced by vector frames on the base space, this method directly invokes- set_change_of_frame()of the class- DifferentiableManifold.- INPUT: - frame1– frame 1, denoted \((e_i)\) below
- frame2– frame 2, denoted \((f_i)\) below
- change_of_frame– instance of class- FreeModuleAutomorphismdescribing the automorphism \(P\) that relates the basis \((e_i)\) to the basis \((f_i)\) according to \(f_i = P(e_i)\)
- compute_inverse– boolean (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 dictionary- self._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] - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2) >>> e = M.vector_frame('e') >>> f = M.vector_frame('f') >>> a = M.automorphism_field() >>> a[e,:] = [[Integer(1),Integer(2)],[Integer(0),Integer(3)]] >>> TM = M.tangent_bundle() >>> TM.set_change_of_frame(e, f, a) >>> f[Integer(0)].display(e) f_0 = e_0 >>> f[Integer(1)].display(e) f_1 = 2 e_0 + 3 e_1 >>> e[Integer(0)].display(f) e_0 = f_0 >>> e[Integer(1)].display(f) e_1 = -2/3 f_0 + 1/3 f_1 >>> TM.change_of_frame(e,f)[e,:] [1 2] [0 3] 
 - set_default_frame(frame)[source]¶
- 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–- VectorFramea 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)) - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2) >>> TM = M.tangent_bundle() >>> e = TM.vector_frame('e') >>> TM.default_frame() Coordinate frame (M, (∂/∂x,∂/∂y)) >>> TM.set_default_frame(e) >>> TM.default_frame() Vector frame (M, (e_0,e_1)) >>> M.default_frame() Vector frame (M, (e_0,e_1)) 
 - set_orientation(orientation)[source]¶
- Set the preferred orientation of - self.- INPUT: - orientation– a vector frame or a list of vector frames, covering the base space of- self
 - 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))] - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2) >>> T11 = M.tensor_bundle(Integer(1), Integer(1)) >>> e = T11.local_frame('e'); e Vector frame (M, (e_0,e_1)) >>> T11.set_orientation(e) >>> 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))] - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> U = M.open_subset('U'); V = M.open_subset('V') >>> M.declare_union(U, V) >>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2); c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2) >>> T12 = M.tensor_bundle(Integer(1), Integer(2)) >>> e = T12.local_frame('e', domain=U) >>> f = T12.local_frame('f', domain=V) >>> T12.set_orientation([e, f]) >>> T12.orientation() [Vector frame (U, (e_0,e_1)), Vector frame (V, (f_0,f_1))] 
 - transition(chart1, chart2)[source]¶
- 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 class- TopologicalManifold.- INPUT: - chart1– chart 1
- chart2– chart 2
 - OUTPUT: - instance of - CoordChangerepresenting 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)) - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2) >>> c_uv = M.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2) >>> 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)) >>> TM = M.tangent_bundle() >>> 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()[source]¶
- 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 class- TopologicalManifold.- 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))} - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> c_xy = M.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2) >>> c_uv = M.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2) >>> xy_to_uv = c_xy.transition_map(c_uv, [x+y, x-y]) >>> TM = M.tangent_bundle() >>> TM.transitions() {(Chart (M, (x, y)), Chart (M, (u, v))): Change of coordinates from Chart (M, (x, y)) to Chart (M, (u, v))} >>> uv_to_xy = xy_to_uv.inverse() >>> 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))} >>> c_rs = M.chart(names=('r', 's',)); (r, s,) = c_rs._first_ngens(2) >>> uv_to_rs = c_uv.transition_map(c_rs, [-u+Integer(2)*v, Integer(3)*u-v]) >>> 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))} >>> xy_to_rs = uv_to_rs * xy_to_uv >>> 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)[source]¶
- Return a trivialization of - selfin 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 class- TopologicalManifold.- INPUT: - coordinates– (default:- ''(empty string)) string defining the coordinate symbols, ranges and possible periodicities, see below
- names– (default:- None) unused argument, except if- coordinatesis 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': SymPy
- None: the current calculus method defined on the manifold is used (cf.- set_calculus_method())
 
 - The coordinates declared in the string - coordinatesare 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 bounds- aand- bcan be- +/-Infinity,- Inf,- infinity,- infor- oo. 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, where- Tis the period, or, for manifolds over \(\RR\) only, as the keyword- periodic(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 - coordinatesmust 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 argument- coordinatescan be omitted when the shortcut operator- <,>is used to declare the trivialization.- OUTPUT: - the created chart, as an instance of - Chartor one of its subclasses, like- RealDiffChartfor 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) - >>> from sage.all import * >>> M = Manifold(Integer(2), 'M') >>> TM = M.tangent_bundle() >>> X = TM.trivialization('x y'); X Chart (M, (x, y)) >>> X[Integer(0)] x >>> X[Integer(1)] y >>> X[:] (x, y) 
 - vector_frame(*args, **kwargs)[source]¶
- 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 \[p \mapsto \Big(\underbrace{e^*(p), \dots, e^*(p)}_{k\ \; \text{times}}, \underbrace{e(p), \dots, e(p)}_{l\ \; \text{times}}\Big) \in T^{(k,l)}_q N ,\]- 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 - VectorFramefor 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 be- Noneonly if- from_frameis not- None(see below)
- vector_fields– tuple or list of \(n\) linearly independent vector fields on- domain(\(n\) being the dimension of- domain) defining the vector frame; can be omitted if the vector frame is created from scratch or if- from_frameis not- None
- 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; if- None,- symbolis used in place of- latex_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 if- symbolis a single string) tuple of strings representing the indices labelling the vector fields of the frame; if- None, the indices will be generated as integers within the range declared on- self
- latex_indices– (default:- None) tuple of strings representing the indices for the LaTeX symbols of the vector fields; if- None,- indicesis used instead
- symbol_dual– (default:- None) same as- symbolbut for the dual coframe; if- None,- symbolmust be a string and is used for the common base of the symbols of the elements of the dual coframe
- latex_symbol_dual– (default:- None) same as- latex_symbolbut for the dual coframe
- domain– (default:- None) domain on which the local frame is defined; if- Noneis provided, the base space of- selfis 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 - >>> from sage.all import * >>> M = Manifold(Integer(3), 'M') >>> TM = M.tangent_bundle() >>> e = TM.local_frame('e'); e Vector frame (M, (e_0,e_1,e_2)) >>> e[Integer(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 - >>> from sage.all import * >>> U = M.open_subset('U') >>> f = TM.local_frame('f', domain=U); f Vector frame (U, (f_0,f_1,f_2)) >>> f[Integer(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.