Local Frames#
The class LocalFrame
implements local frames on vector bundles
(see TopologicalVectorBundle
or
DifferentiableVectorBundle
).
For \(k=0,1,\dots\), a local frame on a vector bundle \(E \to M\) of class \(C^k\) and rank \(n\) is a local section \((e_1,\dots,e_n):U \to E^n\) of class \(C^k\) defined on some subset \(U\) of the base space \(M\), such that \(e(p)\) is a basis of the fiber \(E_p\) for any \(p \in U\).
AUTHORS:
Michael Jung (2019): initial version
EXAMPLES:
Defining a global frame on a topological vector bundle of rank 3:
sage: M = Manifold(3, 'M', structure='top')
sage: E = M.vector_bundle(3, 'E')
sage: e = E.local_frame('e'); e
Local frame (E|_M, (e_0,e_1,e_2))
This frame is now the default frame of the corresponding section module and saved in the vector bundle:
sage: e in E.frames()
True
sage: sec_module = E.section_module(); sec_module
Free module C^0(M;E) of sections on the 3-dimensional topological manifold M
with values in the real vector bundle E of rank 3
sage: sec_module.default_basis()
Local frame (E|_M, (e_0,e_1,e_2))
However, the default frame can be changed:
sage: sec_module.set_default_basis(e)
sage: sec_module.default_basis()
Local frame (E|_M, (e_0,e_1,e_2))
The elements of a local frame are local sections in the vector bundle:
sage: for vec in e:
....: print(vec)
Section e_0 on the 3-dimensional topological manifold M with values in the
real vector bundle E of rank 3
Section e_1 on the 3-dimensional topological manifold M with values in the
real vector bundle E of rank 3
Section e_2 on the 3-dimensional topological manifold M with values in the
real vector bundle E of rank 3
Each element of a vector frame can be accessed by its index:
sage: e[0]
Section e_0 on the 3-dimensional topological manifold M with values in the
real vector bundle E of rank 3
The slice operator :
can be used to access to more than one element:
sage: e[0:2]
(Section e_0 on the 3-dimensional topological manifold M with values in the
real vector bundle E of rank 3,
Section e_1 on the 3-dimensional topological manifold M with values in the
real vector bundle E of rank 3)
sage: e[:]
(Section e_0 on the 3-dimensional topological manifold M with values in the
real vector bundle E of rank 3,
Section e_1 on the 3-dimensional topological manifold M with values in the
real vector bundle E of rank 3,
Section e_2 on the 3-dimensional topological manifold M with values in the
real vector bundle E of rank 3)
The index range depends on the starting index defined on the manifold:
sage: M = Manifold(3, 'M', structure='top', start_index=1)
sage: c_xyz.<x,y,z> = M.chart()
sage: U = M.open_subset('U')
sage: c_xyz_U = c_xyz.restrict(U)
sage: E = M.vector_bundle(3, 'E')
sage: e = E.local_frame('e', domain=U); e
Local frame (E|_U, (e_1,e_2,e_3))
sage: [e[i] for i in M.irange()]
[Section e_1 on the Open subset U of the 3-dimensional topological manifold
M with values in the real vector bundle E of rank 3,
Section e_2 on the Open subset U of the 3-dimensional topological manifold
M with values in the real vector bundle E of rank 3,
Section e_3 on the Open subset U of the 3-dimensional topological manifold
M with values in the real vector bundle E of rank 3]
sage: e[1], e[2], e[3]
(Section e_1 on the Open subset U of the 3-dimensional topological manifold
M with values in the real vector bundle E of rank 3,
Section e_2 on the Open subset U of the 3-dimensional topological manifold
M with values in the real vector bundle E of rank 3,
Section e_3 on the Open subset U of the 3-dimensional topological manifold
M with values in the real vector bundle E of rank 3)
Let us check that the local sections e[i]
are indeed the frame vectors
from their components with respect to the frame \(e\):
sage: e[1].comp(e)[:]
[1, 0, 0]
sage: e[2].comp(e)[:]
[0, 1, 0]
sage: e[3].comp(e)[:]
[0, 0, 1]
Defining a local frame on a vector bundle, the dual coframe is automatically created, which, by default, bares the same name (here \(e\)):
sage: E.coframes()
[Local coframe (E|_U, (e^1,e^2,e^3))]
sage: e_dual = E.coframes()[0] ; e_dual
Local coframe (E|_U, (e^1,e^2,e^3))
sage: e_dual is e.coframe()
True
Let us check that the coframe \((e^i)\) is indeed the dual of the vector frame \((e_i)\):
sage: e_dual[1](e[1]) # linear form e^1 applied to local section e_1
Scalar field e^1(e_1) on the Open subset U of the 3-dimensional topological
manifold M
sage: e_dual[1](e[1]).expr() # the explicit expression of e^1(e_1)
1
sage: e_dual[1](e[1]).expr(), e_dual[1](e[2]).expr(), e_dual[1](e[3]).expr()
(1, 0, 0)
sage: e_dual[2](e[1]).expr(), e_dual[2](e[2]).expr(), e_dual[2](e[3]).expr()
(0, 1, 0)
sage: e_dual[3](e[1]).expr(), e_dual[3](e[2]).expr(), e_dual[3](e[3]).expr()
(0, 0, 1)
Via bundle automorphisms, a new frame can be created from an existing one:
sage: sec_module_U = E.section_module(domain=U)
sage: change_frame = sec_module_U.automorphism()
sage: change_frame[:] = [[0,1,0],[0,0,1],[1,0,0]]
sage: f = e.new_frame(change_frame, 'f'); f
Local frame (E|_U, (f_1,f_2,f_3))
A copy of this automorphism and its inverse is now part of the vector bundle’s frame changes:
sage: E.change_of_frame(e, f)
Automorphism of the Free module C^0(U;E) of sections on the Open subset U of
the 3-dimensional topological manifold M with values in the real vector
bundle E of rank 3
sage: E.change_of_frame(e, f) == change_frame
True
sage: E.change_of_frame(f, e) == change_frame.inverse()
True
Let us check the components of \(f\) with respect to the frame \(e\):
sage: f[1].comp(e)[:]
[0, 0, 1]
sage: f[2].comp(e)[:]
[1, 0, 0]
sage: f[3].comp(e)[:]
[0, 1, 0]
- class sage.manifolds.local_frame.LocalCoFrame(frame, symbol, latex_symbol=None, indices=None, latex_indices=None)#
Bases:
sage.tensor.modules.free_module_basis.FreeModuleCoBasis
Local coframe on a vector bundle.
A local coframe on a vector bundle \(E \to M\) of class \(C^k\) is a local section \(e^*: U \to E^n\) of class \(C^k\) on some subset \(U\) of the base space \(M\), such that \(e^*(p)\) is a basis of the fiber \(E^*_p\) of the dual bundle for any \(p \in U\).
INPUT:
frame
– the local frame dual to the coframesymbol
– either a string, to be used as a common base for the symbols of the linear forms constituting the coframe, or a tuple of strings, representing the individual symbols of the linear formslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the linear forms constituting the coframe, or a tuple of strings, representing the individual LaTeX symbols of the linear forms; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the linear forms of the coframe; ifNone
, the indices will be generated as integers within the range declared on the coframe’s domainlatex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the linear forms of the coframe; ifNone
,indices
is used instead
EXAMPLES:
Local coframe on a topological vector bundle of rank 3:
sage: M = Manifold(3, 'M', structure='top', start_index=1) sage: X.<x,y,z> = M.chart() sage: E = M.vector_bundle(3, 'E') sage: e = E.local_frame('e') sage: from sage.manifolds.local_frame import LocalCoFrame sage: f = LocalCoFrame(e, 'f'); f Local coframe (E|_M, (f^1,f^2,f^3))
The local coframe can also be obtained by using the method
dual_basis()
orcoframe()
:sage: e_dual = e.dual_basis(); e_dual Local coframe (E|_M, (e^1,e^2,e^3)) sage: e_dual is e.coframe() True sage: e_dual is f False sage: e_dual[:] == f[:] True sage: f[1].display(e) f^1 = e^1
The consisted linear forms can be obtained via the operator
[]
:sage: f[1], f[2], f[3] (Linear form f^1 on the Free module C^0(M;E) of sections on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3, Linear form f^2 on the Free module C^0(M;E) of sections on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3, Linear form f^3 on the Free module C^0(M;E) of sections on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3)
Checking that \(f\) is the dual of \(e\):
sage: f[1](e[1]).expr(), f[1](e[2]).expr(), f[1](e[3]).expr() (1, 0, 0) sage: f[2](e[1]).expr(), f[2](e[2]).expr(), f[2](e[3]).expr() (0, 1, 0) sage: f[3](e[1]).expr(), f[3](e[2]).expr(), f[3](e[3]).expr() (0, 0, 1)
- at(point)#
Return the value of
self
at a given point on the base space, this value being a basis of the dual vector bundle at this point.INPUT:
point
–ManifoldPoint
; point \(p\) in the domain \(U\) of the coframe (denoted \(f\) hereafter)
OUTPUT:
FreeModuleCoBasis
representing the basis \(f(p)\) of the vector space \(E^*_p\), dual to the vector bundle fiber \(E_p\)
EXAMPLES:
Cobasis of a vector bundle fiber:
sage: M = Manifold(2, 'M', structure='top', start_index=1) sage: X.<x,y> = M.chart() sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e') sage: e_dual = e.coframe(); e_dual Local coframe (E|_M, (e^1,e^2)) sage: p = M.point((-1,2), name='p') sage: e_dual_p = e_dual.at(p) ; e_dual_p Dual basis (e^1,e^2) on the Fiber of E at Point p on the 2-dimensional topological manifold M sage: type(e_dual_p) <class 'sage.tensor.modules.free_module_basis.FreeModuleCoBasis_with_category'> sage: e_dual_p[1] Linear form e^1 on the Fiber of E at Point p on the 2-dimensional topological manifold M sage: e_dual_p[2] Linear form e^2 on the Fiber of E at Point p on the 2-dimensional topological manifold M sage: e_dual_p is e.at(p).dual_basis() True
- set_name(symbol, latex_symbol=None, indices=None, latex_indices=None, index_position='up', include_domain=True)#
Set (or change) the text name and LaTeX name of
self
.INPUT:
symbol
– either a string, to be used as a common base for the symbols of the linear forms constituting the coframe, or a list/tuple of strings, representing the individual symbols of the linear formslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the linear forms constituting the coframe, or a list/tuple of strings, representing the individual LaTeX symbols of the linear forms; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the linear forms of the coframe; 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 linear forms; ifNone
,indices
is used insteadindex_position
– (default:'up'
) determines the position of the indices labelling the linear forms of the coframe; can be either'down'
or'up'
include_domain
– (default:True
) boolean determining whether the name of the domain is included in the beginning of the coframe name
EXAMPLES:
sage: M = Manifold(3, 'M', structure='top') sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e').coframe(); e Local coframe (E|_M, (e^0,e^1)) sage: e.set_name('f'); e Local coframe (E|_M, (f^0,f^1)) sage: e.set_name('e', latex_symbol=r'\epsilon') sage: latex(e) \left(E|_{M}, \left(\epsilon^{0},\epsilon^{1}\right)\right) sage: e.set_name('e', include_domain=False); e Local coframe (e^0,e^1) sage: e.set_name(['a', 'b'], latex_symbol=[r'\alpha', r'\beta']); e Local coframe (E|_M, (a,b)) sage: latex(e) \left(E|_{M}, \left(\alpha,\beta\right)\right) sage: e.set_name('e', indices=['x','y'], ....: latex_indices=[r'\xi', r'\zeta']); e Local coframe (E|_M, (e^x,e^y)) sage: latex(e) \left(E|_{M}, \left(e^{\xi},e^{\zeta}\right)\right)
- class sage.manifolds.local_frame.LocalFrame(section_module, symbol, latex_symbol=None, indices=None, latex_indices=None, symbol_dual=None, latex_symbol_dual=None)#
Bases:
sage.tensor.modules.free_module_basis.FreeModuleBasis
Local frame on a vector bundle.
A local frame on a vector bundle \(E \to M\) of class \(C^k\) is a local section \((e_1,\dots,e_n):U \to E^n\) of class \(C^k\) defined on some subset \(U\) of the base space \(M\), such that \(e(p)\) is a basis of the fiber \(E_p\) for any \(p \in U\).
For each instantiation of a local frame, a local coframe is automatically created, as an instance of the class
LocalCoFrame
. It is returned by the methodcoframe()
.INPUT:
section_module
– free module of local sections over \(U\) in the given vector bundle \(E \to M\)symbol
– either a string, to be used as a common base for the symbols of the local sections constituting the local frame, or a tuple of strings, representing the individual symbols of the local sectionslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the local sections constituting the local frame, or a tuple of strings, representing the individual LaTeX symbols of the local sections; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the local sections of the frame; ifNone
, the indices will be generated as integers within the range declared on the local frame’s domainlatex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the local sections; 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 coframe
EXAMPLES:
Defining a local frame on a 3-dimensional vector bundle over a 3-dimensional manifold:
sage: M = Manifold(3, 'M', start_index=1, structure='top') sage: E = M.vector_bundle(3, 'E') sage: e = E.local_frame('e'); e Local frame (E|_M, (e_1,e_2,e_3)) sage: latex(e) \left(E|_{M}, \left(e_{1},e_{2},e_{3}\right)\right)
The individual elements of the vector frame are accessed via square brackets, with the possibility to invoke the slice operator ‘
:
’ to get more than a single element:sage: e[2] Section e_2 on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3 sage: e[1:3] (Section e_1 on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3, Section e_2 on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3) sage: e[:] (Section e_1 on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3, Section e_2 on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3, Section e_3 on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3)
The LaTeX symbol can be specified:
sage: eps = E.local_frame('eps', latex_symbol=r'\epsilon') sage: latex(eps) \left(E|_{M}, \left(\epsilon_{1},\epsilon_{2},\epsilon_{3}\right)\right)
By default, the elements of the local frame are labelled by integers within the range specified at the manifold declaration. It is however possible to fully customize the labels, via the argument
indices
:sage: u = E.local_frame('u', indices=('x', 'y', 'z')) ; u Local frame (E|_M, (u_x,u_y,u_z)) sage: u[1] Section u_x on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3 sage: u.coframe() Local coframe (E|_M, (u^x,u^y,u^z))
The LaTeX format of the indices can be adjusted:
sage: v = E.local_frame('v', indices=('a', 'b', 'c'), ....: latex_indices=(r'\alpha', r'\beta', r'\gamma')) sage: v Local frame (E|_M, (v_a,v_b,v_c)) sage: latex(v) \left(E|_{M}, \left(v_{\alpha},v_{\beta},v_{\gamma}\right)\right) sage: latex(v.coframe()) \left(E|_{M}, \left(v^{\alpha},v^{\beta},v^{\gamma}\right)\right)
The symbol of each element of the local frame can also be freely chosen, by providing a tuple of symbols as the first argument of
local_frame
; it is then mandatory to specify as well some symbols for the dual coframe:sage: h = E.local_frame(('a', 'b', 'c'), symbol_dual=('A', 'B', 'C')); h Local frame (E|_M, (a,b,c)) sage: h[1] Section a on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3 sage: h.coframe() Local coframe (E|_M, (A,B,C)) sage: h.coframe()[1] Linear form A on the Free module C^0(M;E) of sections on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3
Local frames are bases of free modules formed by local sections:
sage: N = Manifold(2, 'N', structure='top', start_index=1) sage: X.<x,y> = N.chart() sage: U = N.open_subset('U') sage: F = N.vector_bundle(2, 'F') sage: f = F.local_frame('f', domain=U) sage: f.module() Free module C^0(U;F) of sections on the Open subset U of the 2-dimensional topological manifold N with values in the real vector bundle F of rank 2 sage: f.module().base_ring() Algebra of scalar fields on the Open subset U of the 2-dimensional topological manifold N sage: f.module() is F.section_module(domain=f.domain()) True sage: f in F.section_module(domain=U).bases() True
The value of the local frame at a given point is a basis of the corresponding fiber:
sage: X_U = X.restrict(U) # We need coordinates on the subset sage: p = N((0,1), name='p') ; p Point p on the 2-dimensional topological manifold N sage: f.at(p) Basis (f_1,f_2) on the Fiber of F at Point p on the 2-dimensional topological manifold N
- at(point)#
Return the value of
self
at a given point, this value being a basis of the vector bundle fiber at the point.INPUT:
point
–ManifoldPoint
; point \(p\) in the domain \(U\) of the local frame (denoted \(e\) hereafter)
OUTPUT:
FreeModuleBasis
representing the basis \(e(p)\) of the vector bundle fiber \(E_p\)
EXAMPLES:
Basis of a fiber of a trivial vector bundle:
sage: M = Manifold(2, 'M', structure='top') sage: X.<x,y> = M.chart() sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e'); e Local frame (E|_M, (e_0,e_1)) sage: p = M.point((-1,2), name='p') sage: ep = e.at(p) ; ep Basis (e_0,e_1) on the Fiber of E at Point p on the 2-dimensional topological manifold M sage: type(ep) <class 'sage.tensor.modules.free_module_basis.FreeModuleBasis_with_category'> sage: ep[0] Vector e_0 in the fiber of E at Point p on the 2-dimensional topological manifold M sage: ep[1] Vector e_1 in the fiber of E at Point p on the 2-dimensional topological manifold M
Note that the symbols used to denote the vectors are same as those for the vector fields of the frame. At this stage,
ep
is the unique basis on fiber atp
:sage: Ep = E.fiber(p) sage: Ep.bases() [Basis (e_0,e_1) on the Fiber of E at Point p on the 2-dimensional topological manifold M]
Let us consider another local frame:
sage: aut = E.section_module().automorphism() sage: aut[:] = [[1+y^2, 0], [0, 2]] sage: f = e.new_frame(aut, 'f') ; f Local frame (E|_M, (f_0,f_1)) sage: fp = f.at(p) ; fp Basis (f_0,f_1) on the Fiber of E at Point p on the 2-dimensional topological manifold M
There are now two bases on the fiber:
sage: Ep.bases() [Basis (e_0,e_1) on the Fiber of E at Point p on the 2-dimensional topological manifold M, Basis (f_0,f_1) on the Fiber of E at Point p on the 2-dimensional topological manifold M]
Moreover, the changes of bases in the tangent space have been computed from the known relation between the frames
e
andf
(via the automorphismaut
defined above):sage: Ep.change_of_basis(ep, fp) Automorphism of the Fiber of E at Point p on the 2-dimensional topological manifold M sage: Ep.change_of_basis(ep, fp).display() 5 e_0⊗e^0 + 2 e_1⊗e^1 sage: Ep.change_of_basis(fp, ep) Automorphism of the Fiber of E at Point p on the 2-dimensional topological manifold M sage: Ep.change_of_basis(fp, ep).display() 1/5 e_0⊗e^0 + 1/2 e_1⊗e^1
The dual bases:
sage: e.coframe() Local coframe (E|_M, (e^0,e^1)) sage: ep.dual_basis() Dual basis (e^0,e^1) on the Fiber of E at Point p on the 2-dimensional topological manifold M sage: ep.dual_basis() is e.coframe().at(p) True sage: f.coframe() Local coframe (E|_M, (f^0,f^1)) sage: fp.dual_basis() Dual basis (f^0,f^1) on the Fiber of E at Point p on the 2-dimensional topological manifold M sage: fp.dual_basis() is f.coframe().at(p) True
- base_space()#
Return the base space on which the overlying vector bundle is defined.
EXAMPLES:
sage: M = Manifold(3, 'M', structure='top') sage: U = M.open_subset('U') sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e', domain=U) sage: e.base_space() 3-dimensional topological manifold M
- coframe()#
Return the coframe of
self
.EXAMPLES:
sage: M = Manifold(2, 'M', structure='top') sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e'); e Local frame (E|_M, (e_0,e_1)) sage: e.coframe() Local coframe (E|_M, (e^0,e^1))
- domain()#
Return the domain on which
self
is defined.EXAMPLES:
sage: M = Manifold(3, 'M', structure='top') sage: U = M.open_subset('U') sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e', domain=U); e Local frame (E|_U, (e_0,e_1)) sage: e.domain() Open subset U of the 3-dimensional topological manifold M
- new_frame(change_of_frame, symbol, latex_symbol=None, indices=None, latex_indices=None, symbol_dual=None, latex_symbol_dual=None)#
Define a new local frame from
self
.The new local frame is defined from vector bundle automorphisms; its module is the same as that of the current frame.
INPUT:
change_of_frame
–FreeModuleAutomorphism
; vector bundle automorphisms \(P\) that relates the current frame \((e_i)\) to the new frame \((f_i)\) according to \(f_i = P(e_i)\)symbol
– either a string, to be used as a common base for the symbols of the sections constituting the local frame, or a list/tuple of strings, representing the individual symbols of the sectionslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the sections constituting the local frame, or a list/tuple of strings, representing the individual LaTeX symbols of the sections; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the sections 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 sections; 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 coframe
OUTPUT:
the new frame \((f_i)\), as an instance of
LocalFrame
EXAMPLES:
Orthogonal transformation of a frame on the 2-dimensional trivial vector bundle over the Euclidean plane:
sage: M = Manifold(2, 'R^2', structure='top', start_index=1) sage: c_cart.<x,y> = M.chart() sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e'); e Local frame (E|_R^2, (e_1,e_2)) sage: orth = E.section_module().automorphism() sage: orth[:] = [[sqrt(3)/2, -1/2], [1/2, sqrt(3)/2]] sage: f = e.new_frame(orth, 'f') sage: f[1][:] [1/2*sqrt(3), 1/2] sage: f[2][:] [-1/2, 1/2*sqrt(3)] sage: a = E.change_of_frame(e,f) sage: a[:] [1/2*sqrt(3) -1/2] [ 1/2 1/2*sqrt(3)] sage: a == orth True sage: a is orth False sage: a._components # random (dictionary output) {Local frame (E|_D_0, (e_1,e_2)): 2-indices components w.r.t. Local frame (E|_D_0, (e_1,e_2)), Local frame (E|_D_0, (f_1,f_2)): 2-indices components w.r.t. Local frame (E|_D_0, (f_1,f_2))} sage: a.comp(f)[:] [1/2*sqrt(3) -1/2] [ 1/2 1/2*sqrt(3)] sage: a1 = E.change_of_frame(f,e) sage: a1[:] [1/2*sqrt(3) 1/2] [ -1/2 1/2*sqrt(3)] sage: a1 == orth.inverse() True sage: a1 is orth.inverse() False sage: e[1].comp(f)[:] [1/2*sqrt(3), -1/2] sage: e[2].comp(f)[:] [1/2, 1/2*sqrt(3)]
- restrict(subdomain)#
Return the restriction of
self
to some open subset of its domain.If the restriction has not been defined yet, it is constructed here.
INPUT:
subdomain
– open subset \(V\) of the current frame domain \(U\)
OUTPUT:
the restriction of the current frame to \(V\) as a
LocalFrame
EXAMPLES:
Restriction of a frame defined on \(\RR^2\) to the unit disk:
sage: M = Manifold(2, 'R^2', structure='top', start_index=1) sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2 sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e'); e Local frame (E|_R^2, (e_1,e_2)) sage: a = E.section_module().automorphism() sage: a[:] = [[1-y^2,0], [1+x^2, 2]] sage: f = e.new_frame(a, 'f'); f Local frame (E|_R^2, (f_1,f_2)) sage: U = M.open_subset('U', coord_def={c_cart: x^2+y^2<1}) sage: e_U = e.restrict(U); e_U Local frame (E|_U, (e_1,e_2)) sage: f_U = f.restrict(U) ; f_U Local frame (E|_U, (f_1,f_2))
The vectors of the restriction have the same symbols as those of the original frame:
sage: f_U[1].display() f_1 = (-y^2 + 1) e_1 + (x^2 + 1) e_2 sage: f_U[2].display() f_2 = 2 e_2
Actually, the components are the restrictions of the original frame vectors:
sage: f_U[1] is f[1].restrict(U) True sage: f_U[2] is f[2].restrict(U) True
- set_name(symbol, latex_symbol=None, indices=None, latex_indices=None, index_position='down', include_domain=True)#
Set (or change) the text name and LaTeX name of
self
.INPUT:
symbol
– either a string, to be used as a common base for the symbols of the local sections constituting the local frame, or a list/tuple of strings, representing the individual symbols of the local sectionslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the local sections constituting the local frame, or a list/tuple of strings, representing the individual LaTeX symbols of the local sections; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the local sections 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 local sections; ifNone
,indices
is used insteadindex_position
– (default:'down'
) determines the position of the indices labelling the local sections of the frame; can be either'down'
or'up'
include_domain
– (default:True
) boolean determining whether the name of the domain is included in the beginning of the vector frame name
EXAMPLES:
sage: M = Manifold(3, 'M', structure='top') sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e'); e Local frame (E|_M, (e_0,e_1)) sage: e.set_name('f'); e Local frame (E|_M, (f_0,f_1)) sage: e.set_name('e', include_domain=False); e Local frame (e_0,e_1) sage: e.set_name(['a', 'b']); e Local frame (E|_M, (a,b)) sage: e.set_name('e', indices=['x', 'y']); e Local frame (E|_M, (e_x,e_y)) sage: e.set_name('e', latex_symbol=r'\epsilon') sage: latex(e) \left(E|_{M}, \left(\epsilon_{0},\epsilon_{1}\right)\right) sage: e.set_name('e', latex_symbol=[r'\alpha', r'\beta']) sage: latex(e) \left(E|_{M}, \left(\alpha,\beta\right)\right) sage: e.set_name('e', latex_symbol='E', ....: latex_indices=[r'\alpha', r'\beta']) sage: latex(e) \left(E|_{M}, \left(E_{\alpha},E_{\beta}\right)\right)
- vector_bundle()#
Return the vector bundle on which
self
is defined.EXAMPLES:
sage: M = Manifold(3, 'M', structure='top') sage: U = M.open_subset('U') sage: E = M.vector_bundle(2, 'E') sage: e = E.local_frame('e', domain=U) sage: e.vector_bundle() Topological real vector bundle E -> M of rank 2 over the base space 3-dimensional topological manifold M sage: e.vector_bundle() is E True
- class sage.manifolds.local_frame.TrivializationCoFrame(triv_frame, symbol, latex_symbol=None, indices=None, latex_indices=None)#
Bases:
sage.manifolds.local_frame.LocalCoFrame
Trivialization coframe on a vector bundle.
A trivialization coframe is the coframe of the trivialization frame induced by a trivialization (see:
TrivializationFrame
).More precisely, a trivialization frame on a vector bundle \(E \to M\) of class \(C^k\) and rank \(n\) over the topological field \(K\) and over a topological manifold \(M\) is a local coframe induced by a local trivialization \(\varphi:E|_U \to U \times K^n\) of the domain \(U \in M\). Namely, the local dual sections
\[\varphi^*e^i := \varphi(\;\cdot\;, e^i)\]on \(U\) induce a local frame \((\varphi^*e^1, \dots, \varphi^*e^n)\), where \((e^1, \dots, e^n)\) is the dual of the standard basis of \(K^n\).
INPUT:
triv_frame
– trivialization frame dual to the trivialization coframesymbol
– either a string, to be used as a common base for the symbols of the dual sections constituting the coframe, or a tuple of strings, representing the individual symbols of the dual sectionslatex_symbol
– (default:None
) either a string, to be used as a common base for the LaTeX symbols of the dual sections constituting the coframe, or a tuple of strings, representing the individual LaTeX symbols of the dual sections; ifNone
,symbol
is used in place oflatex_symbol
indices
– (default:None
; used only ifsymbol
is a single string) tuple of strings representing the indices labelling the dual sections of the coframe; ifNone
, the indices will be generated as integers within the range declared on the local frame’s domainlatex_indices
– (default:None
) tuple of strings representing the indices for the LaTeX symbols of the dual sections of the coframe; ifNone
,indices
is used instead
EXAMPLES:
Trivialization coframe on a trivial vector bundle of rank 3:
sage: M = Manifold(3, 'M', start_index=1, structure='top') sage: X.<x,y,z> = M.chart() sage: E = M.vector_bundle(3, 'E') sage: phi = E.trivialization('phi'); phi Trivialization (phi, E|_M) sage: E.frames() [Trivialization frame (E|_M, ((phi^*e_1),(phi^*e_2),(phi^*e_3)))] sage: E.coframes() [Trivialization coframe (E|_M, ((phi^*e^1),(phi^*e^2),(phi^*e^3)))] sage: f = E.coframes()[0] ; f Trivialization coframe (E|_M, ((phi^*e^1),(phi^*e^2),(phi^*e^3)))
The linear forms composing the coframe are obtained via the operator
[]
:sage: f[1] Linear form (phi^*e^1) on the Free module C^0(M;E) of sections on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3 sage: f[2] Linear form (phi^*e^2) on the Free module C^0(M;E) of sections on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3 sage: f[3] Linear form (phi^*e^3) on the Free module C^0(M;E) of sections on the 3-dimensional topological manifold M with values in the real vector bundle E of rank 3 sage: f[1][:] [1, 0, 0] sage: f[2][:] [0, 1, 0] sage: f[3][:] [0, 0, 1]
The coframe is the dual of the trivialization frame:
sage: e = phi.frame() ; e Trivialization frame (E|_M, ((phi^*e_1),(phi^*e_2),(phi^*e_3))) sage: f[1](e[1]).expr(), f[1](e[2]).expr(), f[1](e[3]).expr() (1, 0, 0) sage: f[2](e[1]).expr(), f[2](e[2]).expr(), f[2](e[3]).expr() (0, 1, 0) sage: f[3](e[1]).expr(), f[3](e[2]).expr(), f[3](e[3]).expr() (0, 0, 1)
- class sage.manifolds.local_frame.TrivializationFrame(trivialization)#
Bases:
sage.manifolds.local_frame.LocalFrame
Trivialization frame on a topological vector bundle.
A trivialization frame on a topological vector bundle \(E \to M\) of rank \(n\) over the topological field \(K\) and over a topological manifold \(M\) is a local frame induced by a local trivialization \(\varphi:E|_U \to U \times K^n\) of the domain \(U \in M\). More precisely, the local sections
\[\varphi^*e_i := \varphi(\;\cdot\;, e_i)\]on \(U\) induce a local frame \((\varphi^*e_1, \dots, \varphi^*e_n)\), where \((e_1, \dots, e_n)\) is the standard basis of \(K^n\).
INPUT:
trivialization
– the trivialization defined on the vector bundle
EXAMPLES:
sage: M = Manifold(3, 'M') sage: U = M.open_subset('U') sage: E = M.vector_bundle(2, 'E') sage: phi_U = E.trivialization('phi_U', domain=U) sage: phi_U.frame() Trivialization frame (E|_U, ((phi_U^*e_1),(phi_U^*e_2))) sage: latex(phi_U.frame()) \left(E|_{U}, \left(\left(phi_U^* e_{ 1 }\right),\left(phi_U^* e_{ 2 }\right)\right)\right)
- trivialization()#
Return the underlying trivialization of
self
.EXAMPLES:
sage: M = Manifold(3, 'M') sage: U = M.open_subset('U') sage: E = M.vector_bundle(2, 'E') sage: phi_U = E.trivialization('phi_U', domain=U) sage: e = phi_U.frame() sage: e.trivialization() Trivialization (phi_U, E|_U) sage: e.trivialization() is phi_U True