Sections#

The class Section implements sections on vector bundles. The derived class TrivialSection is devoted to sections on trivial parts of a vector bundle.

AUTHORS:

  • Michael Jung (2019): initial version

class sage.manifolds.section.Section(section_module, name=None, latex_name=None)[source]#

Bases: ModuleElementWithMutability

Section in a vector bundle.

An instance of this class is a section in a vector bundle \(E \to M\) of class \(C^k\), where \(E|_U\) is not manifestly trivial. More precisely, a (local) section on a subset \(U \in M\) is a map of class \(C^k\)

\[s: U \longrightarrow E\]

such that

\[\forall p \in U,\ s(p) \in E_p\]

where \(E_p\) denotes the vector bundle fiber of \(E\) over the point \(p \in U\).

If \(E|_U\) is trivial, the class TrivialSection should be used instead.

This is a Sage element class, the corresponding parent class being SectionModule.

INPUT:

  • section_module – module \(C^k(U;E)\) of sections on \(E\) over \(U\) (cf. SectionModule)

  • name – (default: None) name given to the section

  • latex_name – (default: None) LaTeX symbol to denote the section; if none is provided, the LaTeX symbol is set to name

EXAMPLES:

A section on a non-trivial rank 2 vector bundle over a non-trivial 2-manifold:

sage: M = Manifold(2, 'M', structure='top')
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V)   # M is the union of U and V
sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart()
sage: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
....:                    intersection_name='W', restrictions1= x>0,
....:                    restrictions2= u+v>0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V)
sage: E = M.vector_bundle(2, 'E') # define the vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,x],[x,0]]) # transition map between trivializations
sage: fU = phi_U.frame(); fV = phi_V.frame() # define induced frames
sage: s = E.section(name='s'); s
Section s on the 2-dimensional topological manifold M with values in the
 real vector bundle E of rank 2

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> 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)
>>> xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
...                    intersection_name='W', restrictions1= x>Integer(0),
...                    restrictions2= u+v>Integer(0))
>>> uv_to_xy = xy_to_uv.inverse()
>>> W = U.intersection(V)
>>> E = M.vector_bundle(Integer(2), 'E') # define the vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),x],[x,Integer(0)]]) # transition map between trivializations
>>> fU = phi_U.frame(); fV = phi_V.frame() # define induced frames
>>> s = E.section(name='s'); s
Section s on the 2-dimensional topological manifold M with values in the
 real vector bundle E of rank 2

The parent of \(s\) is not a free module, since \(E\) is not trivial:

sage: isinstance(s.parent(), FiniteRankFreeModule)
False

>>> from sage.all import *
>>> isinstance(s.parent(), FiniteRankFreeModule)
False

To fully define \(s\), we have to specify its components in some local frames defined on the trivial parts of \(E\). The components consist of scalar fields defined on the corresponding domain. Let us start with \(E|_U\):

sage: s[fU,:] = [x^2, 1-y]
sage: s.display(fU)
s = x^2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)

>>> from sage.all import *
>>> s[fU,:] = [x**Integer(2), Integer(1)-y]
>>> s.display(fU)
s = x^2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)

To set the components of \(s\) on \(V\) consistently, we copy the expressions of the components in the common subset \(W\):

sage: fUW = fU.restrict(W); fVW = fV.restrict(W)
sage: c_uvW = c_uv.restrict(W)
sage: s[fV,0] = s[fVW,0,c_uvW].expr()  # long time
sage: s[fV,1] = s[fVW,1,c_uvW].expr()  # long time

>>> from sage.all import *
>>> fUW = fU.restrict(W); fVW = fV.restrict(W)
>>> c_uvW = c_uv.restrict(W)
>>> s[fV,Integer(0)] = s[fVW,Integer(0),c_uvW].expr()  # long time
>>> s[fV,Integer(1)] = s[fVW,Integer(1),c_uvW].expr()  # long time

Actually, the operation above can be performed in a single line by means of the method add_comp_by_continuation():

sage: s.add_comp_by_continuation(fV, W, chart=c_uv)

>>> from sage.all import *
>>> s.add_comp_by_continuation(fV, W, chart=c_uv)

At this stage, \(s\) is fully defined, having components in frames fU and fV and the union of the domains of fU and fV being the whole manifold:

sage: s.display(fV)
s = (-1/4*u^2 + 1/4*v^2 + 1/2*u + 1/2*v) (phi_V^*e_1)
    + (1/8*u^3 + 3/8*u^2*v + 3/8*u*v^2 + 1/8*v^3) (phi_V^*e_2)

>>> from sage.all import *
>>> s.display(fV)
s = (-1/4*u^2 + 1/4*v^2 + 1/2*u + 1/2*v) (phi_V^*e_1)
    + (1/8*u^3 + 3/8*u^2*v + 3/8*u*v^2 + 1/8*v^3) (phi_V^*e_2)

Sections can be pointwisely added:

sage: t = E.section([x,y], frame=fU, name='t'); t
Section t on the 2-dimensional topological manifold M with values in the
 real vector bundle E of rank 2
sage: t.add_comp_by_continuation(fV, W, chart=c_uv)
sage: t.display(fV)
t = (1/4*u^2 - 1/4*v^2) (phi_V^*e_1) + (1/4*u^2 + 1/2*u*v + 1/4*v^2) (phi_V^*e_2)
sage: a = s + t; a
Section s+t on the 2-dimensional topological manifold M with values
 in the real vector bundle E of rank 2
sage: a.display(fU)
s+t = (x^2 + x) (phi_U^*e_1) + (phi_U^*e_2)
sage: a.display(fV)
s+t = (1/2*u + 1/2*v) (phi_V^*e_1) + (1/8*u^3 + 1/8*(3*u + 2)*v^2
      + 1/8*v^3 + 1/4*u^2 + 1/8*(3*u^2 + 4*u)*v) (phi_V^*e_2)

>>> from sage.all import *
>>> t = E.section([x,y], frame=fU, name='t'); t
Section t on the 2-dimensional topological manifold M with values in the
 real vector bundle E of rank 2
>>> t.add_comp_by_continuation(fV, W, chart=c_uv)
>>> t.display(fV)
t = (1/4*u^2 - 1/4*v^2) (phi_V^*e_1) + (1/4*u^2 + 1/2*u*v + 1/4*v^2) (phi_V^*e_2)
>>> a = s + t; a
Section s+t on the 2-dimensional topological manifold M with values
 in the real vector bundle E of rank 2
>>> a.display(fU)
s+t = (x^2 + x) (phi_U^*e_1) + (phi_U^*e_2)
>>> a.display(fV)
s+t = (1/2*u + 1/2*v) (phi_V^*e_1) + (1/8*u^3 + 1/8*(3*u + 2)*v^2
      + 1/8*v^3 + 1/4*u^2 + 1/8*(3*u^2 + 4*u)*v) (phi_V^*e_2)

and multiplied by scalar fields:

sage: f = M.scalar_field(y^2-x^2, name='f')
sage: f.add_expr_by_continuation(c_uv, W)
sage: f.display()
f: M → ℝ
on U: (x, y) ↦ -x^2 + y^2
on V: (u, v) ↦ -u*v
sage: b = f*s; b
Section f*s on the 2-dimensional topological manifold M with values
 in the real vector bundle E of rank 2
sage: b.display(fU)
f*s = (-x^4 + x^2*y^2) (phi_U^*e_1) + (x^2*y - y^3 - x^2 + y^2) (phi_U^*e_2)
sage: b.display(fV)
f*s = (-1/4*u*v^3 - 1/2*u*v^2 + 1/4*(u^3 - 2*u^2)*v) (phi_V^*e_1)
      + (-1/8*u^4*v - 3/8*u^3*v^2 - 3/8*u^2*v^3 - 1/8*u*v^4) (phi_V^*e_2)

>>> from sage.all import *
>>> f = M.scalar_field(y**Integer(2)-x**Integer(2), name='f')
>>> f.add_expr_by_continuation(c_uv, W)
>>> f.display()
f: M → ℝ
on U: (x, y) ↦ -x^2 + y^2
on V: (u, v) ↦ -u*v
>>> b = f*s; b
Section f*s on the 2-dimensional topological manifold M with values
 in the real vector bundle E of rank 2
>>> b.display(fU)
f*s = (-x^4 + x^2*y^2) (phi_U^*e_1) + (x^2*y - y^3 - x^2 + y^2) (phi_U^*e_2)
>>> b.display(fV)
f*s = (-1/4*u*v^3 - 1/2*u*v^2 + 1/4*(u^3 - 2*u^2)*v) (phi_V^*e_1)
      + (-1/8*u^4*v - 3/8*u^3*v^2 - 3/8*u^2*v^3 - 1/8*u*v^4) (phi_V^*e_2)

The domain on which the section should be defined, can be stated via the domain option in section():

sage: cU = E.section([1,x], domain=U, name='c'); cU
Section c on the Open subset U of the 2-dimensional topological manifold
 M with values in the real vector bundle E of rank 2
sage: cU.display()
c = (phi_U^*e_1) + x (phi_U^*e_2)

>>> from sage.all import *
>>> cU = E.section([Integer(1),x], domain=U, name='c'); cU
Section c on the Open subset U of the 2-dimensional topological manifold
 M with values in the real vector bundle E of rank 2
>>> cU.display()
c = (phi_U^*e_1) + x (phi_U^*e_2)

Since \(E|_U\) is trivial, cU now belongs to the free module:

sage: isinstance(cU.parent(), FiniteRankFreeModule)
True

>>> from sage.all import *
>>> isinstance(cU.parent(), FiniteRankFreeModule)
True

Omitting the domain option, the section is defined on the whole base space:

sage: c = E.section(name='c'); c
Section c on the 2-dimensional topological manifold M with values in the
 real vector bundle E of rank 2

>>> from sage.all import *
>>> c = E.section(name='c'); c
Section c on the 2-dimensional topological manifold M with values in the
 real vector bundle E of rank 2

Via set_restriction(), cU can be defined as the restriction of c to \(U\):

sage: c.set_restriction(cU)
sage: c.display(fU)
c = (phi_U^*e_1) + x (phi_U^*e_2)
sage: c.restrict(U) == cU
True

>>> from sage.all import *
>>> c.set_restriction(cU)
>>> c.display(fU)
c = (phi_U^*e_1) + x (phi_U^*e_2)
>>> c.restrict(U) == cU
True

Notice that the zero section is immutable, and therefore its components cannot be changed:

sage: zer = E.section_module().zero()
sage: zer.is_immutable()
True
sage: zer.set_comp()
Traceback (most recent call last):
...
ValueError: the components of an immutable element cannot be
 changed

>>> from sage.all import *
>>> zer = E.section_module().zero()
>>> zer.is_immutable()
True
>>> zer.set_comp()
Traceback (most recent call last):
...
ValueError: the components of an immutable element cannot be
 changed

Other sections can be declared immutable, too:

sage: c.is_immutable()
False
sage: c.set_immutable()
sage: c.is_immutable()
True
sage: c.set_comp()
Traceback (most recent call last):
...
ValueError: the components of an immutable element cannot be
 changed
sage: c.set_name('b')
Traceback (most recent call last):
...
ValueError: the name of an immutable element cannot be changed

>>> from sage.all import *
>>> c.is_immutable()
False
>>> c.set_immutable()
>>> c.is_immutable()
True
>>> c.set_comp()
Traceback (most recent call last):
...
ValueError: the components of an immutable element cannot be
 changed
>>> c.set_name('b')
Traceback (most recent call last):
...
ValueError: the name of an immutable element cannot be changed
add_comp(basis=None)[source]#

Return the components of self in a given local frame for assignment.

The components with respect to other frames having the same domain as the provided local frame are kept. To delete them, use the method set_comp() instead.

INPUT:

  • basis – (default: None) local frame in which the components are defined; if None, the components are assumed to refer to the section domain’s default frame

OUTPUT:

  • components in the given frame, as a Components; if such components did not exist previously, they are created

EXAMPLES:

sage: S2 = Manifold(2, 'S^2', structure='top', start_index=1)
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: xy_to_uv = stereoN.transition_map(stereoS, (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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E') # define vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,1],[1,0]])
sage: fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
sage: s = E.section(name='s')
sage: s.add_comp(fS)
1-index components w.r.t. Trivialization frame (E|_V, ((phi_V^*e_1),(phi_V^*e_2)))
sage: s.add_comp(fS)[1] = u+v
sage: s.display(fS)
s = (u + v) (phi_V^*e_1)

>>> from sage.all import *
>>> S2 = Manifold(Integer(2), 'S^2', structure='top', start_index=Integer(1))
>>> U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
>>> S2.declare_union(U,V)
>>> stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)# stereographic coordinates from the North pole
>>> stereoS = V.chart(names=('u', 'v',)); (u, v,) = stereoS._first_ngens(2)# stereographic coordinates from the South pole
>>> xy_to_uv = stereoN.transition_map(stereoS, (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))
>>> W = U.intersection(V)
>>> uv_to_xy = xy_to_uv.inverse()
>>> E = S2.vector_bundle(Integer(2), 'E') # define vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),Integer(1)],[Integer(1),Integer(0)]])
>>> fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
>>> s = E.section(name='s')
>>> s.add_comp(fS)
1-index components w.r.t. Trivialization frame (E|_V, ((phi_V^*e_1),(phi_V^*e_2)))
>>> s.add_comp(fS)[Integer(1)] = u+v
>>> s.display(fS)
s = (u + v) (phi_V^*e_1)

Setting the components in a new frame:

sage: e = E.local_frame('e', domain=V)
sage: s.add_comp(e)
1-index components w.r.t. Local frame (E|_V, (e_1,e_2))
sage: s.add_comp(e)[1] = u*v
sage: s.display(e)
s = u*v e_1

>>> from sage.all import *
>>> e = E.local_frame('e', domain=V)
>>> s.add_comp(e)
1-index components w.r.t. Local frame (E|_V, (e_1,e_2))
>>> s.add_comp(e)[Integer(1)] = u*v
>>> s.display(e)
s = u*v e_1

The components with respect to fS are kept:

sage: s.display(fS)
s = (u + v) (phi_V^*e_1)

>>> from sage.all import *
>>> s.display(fS)
s = (u + v) (phi_V^*e_1)
add_comp_by_continuation(frame, subdomain, chart=None)[source]#

Set components with respect to a local frame by continuation of the coordinate expression of the components in a subframe.

The continuation is performed by demanding that the components have the same coordinate expression as those on the restriction of the frame to a given subdomain.

INPUT:

  • frame – local frame \(e\) in which the components are to be set

  • subdomain – open subset of \(e\)’s domain in which the components are known or can be evaluated from other components

  • chart – (default: None) coordinate chart on \(e\)’s domain in which the extension of the expression of the components is to be performed; if None, the default’s chart of \(e\)’s domain is assumed

EXAMPLES:

Components of a vector field on the sphere \(S^2\):

sage: S2 = Manifold(2, 'S^2', structure='top', start_index=1)
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: xy_to_uv = stereoN.transition_map(stereoS,
....:                                   (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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E') # define vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,1],[1,0]])
sage: fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
sage: a = E.section({fN: [x, 2+y]}, name='a')

>>> from sage.all import *
>>> S2 = Manifold(Integer(2), 'S^2', structure='top', start_index=Integer(1))
>>> U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
>>> S2.declare_union(U,V)
>>> stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)# stereographic coordinates from the North pole
>>> stereoS = V.chart(names=('u', 'v',)); (u, v,) = stereoS._first_ngens(2)# stereographic coordinates from the South pole
>>> xy_to_uv = stereoN.transition_map(stereoS,
...                                   (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))
>>> W = U.intersection(V)
>>> uv_to_xy = xy_to_uv.inverse()
>>> E = S2.vector_bundle(Integer(2), 'E') # define vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),Integer(1)],[Integer(1),Integer(0)]])
>>> fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
>>> a = E.section({fN: [x, Integer(2)+y]}, name='a')

At this stage, the section has been defined only on the open subset U (through its components in the frame fN):

sage: a.display(fN)
a = x (phi_U^*e_1) + (y + 2) (phi_U^*e_2)

>>> from sage.all import *
>>> a.display(fN)
a = x (phi_U^*e_1) + (y + 2) (phi_U^*e_2)

The components with respect to the restriction of fS to the common subdomain W, in terms of the (u,v) coordinates, are obtained by a change-of-frame formula on W:

sage: a.display(fS.restrict(W), stereoS.restrict(W))
a = (2*u^2 + 2*v^2 + v)/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2)
 (phi_V^*e_2)

>>> from sage.all import *
>>> a.display(fS.restrict(W), stereoS.restrict(W))
a = (2*u^2 + 2*v^2 + v)/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2)
 (phi_V^*e_2)

The continuation consists in extending the definition of the vector field to the whole open subset V by demanding that the components in the frame eV have the same coordinate expression as the above one:

sage: a.add_comp_by_continuation(fS, W, chart=stereoS)

>>> from sage.all import *
>>> a.add_comp_by_continuation(fS, W, chart=stereoS)

We have then:

sage: a.display(fS)
a = (2*u^2 + 2*v^2 + v)/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2)
 (phi_V^*e_2)

>>> from sage.all import *
>>> a.display(fS)
a = (2*u^2 + 2*v^2 + v)/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2)
 (phi_V^*e_2)

and \(a\) is defined on the entire manifold \(S^2\).

add_expr_from_subdomain(frame, subdomain)[source]#

Add an expression to an existing component from a subdomain.

INPUT:

  • frame – local frame \(e\) in which the components are to be set

  • subdomain – open subset of \(e\)’s domain in which the components have additional expressions.

EXAMPLES:

We are going to consider a section on the trivial rank 2 vector bundle over the 2-sphere:

sage: S2 = Manifold(2, 'S^2', structure='top', start_index=1)
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: xy_to_uv = stereoN.transition_map(stereoS,
....:              (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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E') # define vector bundle
sage: e = E.local_frame('e') # frame to trivialize E
sage: eU = e.restrict(U); eV = e.restrict(V); eW = e.restrict(W) # this step is essential since U, V and W must be trivial

>>> from sage.all import *
>>> S2 = Manifold(Integer(2), 'S^2', structure='top', start_index=Integer(1))
>>> U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
>>> S2.declare_union(U,V)
>>> stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)# stereographic coordinates from the North pole
>>> stereoS = V.chart(names=('u', 'v',)); (u, v,) = stereoS._first_ngens(2)# stereographic coordinates from the South pole
>>> xy_to_uv = stereoN.transition_map(stereoS,
...              (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))
>>> W = U.intersection(V)
>>> uv_to_xy = xy_to_uv.inverse()
>>> E = S2.vector_bundle(Integer(2), 'E') # define vector bundle
>>> e = E.local_frame('e') # frame to trivialize E
>>> eU = e.restrict(U); eV = e.restrict(V); eW = e.restrict(W) # this step is essential since U, V and W must be trivial

To define a section s on \(S^2\), we first set the components on U:

sage: s = E.section(name='s')
sage: sU = s.restrict(U)
sage: sU[:] = [x, y]

>>> from sage.all import *
>>> s = E.section(name='s')
>>> sU = s.restrict(U)
>>> sU[:] = [x, y]

But because E is trivial, these components can be extended with respect to the global frame e onto \(S^2\):

sage: s.add_comp_by_continuation(e, U)

>>> from sage.all import *
>>> s.add_comp_by_continuation(e, U)

One can see that s is not yet fully defined: the components (scalar fields) do not have values on the whole manifold:

sage: sorted(s._components.values())[0]._comp[(1,)].display()
S^2 → ℝ
on U: (x, y) ↦ x
on W: (u, v) ↦ u/(u^2 + v^2)

>>> from sage.all import *
>>> sorted(s._components.values())[Integer(0)]._comp[(Integer(1),)].display()
S^2 → ℝ
on U: (x, y) ↦ x
on W: (u, v) ↦ u/(u^2 + v^2)

To fix that, we extend the components from W to V first, using add_comp_by_continuation():

sage: s.add_comp_by_continuation(eV, W, stereoS)

>>> from sage.all import *
>>> s.add_comp_by_continuation(eV, W, stereoS)

Then, the expression on the subdomain V is added to the components on \(S^2\) already known by:

sage: s.add_expr_from_subdomain(e, V)

>>> from sage.all import *
>>> s.add_expr_from_subdomain(e, V)

The definition of s is now complete:

sage: sorted(s._components.values())[0]._comp[(2,)].display()
S^2 → ℝ
on U: (x, y) ↦ y
on V: (u, v) ↦ v/(u^2 + v^2)

>>> from sage.all import *
>>> sorted(s._components.values())[Integer(0)]._comp[(Integer(2),)].display()
S^2 → ℝ
on U: (x, y) ↦ y
on V: (u, v) ↦ v/(u^2 + v^2)
at(point)[source]#

Value of self at a point of its domain.

If the current section is

\[s:\ U \longrightarrow E ,\]

then for any point \(p \in U\), \(s(p)\) is a vector in the fiber \(E_p\) of \(E\) at \(p\).

INPUT:

  • point – ManifoldPoint; point \(p\) in the domain of the section \(U\)

OUTPUT:

EXAMPLES:

Vector on a rank 2 vector bundle fiber over a non-parallelizable 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='top')
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: E = M.vector_bundle(2, 'E') # define vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,x],[x,0]])
sage: fU = phi_U.frame(); fV = phi_V.frame() # get induced frames
sage: s = E.section({fU: [1+y, x]}, name='s')
sage: s.add_comp_by_continuation(fV, W, chart=c_uv)
sage: s.display(fU)
s = (y + 1) (phi_U^*e_1) + x (phi_U^*e_2)
sage: s.display(fV)
s = (1/4*u^2 + 1/2*u*v + 1/4*v^2) (phi_V^*e_1) + (1/4*u^2 - 1/4*v^2
 + 1/2*u + 1/2*v) (phi_V^*e_2)
sage: p = M.point((2,3), chart=c_xy, name='p')
sage: sp = s.at(p) ; sp
Vector s in the fiber of E at Point p on the 2-dimensional
 topological manifold M
sage: sp.parent()
Fiber of E at Point p on the 2-dimensional topological manifold M
sage: sp.display(fU.at(p))
s = 4 (phi_U^*e_1) + 2 (phi_U^*e_2)
sage: sp.display(fV.at(p))
s = 4 (phi_V^*e_1) + 8 (phi_V^*e_2)
sage: p.coord(c_uv) # to check the above expression
(5, -1)

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> 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)
>>> E = M.vector_bundle(Integer(2), 'E') # define vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),x],[x,Integer(0)]])
>>> fU = phi_U.frame(); fV = phi_V.frame() # get induced frames
>>> s = E.section({fU: [Integer(1)+y, x]}, name='s')
>>> s.add_comp_by_continuation(fV, W, chart=c_uv)
>>> s.display(fU)
s = (y + 1) (phi_U^*e_1) + x (phi_U^*e_2)
>>> s.display(fV)
s = (1/4*u^2 + 1/2*u*v + 1/4*v^2) (phi_V^*e_1) + (1/4*u^2 - 1/4*v^2
 + 1/2*u + 1/2*v) (phi_V^*e_2)
>>> p = M.point((Integer(2),Integer(3)), chart=c_xy, name='p')
>>> sp = s.at(p) ; sp
Vector s in the fiber of E at Point p on the 2-dimensional
 topological manifold M
>>> sp.parent()
Fiber of E at Point p on the 2-dimensional topological manifold M
>>> sp.display(fU.at(p))
s = 4 (phi_U^*e_1) + 2 (phi_U^*e_2)
>>> sp.display(fV.at(p))
s = 4 (phi_V^*e_1) + 8 (phi_V^*e_2)
>>> p.coord(c_uv) # to check the above expression
(5, -1)
base_module()[source]#

Return the section module on which self acts as a section.

OUTPUT:

EXAMPLES:

sage: M = Manifold(3, 'M', structure='top')
sage: U = M.open_subset('U')
sage: E = M.vector_bundle(2, 'E')
sage: s = E.section(domain=U)
sage: s.base_module()
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 2

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='top')
>>> U = M.open_subset('U')
>>> E = M.vector_bundle(Integer(2), 'E')
>>> s = E.section(domain=U)
>>> s.base_module()
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 2
comp(basis=None, from_basis=None)[source]#

Return the components in a given local frame.

If the components are not known already, they are computed by the change-of-basis formula from components in another local frame.

INPUT:

  • basis – (default: None) local frame in which the components are required; if none is provided, the components are assumed to refer to the section module’s default frame on the corresponding domain

  • from_basis – (default: None) local frame from which the required components are computed, via the change-of-basis formula, if they are not known already in the basis basis

OUTPUT:

  • components in the local frame basis, as a Components

EXAMPLES:

Components of a section defined on a rank 2 vector bundle over two open subsets:

sage: M = Manifold(2, 'M', structure='top')
sage: X.<x, y> = M.chart()
sage: U = M.open_subset('U'); V = M.open_subset('V')
sage: M.declare_union(U, V)
sage: XU = X.restrict(U); XV = X.restrict(V)
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: f = E.local_frame('f', domain=V); f
Local frame (E|_V, (f_0,f_1))
sage: s = E.section(name='s')
sage: s[e,:] = - x + y^3, 2+x
sage: s[f,0] = x^2
sage: s[f,1] = x+y
sage: s.comp(e)
1-index components w.r.t. Local frame (E|_U, (e_0,e_1))
sage: s.comp(e)[:]
[y^3 - x, x + 2]
sage: s.comp(f)
1-index components w.r.t. Local frame (E|_V, (f_0,f_1))
sage: s.comp(f)[:]
[x^2, x + y]

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> U = M.open_subset('U'); V = M.open_subset('V')
>>> M.declare_union(U, V)
>>> XU = X.restrict(U); XV = X.restrict(V)
>>> E = M.vector_bundle(Integer(2), 'E')
>>> e = E.local_frame('e', domain=U); e
Local frame (E|_U, (e_0,e_1))
>>> f = E.local_frame('f', domain=V); f
Local frame (E|_V, (f_0,f_1))
>>> s = E.section(name='s')
>>> s[e,:] = - x + y**Integer(3), Integer(2)+x
>>> s[f,Integer(0)] = x**Integer(2)
>>> s[f,Integer(1)] = x+y
>>> s.comp(e)
1-index components w.r.t. Local frame (E|_U, (e_0,e_1))
>>> s.comp(e)[:]
[y^3 - x, x + 2]
>>> s.comp(f)
1-index components w.r.t. Local frame (E|_V, (f_0,f_1))
>>> s.comp(f)[:]
[x^2, x + y]

Since e is the default frame of E|_U, the argument e can be omitted after restricting:

sage: e is E.section_module(domain=U).default_frame()
True
sage: s.restrict(U).comp() is s.comp(e)
True

>>> from sage.all import *
>>> e is E.section_module(domain=U).default_frame()
True
>>> s.restrict(U).comp() is s.comp(e)
True
copy(name=None, latex_name=None)[source]#

Return an exact copy of self.

INPUT:

  • name – (default: None) name given to the copy

  • latex_name – (default: None) LaTeX symbol to denote the copy; if none is provided, the LaTeX symbol is set to name

Note

The name and the derived quantities are not copied.

EXAMPLES:

Copy of a section on a rank 2 vector bundle over a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='top')
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V)   # M is the union of U and V
sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart()
sage: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
....:                    intersection_name='W', restrictions1= x>0,
....:                    restrictions2= u+v>0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V)
sage: E = M.vector_bundle(2, 'E') # define vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,x],[x,0]])
sage: fU = phi_U.frame(); fV = phi_V.frame()
sage: s = E.section(name='s')
sage: s[fU,:] = [2, 1-y]
sage: s.add_comp_by_continuation(fV, U.intersection(V), c_uv)
sage: t = s.copy(); t
Section on the 2-dimensional topological manifold M with values in
 the real vector bundle E of rank 2
sage: t.display(fU)
2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
sage: t == s
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> 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)
>>> xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
...                    intersection_name='W', restrictions1= x>Integer(0),
...                    restrictions2= u+v>Integer(0))
>>> uv_to_xy = xy_to_uv.inverse()
>>> W = U.intersection(V)
>>> E = M.vector_bundle(Integer(2), 'E') # define vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),x],[x,Integer(0)]])
>>> fU = phi_U.frame(); fV = phi_V.frame()
>>> s = E.section(name='s')
>>> s[fU,:] = [Integer(2), Integer(1)-y]
>>> s.add_comp_by_continuation(fV, U.intersection(V), c_uv)
>>> t = s.copy(); t
Section on the 2-dimensional topological manifold M with values in
 the real vector bundle E of rank 2
>>> t.display(fU)
2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
>>> t == s
True

If the original section is modified, the copy is not:

sage: s[fU,0] = -1
sage: s.display(fU)
s = -(phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
sage: t.display(fU)
2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
sage: t == s
False

>>> from sage.all import *
>>> s[fU,Integer(0)] = -Integer(1)
>>> s.display(fU)
s = -(phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
>>> t.display(fU)
2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
>>> t == s
False
copy_from(other)[source]#

Make self a copy of other.

INPUT:

  • other – other section, in the same module as self

Note

While the derived quantities are not copied, the name is kept.

Warning

All previous defined components and restrictions will be deleted!

EXAMPLES:

sage: M = Manifold(2, 'M', structure='top')
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V)   # M is the union of U and V
sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart()
sage: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
....:                    intersection_name='W', restrictions1= x>0,
....:                    restrictions2= u+v>0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V)
sage: E = M.vector_bundle(2, 'E') # define vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,x],[x,0]])
sage: fU = phi_U.frame(); fV = phi_V.frame()
sage: s = E.section(name='s')
sage: s[fU,:] = [2, 1-y]
sage: s.add_comp_by_continuation(fV, U.intersection(V), c_uv)
sage: t = E.section(name='t')
sage: t.copy_from(s)
sage: t.display(fU)
t = 2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
sage: s == t
True

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> 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)
>>> xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
...                    intersection_name='W', restrictions1= x>Integer(0),
...                    restrictions2= u+v>Integer(0))
>>> uv_to_xy = xy_to_uv.inverse()
>>> W = U.intersection(V)
>>> E = M.vector_bundle(Integer(2), 'E') # define vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),x],[x,Integer(0)]])
>>> fU = phi_U.frame(); fV = phi_V.frame()
>>> s = E.section(name='s')
>>> s[fU,:] = [Integer(2), Integer(1)-y]
>>> s.add_comp_by_continuation(fV, U.intersection(V), c_uv)
>>> t = E.section(name='t')
>>> t.copy_from(s)
>>> t.display(fU)
t = 2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
>>> s == t
True

If the original section is modified, the copy is not:

sage: s[fU,0] = -1
sage: s.display(fU)
s = -(phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
sage: t.display(fU)
t = 2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
sage: s == t
False

>>> from sage.all import *
>>> s[fU,Integer(0)] = -Integer(1)
>>> s.display(fU)
s = -(phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
>>> t.display(fU)
t = 2 (phi_U^*e_1) + (-y + 1) (phi_U^*e_2)
>>> s == t
False
disp(frame=None, chart=None)[source]#

Display the section in terms of its expansion with respect to a given local frame.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

INPUT:

  • frame – (default: None) local frame with respect to which the section is expanded; if frame is None and chart is not None, the default frame in the corresponding section module is assumed

  • chart – (default: None) chart with respect to which the components of the section in the selected frame are expressed; if None, the default chart of the local frame domain is assumed

EXAMPLES:

Display of section on a rank 2 vector bundle over the 2-sphere:

sage: S2 = Manifold(2, 'S^2', structure='top', start_index=1)
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: xy_to_uv = stereoN.transition_map(stereoS,
....:                                   (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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E') # define vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,1],[1,0]])
sage: fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
sage: s = E.section(name='s')
sage: s[fN,:] = [x, y]
sage: s.add_comp_by_continuation(fS, W, stereoS)
sage: s.display(fN)
s = x (phi_U^*e_1) + y (phi_U^*e_2)
sage: s.display(fS)
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

>>> from sage.all import *
>>> S2 = Manifold(Integer(2), 'S^2', structure='top', start_index=Integer(1))
>>> U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
>>> S2.declare_union(U,V)
>>> stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)# stereographic coordinates from the North pole
>>> stereoS = V.chart(names=('u', 'v',)); (u, v,) = stereoS._first_ngens(2)# stereographic coordinates from the South pole
>>> xy_to_uv = stereoN.transition_map(stereoS,
...                                   (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))
>>> W = U.intersection(V)
>>> uv_to_xy = xy_to_uv.inverse()
>>> E = S2.vector_bundle(Integer(2), 'E') # define vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),Integer(1)],[Integer(1),Integer(0)]])
>>> fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
>>> s = E.section(name='s')
>>> s[fN,:] = [x, y]
>>> s.add_comp_by_continuation(fS, W, stereoS)
>>> s.display(fN)
s = x (phi_U^*e_1) + y (phi_U^*e_2)
>>> s.display(fS)
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

Since fN is the default frame on E|_U, the argument fN can be omitted after restricting:

sage: fN is E.section_module(domain=U).default_frame()
True
sage: s.restrict(U).display()
s = x (phi_U^*e_1) + y (phi_U^*e_2)

>>> from sage.all import *
>>> fN is E.section_module(domain=U).default_frame()
True
>>> s.restrict(U).display()
s = x (phi_U^*e_1) + y (phi_U^*e_2)

Similarly, since fS is V’s default frame, the argument fS can be omitted when considering the restriction of s to V:

sage: s.restrict(V).display()
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

>>> from sage.all import *
>>> s.restrict(V).display()
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

The second argument comes into play whenever the frame’s domain is covered by two distinct charts. Since stereoN.restrict(W) is the default chart on W, the second argument can be omitted for the expression in this chart:

sage: s.display(fS.restrict(W))
s = y (phi_V^*e_1) + x (phi_V^*e_2)

>>> from sage.all import *
>>> s.display(fS.restrict(W))
s = y (phi_V^*e_1) + x (phi_V^*e_2)

To get the expression in the other chart, the second argument must be used:

sage: s.display(fN.restrict(W), stereoS.restrict(W))
s = u/(u^2 + v^2) (phi_U^*e_1) + v/(u^2 + v^2) (phi_U^*e_2)

>>> from sage.all import *
>>> s.display(fN.restrict(W), stereoS.restrict(W))
s = u/(u^2 + v^2) (phi_U^*e_1) + v/(u^2 + v^2) (phi_U^*e_2)

One can ask for the display with respect to a frame in which s has not been initialized yet (this will automatically trigger the use of the change-of-frame formula for tensors):

sage: a = E.section_module(domain=U).automorphism()
sage: a[:] = [[1+x^2,0],[0,1+y^2]]
sage: e = fN.new_frame(a, 'e')
sage: [e[i].display() for i in S2.irange()]
[e_1 = (x^2 + 1) (phi_U^*e_1), e_2 = (y^2 + 1) (phi_U^*e_2)]
sage: s.display(e)
s = x/(x^2 + 1) e_1 + y/(y^2 + 1) e_2

>>> from sage.all import *
>>> a = E.section_module(domain=U).automorphism()
>>> a[:] = [[Integer(1)+x**Integer(2),Integer(0)],[Integer(0),Integer(1)+y**Integer(2)]]
>>> e = fN.new_frame(a, 'e')
>>> [e[i].display() for i in S2.irange()]
[e_1 = (x^2 + 1) (phi_U^*e_1), e_2 = (y^2 + 1) (phi_U^*e_2)]
>>> s.display(e)
s = x/(x^2 + 1) e_1 + y/(y^2 + 1) e_2

A shortcut of display() is disp():

sage: s.disp(fS)
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

>>> from sage.all import *
>>> s.disp(fS)
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)
display(frame=None, chart=None)[source]#

Display the section in terms of its expansion with respect to a given local frame.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

INPUT:

  • frame – (default: None) local frame with respect to which the section is expanded; if frame is None and chart is not None, the default frame in the corresponding section module is assumed

  • chart – (default: None) chart with respect to which the components of the section in the selected frame are expressed; if None, the default chart of the local frame domain is assumed

EXAMPLES:

Display of section on a rank 2 vector bundle over the 2-sphere:

sage: S2 = Manifold(2, 'S^2', structure='top', start_index=1)
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: xy_to_uv = stereoN.transition_map(stereoS,
....:                                   (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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E') # define vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,1],[1,0]])
sage: fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
sage: s = E.section(name='s')
sage: s[fN,:] = [x, y]
sage: s.add_comp_by_continuation(fS, W, stereoS)
sage: s.display(fN)
s = x (phi_U^*e_1) + y (phi_U^*e_2)
sage: s.display(fS)
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

>>> from sage.all import *
>>> S2 = Manifold(Integer(2), 'S^2', structure='top', start_index=Integer(1))
>>> U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
>>> S2.declare_union(U,V)
>>> stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)# stereographic coordinates from the North pole
>>> stereoS = V.chart(names=('u', 'v',)); (u, v,) = stereoS._first_ngens(2)# stereographic coordinates from the South pole
>>> xy_to_uv = stereoN.transition_map(stereoS,
...                                   (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))
>>> W = U.intersection(V)
>>> uv_to_xy = xy_to_uv.inverse()
>>> E = S2.vector_bundle(Integer(2), 'E') # define vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),Integer(1)],[Integer(1),Integer(0)]])
>>> fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
>>> s = E.section(name='s')
>>> s[fN,:] = [x, y]
>>> s.add_comp_by_continuation(fS, W, stereoS)
>>> s.display(fN)
s = x (phi_U^*e_1) + y (phi_U^*e_2)
>>> s.display(fS)
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

Since fN is the default frame on E|_U, the argument fN can be omitted after restricting:

sage: fN is E.section_module(domain=U).default_frame()
True
sage: s.restrict(U).display()
s = x (phi_U^*e_1) + y (phi_U^*e_2)

>>> from sage.all import *
>>> fN is E.section_module(domain=U).default_frame()
True
>>> s.restrict(U).display()
s = x (phi_U^*e_1) + y (phi_U^*e_2)

Similarly, since fS is V’s default frame, the argument fS can be omitted when considering the restriction of s to V:

sage: s.restrict(V).display()
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

>>> from sage.all import *
>>> s.restrict(V).display()
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

The second argument comes into play whenever the frame’s domain is covered by two distinct charts. Since stereoN.restrict(W) is the default chart on W, the second argument can be omitted for the expression in this chart:

sage: s.display(fS.restrict(W))
s = y (phi_V^*e_1) + x (phi_V^*e_2)

>>> from sage.all import *
>>> s.display(fS.restrict(W))
s = y (phi_V^*e_1) + x (phi_V^*e_2)

To get the expression in the other chart, the second argument must be used:

sage: s.display(fN.restrict(W), stereoS.restrict(W))
s = u/(u^2 + v^2) (phi_U^*e_1) + v/(u^2 + v^2) (phi_U^*e_2)

>>> from sage.all import *
>>> s.display(fN.restrict(W), stereoS.restrict(W))
s = u/(u^2 + v^2) (phi_U^*e_1) + v/(u^2 + v^2) (phi_U^*e_2)

One can ask for the display with respect to a frame in which s has not been initialized yet (this will automatically trigger the use of the change-of-frame formula for tensors):

sage: a = E.section_module(domain=U).automorphism()
sage: a[:] = [[1+x^2,0],[0,1+y^2]]
sage: e = fN.new_frame(a, 'e')
sage: [e[i].display() for i in S2.irange()]
[e_1 = (x^2 + 1) (phi_U^*e_1), e_2 = (y^2 + 1) (phi_U^*e_2)]
sage: s.display(e)
s = x/(x^2 + 1) e_1 + y/(y^2 + 1) e_2

>>> from sage.all import *
>>> a = E.section_module(domain=U).automorphism()
>>> a[:] = [[Integer(1)+x**Integer(2),Integer(0)],[Integer(0),Integer(1)+y**Integer(2)]]
>>> e = fN.new_frame(a, 'e')
>>> [e[i].display() for i in S2.irange()]
[e_1 = (x^2 + 1) (phi_U^*e_1), e_2 = (y^2 + 1) (phi_U^*e_2)]
>>> s.display(e)
s = x/(x^2 + 1) e_1 + y/(y^2 + 1) e_2

A shortcut of display() is disp():

sage: s.disp(fS)
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)

>>> from sage.all import *
>>> s.disp(fS)
s = v/(u^2 + v^2) (phi_V^*e_1) + u/(u^2 + v^2) (phi_V^*e_2)
display_comp(frame=None, chart=None, only_nonzero=True)[source]#

Display the section components with respect to a given frame, one per line.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

INPUT:

  • frame – (default: None) local frame with respect to which the section components are defined; if None, then the default frame on the section module is used

  • chart – (default: None) chart specifying the coordinate expression of the components; if None, the default chart of the section domain is used

  • only_nonzero – (default: True) boolean; if True, only nonzero components are displayed

EXAMPLES:

Display of the components of a section defined on two open subsets:

sage: M = Manifold(2, 'M', structure='top')
sage: U = M.open_subset('U')
sage: c_xy.<x, y> = U.chart()
sage: V = M.open_subset('V')
sage: c_uv.<u, v> = V.chart()
sage: M.declare_union(U,V)   # M is the union of U and V
sage: E = M.vector_bundle(2, 'E')
sage: e = E.local_frame('e', domain=U)
sage: f = E.local_frame('f', domain=V)
sage: s = E.section(name='s')
sage: s[e,0] = - x + y^3
sage: s[e,1] = 2+x
sage: s[f,1] = - u*v
sage: s.display_comp(e)
s^0 = y^3 - x
s^1 = x + 2
sage: s.display_comp(f)
s^1 = -u*v

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> U = M.open_subset('U')
>>> c_xy = U.chart(names=('x', 'y',)); (x, y,) = c_xy._first_ngens(2)
>>> V = M.open_subset('V')
>>> c_uv = V.chart(names=('u', 'v',)); (u, v,) = c_uv._first_ngens(2)
>>> M.declare_union(U,V)   # M is the union of U and V
>>> E = M.vector_bundle(Integer(2), 'E')
>>> e = E.local_frame('e', domain=U)
>>> f = E.local_frame('f', domain=V)
>>> s = E.section(name='s')
>>> s[e,Integer(0)] = - x + y**Integer(3)
>>> s[e,Integer(1)] = Integer(2)+x
>>> s[f,Integer(1)] = - u*v
>>> s.display_comp(e)
s^0 = y^3 - x
s^1 = x + 2
>>> s.display_comp(f)
s^1 = -u*v

See documentation of sage.manifolds.section.TrivialSection.display_comp() for more options.

domain()[source]#

Return the manifold on which self is defined.

OUTPUT:

EXAMPLES:

sage: M = Manifold(3, 'M', structure='top')
sage: U = M.open_subset('U')
sage: E = M.vector_bundle(2, 'E')
sage: C0_U = E.section_module(domain=U, force_free=True)
sage: z = C0_U.zero()
sage: z.domain()
Open subset U of the 3-dimensional topological manifold M

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='top')
>>> U = M.open_subset('U')
>>> E = M.vector_bundle(Integer(2), 'E')
>>> C0_U = E.section_module(domain=U, force_free=True)
>>> z = C0_U.zero()
>>> z.domain()
Open subset U of the 3-dimensional topological manifold M
restrict(subdomain)[source]#

Return the restriction of self to some subdomain.

If the restriction has not been defined yet, it is constructed here.

INPUT:

OUTPUT:

  • Section representing the restriction

EXAMPLES:

Restrictions of a section on a rank 2 vector bundle over the 2-sphere:

sage: S2 = Manifold(2, 'S^2', structure='top', start_index=1)
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: xy_to_uv = stereoN.transition_map(stereoS,
....:                                   (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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E') # define vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,x],[y,0]])
sage: fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
sage: fN_W = fN.restrict(W); fS_W = fS.restrict(W) # restrict them
sage: stereoN_W = stereoN.restrict(W) # restrict charts, too
sage: stereoS_W = stereoS.restrict(W)
sage: s = E.section({fN: [1, 0]}, name='s')
sage: s.display(fN)
s = (phi_U^*e_1)
sage: sU = s.restrict(U) ; sU
Section s on the Open subset U of the 2-dimensional topological
 manifold S^2 with values in the real vector bundle E of rank 2
sage: sU.display() # fN is the default frame on U
s = (phi_U^*e_1)
sage: sU == fN[1]
True
sage: sW = s.restrict(W) ; sW
Section s on the Open subset W of the 2-dimensional topological
 manifold S^2 with values in the real vector bundle E of rank 2
sage: sW.display(fN_W)
s = (phi_U^*e_1)
sage: sW.display(fS_W, stereoN_W)
s = y (phi_V^*e_2)
sage: sW.display(fS_W, stereoS_W)
s = v/(u^2 + v^2) (phi_V^*e_2)
sage: sW == fN_W[1]
True

>>> from sage.all import *
>>> S2 = Manifold(Integer(2), 'S^2', structure='top', start_index=Integer(1))
>>> U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
>>> S2.declare_union(U,V)
>>> stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)# stereographic coordinates from the North pole
>>> stereoS = V.chart(names=('u', 'v',)); (u, v,) = stereoS._first_ngens(2)# stereographic coordinates from the South pole
>>> xy_to_uv = stereoN.transition_map(stereoS,
...                                   (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))
>>> W = U.intersection(V)
>>> uv_to_xy = xy_to_uv.inverse()
>>> E = S2.vector_bundle(Integer(2), 'E') # define vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),x],[y,Integer(0)]])
>>> fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
>>> fN_W = fN.restrict(W); fS_W = fS.restrict(W) # restrict them
>>> stereoN_W = stereoN.restrict(W) # restrict charts, too
>>> stereoS_W = stereoS.restrict(W)
>>> s = E.section({fN: [Integer(1), Integer(0)]}, name='s')
>>> s.display(fN)
s = (phi_U^*e_1)
>>> sU = s.restrict(U) ; sU
Section s on the Open subset U of the 2-dimensional topological
 manifold S^2 with values in the real vector bundle E of rank 2
>>> sU.display() # fN is the default frame on U
s = (phi_U^*e_1)
>>> sU == fN[Integer(1)]
True
>>> sW = s.restrict(W) ; sW
Section s on the Open subset W of the 2-dimensional topological
 manifold S^2 with values in the real vector bundle E of rank 2
>>> sW.display(fN_W)
s = (phi_U^*e_1)
>>> sW.display(fS_W, stereoN_W)
s = y (phi_V^*e_2)
>>> sW.display(fS_W, stereoS_W)
s = v/(u^2 + v^2) (phi_V^*e_2)
>>> sW == fN_W[Integer(1)]
True

At this stage, defining the restriction of s to the open subset V fully specifies s:

sage: s.restrict(V)[1] = sW[fS_W, 1, stereoS_W].expr()  # note that fS is the default frame on V
sage: s.restrict(V)[2] = sW[fS_W, 2, stereoS_W].expr()
sage: s.display(fS, stereoS)
s = v/(u^2 + v^2) (phi_V^*e_2)
sage: s.restrict(U).display()
s = (phi_U^*e_1)
sage: s.restrict(V).display()
s = v/(u^2 + v^2) (phi_V^*e_2)

>>> from sage.all import *
>>> s.restrict(V)[Integer(1)] = sW[fS_W, Integer(1), stereoS_W].expr()  # note that fS is the default frame on V
>>> s.restrict(V)[Integer(2)] = sW[fS_W, Integer(2), stereoS_W].expr()
>>> s.display(fS, stereoS)
s = v/(u^2 + v^2) (phi_V^*e_2)
>>> s.restrict(U).display()
s = (phi_U^*e_1)
>>> s.restrict(V).display()
s = v/(u^2 + v^2) (phi_V^*e_2)

The restriction of the section to its own domain is of course itself:

sage: s.restrict(S2) is s
True
sage: sU.restrict(U) is sU
True

>>> from sage.all import *
>>> s.restrict(S2) is s
True
>>> sU.restrict(U) is sU
True
set_comp(basis=None)[source]#

Return the components of self in a given local frame for assignment.

The components with respect to other frames having the same domain as the provided local frame are deleted, in order to avoid any inconsistency. To keep them, use the method add_comp() instead.

INPUT:

  • basis – (default: None) local frame in which the components are defined; if none is provided, the components are assumed to refer to the section domain’s default frame

OUTPUT:

  • components in the given frame, as a Components; if such components did not exist previously, they are created

EXAMPLES:

sage: S2 = Manifold(2, 'S^2', structure='top', start_index=1)
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: xy_to_uv = stereoN.transition_map(stereoS,
....:                                   (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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E') # define vector bundle
sage: phi_U = E.trivialization('phi_U', domain=U) # define trivializations
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: transf = phi_U.transition_map(phi_V, [[0,x],[y,0]])
sage: fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
sage: s = E.section(name='s')
sage: s.set_comp(fS)
1-index components w.r.t. Trivialization frame (E|_V, ((phi_V^*e_1),(phi_V^*e_2)))
sage: s.set_comp(fS)[1] = u+v
sage: s.display(fS)
s = (u + v) (phi_V^*e_1)

>>> from sage.all import *
>>> S2 = Manifold(Integer(2), 'S^2', structure='top', start_index=Integer(1))
>>> U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
>>> S2.declare_union(U,V)
>>> stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)# stereographic coordinates from the North pole
>>> stereoS = V.chart(names=('u', 'v',)); (u, v,) = stereoS._first_ngens(2)# stereographic coordinates from the South pole
>>> xy_to_uv = stereoN.transition_map(stereoS,
...                                   (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))
>>> W = U.intersection(V)
>>> uv_to_xy = xy_to_uv.inverse()
>>> E = S2.vector_bundle(Integer(2), 'E') # define vector bundle
>>> phi_U = E.trivialization('phi_U', domain=U) # define trivializations
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> transf = phi_U.transition_map(phi_V, [[Integer(0),x],[y,Integer(0)]])
>>> fN = phi_U.frame(); fS = phi_V.frame() # get induced frames
>>> s = E.section(name='s')
>>> s.set_comp(fS)
1-index components w.r.t. Trivialization frame (E|_V, ((phi_V^*e_1),(phi_V^*e_2)))
>>> s.set_comp(fS)[Integer(1)] = u+v
>>> s.display(fS)
s = (u + v) (phi_V^*e_1)

Setting the components in a new frame (e):

sage: e = E.local_frame('e', domain=V)
sage: s.set_comp(e)
1-index components w.r.t. Local frame (E|_V, (e_1,e_2))
sage: s.set_comp(e)[1] = u*v
sage: s.display(e)
s = u*v e_1

>>> from sage.all import *
>>> e = E.local_frame('e', domain=V)
>>> s.set_comp(e)
1-index components w.r.t. Local frame (E|_V, (e_1,e_2))
>>> s.set_comp(e)[Integer(1)] = u*v
>>> s.display(e)
s = u*v e_1

Since the frames e and fS are defined on the same domain, the components w.r.t. fS have been erased:

sage: s.display(phi_V.frame())
Traceback (most recent call last):
...
ValueError: no basis could be found for computing the components in
 the Trivialization frame (E|_V, ((phi_V^*e_1),(phi_V^*e_2)))

>>> from sage.all import *
>>> s.display(phi_V.frame())
Traceback (most recent call last):
...
ValueError: no basis could be found for computing the components in
 the Trivialization frame (E|_V, ((phi_V^*e_1),(phi_V^*e_2)))
set_immutable()[source]#

Set self and all restrictions of self immutable.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: U = M.open_subset('U', coord_def={X: x^2+y^2<1})
sage: E = M.vector_bundle(2, 'E')
sage: e = E.local_frame('e')
sage: s = E.section([1+y,x], name='s')
sage: sU = s.restrict(U)
sage: s.set_immutable()
sage: s.is_immutable()
True
sage: sU.is_immutable()
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', coord_def={X: x**Integer(2)+y**Integer(2)<Integer(1)})
>>> E = M.vector_bundle(Integer(2), 'E')
>>> e = E.local_frame('e')
>>> s = E.section([Integer(1)+y,x], name='s')
>>> sU = s.restrict(U)
>>> s.set_immutable()
>>> s.is_immutable()
True
>>> sU.is_immutable()
True
set_name(name=None, latex_name=None)[source]#

Set (or change) the text name and LaTeX name of self.

INPUT:

  • name – string (default: None); name given to the section

  • latex_name – string (default: None); LaTeX symbol to denote the section; if None while name is provided, the LaTeX symbol is set to name

EXAMPLES:

sage: M = Manifold(3, 'M', structure='top')
sage: E = M.vector_bundle(2, 'E')
sage: s = E.section(); s
Section on the 3-dimensional topological manifold M with values in
 the real vector bundle E of rank 2
sage: s.set_name(name='s')
sage: s
Section s on the 3-dimensional topological manifold M with values in
 the real vector bundle E of rank 2
sage: latex(s)
s
sage: s.set_name(latex_name=r'\sigma')
sage: latex(s)
\sigma
sage: s.set_name(name='a')
sage: s
Section a on the 3-dimensional topological manifold M with values in
 the real vector bundle E of rank 2
sage: latex(s)
a

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='top')
>>> E = M.vector_bundle(Integer(2), 'E')
>>> s = E.section(); s
Section on the 3-dimensional topological manifold M with values in
 the real vector bundle E of rank 2
>>> s.set_name(name='s')
>>> s
Section s on the 3-dimensional topological manifold M with values in
 the real vector bundle E of rank 2
>>> latex(s)
s
>>> s.set_name(latex_name=r'\sigma')
>>> latex(s)
\sigma
>>> s.set_name(name='a')
>>> s
Section a on the 3-dimensional topological manifold M with values in
 the real vector bundle E of rank 2
>>> latex(s)
a
set_restriction(rst)[source]#

Define a restriction of self to some subdomain.

INPUT:

  • rst – Section defined on a subdomain of the domain of self

EXAMPLES:

sage: S2 = Manifold(2, 'S^2', structure='top')
sage: U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
sage: S2.declare_union(U,V)
sage: stereoN.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: stereoS.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: xy_to_uv = stereoN.transition_map(stereoS,
....:                                   (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: W = U.intersection(V)
sage: uv_to_xy = xy_to_uv.inverse()
sage: E = S2.vector_bundle(2, 'E')
sage: phi_U = E.trivialization('phi_U', domain=U)
sage: phi_V = E.trivialization('phi_V', domain=V)
sage: s = E.section(name='s')
sage: sU = E.section(domain=U, name='s')
sage: sU[:] = x+y, x
sage: s.set_restriction(sU)
sage: s.display(phi_U.frame())
s = (x + y) (phi_U^*e_1) + x (phi_U^*e_2)
sage: s.restrict(U) == sU
True

>>> from sage.all import *
>>> S2 = Manifold(Integer(2), 'S^2', structure='top')
>>> U = S2.open_subset('U') ; V = S2.open_subset('V') # complement of the North and South pole, respectively
>>> S2.declare_union(U,V)
>>> stereoN = U.chart(names=('x', 'y',)); (x, y,) = stereoN._first_ngens(2)# stereographic coordinates from the North pole
>>> stereoS = V.chart(names=('u', 'v',)); (u, v,) = stereoS._first_ngens(2)# stereographic coordinates from the South pole
>>> xy_to_uv = stereoN.transition_map(stereoS,
...                                   (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))
>>> W = U.intersection(V)
>>> uv_to_xy = xy_to_uv.inverse()
>>> E = S2.vector_bundle(Integer(2), 'E')
>>> phi_U = E.trivialization('phi_U', domain=U)
>>> phi_V = E.trivialization('phi_V', domain=V)
>>> s = E.section(name='s')
>>> sU = E.section(domain=U, name='s')
>>> sU[:] = x+y, x
>>> s.set_restriction(sU)
>>> s.display(phi_U.frame())
s = (x + y) (phi_U^*e_1) + x (phi_U^*e_2)
>>> s.restrict(U) == sU
True
class sage.manifolds.section.TrivialSection(section_module, name=None, latex_name=None)[source]#

Bases: FiniteRankFreeModuleElement, Section

Section in a trivial vector bundle.

An instance of this class is a section in a vector bundle \(E \to M\) of class \(C^k\), where \(E|_U\) is manifestly trivial. More precisely, a (local) section on a subset \(U \in M\) is a map of class \(C^k\)

\[s: U \longrightarrow E\]

such that

\[\forall p \in U,\ s(p) \in E_p\]

where \(E_p\) denotes the vector bundle fiber of \(E\) over the point \(p \in U\). \(E\) being trivial means \(E\) being homeomorphic to \(E \times F\), for \(F\) is the typical fiber of \(E\), namely the underlying topological vector space. By this means, \(s\) can be seen as a map of class \(C^k(U;E)\)

\[s: U \longrightarrow F ,\]

so that the set of all sections \(C^k(U;E)\) becomes a free module over the algebra of scalar fields on \(U\).

Note

If \(E|_U\) is not manifestly trivial, the class Section should be used instead.

This is a Sage element class, the corresponding parent class being SectionFreeModule.

INPUT:

  • section_module – free module \(C^k(U;E)\) of sections on \(E\) over \(U\) (cf. SectionFreeModule)

  • name – (default: None) name given to the section

  • latex_name – (default: None) LaTeX symbol to denote the section; if none is provided, the LaTeX symbol is set to name

EXAMPLES:

A section on a trivial rank 3 vector bundle over the 3-sphere:

sage: M = Manifold(3, 'S^3', structure='top')
sage: U = M.open_subset('U') ; V = M.open_subset('V') # complement of the North and South pole, respectively
sage: M.declare_union(U,V)
sage: stereoN.<x,y,z> = U.chart() # stereographic coordinates from the North pole
sage: stereoS.<u,v,t> = V.chart() # stereographic coordinates from the South pole
sage: xyz_to_uvt = stereoN.transition_map(stereoS,
....:           (x/(x^2+y^2+z^2), y/(x^2+y^2+z^2), z/(x^2+y^2+z^2)),
....:           intersection_name='W',
....:           restrictions1= x^2+y^2+z^2!=0,
....:           restrictions2= u^2+v^2+t^2!=0)
sage: W = U.intersection(V)
sage: uvt_to_xyz = xyz_to_uvt.inverse()
sage: E = M.vector_bundle(3, 'E')
sage: e = E.local_frame('e') # Trivializes E
sage: s = E.section(name='s'); s
Section s on the 3-dimensional topological manifold S^3 with values in
 the real vector bundle E of rank 3
sage: s[e,:] = z^2, x-y, 1-x
sage: s.display()
s = z^2 e_0 + (x - y) e_1 + (-x + 1) e_2

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'S^3', structure='top')
>>> U = M.open_subset('U') ; V = M.open_subset('V') # complement of the North and South pole, respectively
>>> M.declare_union(U,V)
>>> stereoN = U.chart(names=('x', 'y', 'z',)); (x, y, z,) = stereoN._first_ngens(3)# stereographic coordinates from the North pole
>>> stereoS = V.chart(names=('u', 'v', 't',)); (u, v, t,) = stereoS._first_ngens(3)# stereographic coordinates from the South pole
>>> xyz_to_uvt = stereoN.transition_map(stereoS,
...           (x/(x**Integer(2)+y**Integer(2)+z**Integer(2)), y/(x**Integer(2)+y**Integer(2)+z**Integer(2)), z/(x**Integer(2)+y**Integer(2)+z**Integer(2))),
...           intersection_name='W',
...           restrictions1= x**Integer(2)+y**Integer(2)+z**Integer(2)!=Integer(0),
...           restrictions2= u**Integer(2)+v**Integer(2)+t**Integer(2)!=Integer(0))
>>> W = U.intersection(V)
>>> uvt_to_xyz = xyz_to_uvt.inverse()
>>> E = M.vector_bundle(Integer(3), 'E')
>>> e = E.local_frame('e') # Trivializes E
>>> s = E.section(name='s'); s
Section s on the 3-dimensional topological manifold S^3 with values in
 the real vector bundle E of rank 3
>>> s[e,:] = z**Integer(2), x-y, Integer(1)-x
>>> s.display()
s = z^2 e_0 + (x - y) e_1 + (-x + 1) e_2

Since \(E\) is trivial, \(s\) is now element of a free section module:

sage: s.parent()
Free module C^0(S^3;E) of sections on the 3-dimensional topological
 manifold S^3 with values in the real vector bundle E of rank 3
sage: isinstance(s.parent(), FiniteRankFreeModule)
True

>>> from sage.all import *
>>> s.parent()
Free module C^0(S^3;E) of sections on the 3-dimensional topological
 manifold S^3 with values in the real vector bundle E of rank 3
>>> isinstance(s.parent(), FiniteRankFreeModule)
True
add_comp(basis=None)[source]#

Return the components of the section in a given local frame for assignment.

The components with respect to other frames on the same domain are kept. To delete them, use the method set_comp() instead.

INPUT:

  • basis – (default: None) local frame in which the components are defined; if none is provided, the components are assumed to refer to the section module’s default frame

OUTPUT:

  • components in the given frame, as an instance of the class Components; if such components did not exist previously, they are created

EXAMPLES:

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') # makes E trivial
sage: s = E.section(name='s')
sage: s.add_comp(e)
1-index components w.r.t. Local frame (E|_M, (e_0,e_1))
sage: s.add_comp(e)[0] = 2
sage: s.display(e)
s = 2 e_0

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> E = M.vector_bundle(Integer(2), 'E')
>>> e = E.local_frame('e') # makes E trivial
>>> s = E.section(name='s')
>>> s.add_comp(e)
1-index components w.r.t. Local frame (E|_M, (e_0,e_1))
>>> s.add_comp(e)[Integer(0)] = Integer(2)
>>> s.display(e)
s = 2 e_0

Adding components with respect to a new frame (f):

sage: f = E.local_frame('f')
sage: s.add_comp(f)
1-index components w.r.t. Local frame (E|_M, (f_0,f_1))
sage: s.add_comp(f)[0] = x
sage: s.display(f)
s = x f_0

>>> from sage.all import *
>>> f = E.local_frame('f')
>>> s.add_comp(f)
1-index components w.r.t. Local frame (E|_M, (f_0,f_1))
>>> s.add_comp(f)[Integer(0)] = x
>>> s.display(f)
s = x f_0

The components with respect to the frame e are kept:

sage: s.display(e)
s = 2 e_0

>>> from sage.all import *
>>> s.display(e)
s = 2 e_0

Adding components in a frame defined on a subdomain:

sage: U = M.open_subset('U', coord_def={X: x>0})
sage: g = E.local_frame('g', domain=U)
sage: s.add_comp(g)
1-index components w.r.t. Local frame (E|_U, (g_0,g_1))
sage: s.add_comp(g)[0] = 1+y
sage: s.display(g)
s = (y + 1) g_0

>>> from sage.all import *
>>> U = M.open_subset('U', coord_def={X: x>Integer(0)})
>>> g = E.local_frame('g', domain=U)
>>> s.add_comp(g)
1-index components w.r.t. Local frame (E|_U, (g_0,g_1))
>>> s.add_comp(g)[Integer(0)] = Integer(1)+y
>>> s.display(g)
s = (y + 1) g_0

The components previously defined are kept:

sage: s.display(e)
s = 2 e_0
sage: s.display(f)
s = x f_0

>>> from sage.all import *
>>> s.display(e)
s = 2 e_0
>>> s.display(f)
s = x f_0
at(point)[source]#

Value of self at a point of its domain.

If the current section is

\[s:\ U \longrightarrow E ,\]

then for any point \(p\in U\), \(s(p)\) is a vector in the fiber \(E_p\) of \(E\) at the point \(p \in U\).

INPUT:

  • point – ManifoldPoint point \(p\) in the domain of the section \(U\)

OUTPUT:

EXAMPLES:

Vector in a tangent space of a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='top')
sage: X.<x,y> = M.chart()
sage: p = M.point((-2,3), name='p')
sage: E = M.vector_bundle(2, 'E')
sage: e = E.local_frame('e') # makes E trivial
sage: s = E.section(y, x^2, name='s')
sage: s.display()
s = y e_0 + x^2 e_1
sage: sp = s.at(p) ; sp
Vector s in the fiber of E at Point p on the 2-dimensional
 topological manifold M
sage: sp.parent()
Fiber of E at Point p on the 2-dimensional topological manifold M
sage: sp.display()
s = 3 e_0 + 4 e_1

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> p = M.point((-Integer(2),Integer(3)), name='p')
>>> E = M.vector_bundle(Integer(2), 'E')
>>> e = E.local_frame('e') # makes E trivial
>>> s = E.section(y, x**Integer(2), name='s')
>>> s.display()
s = y e_0 + x^2 e_1
>>> sp = s.at(p) ; sp
Vector s in the fiber of E at Point p on the 2-dimensional
 topological manifold M
>>> sp.parent()
Fiber of E at Point p on the 2-dimensional topological manifold M
>>> sp.display()
s = 3 e_0 + 4 e_1
comp(basis=None, from_basis=None)[source]#

Return the components in a given local frame.

If the components are not known already, they are computed by the tensor change-of-basis formula from components in another local frame.

INPUT:

  • basis – (default: None) local frame in which the components are required; if none is provided, the components are assumed to refer to the section module’s default frame

  • from_basis – (default: None) local frame from which the required components are computed, via the tensor change-of-basis formula, if they are not known already in the basis basis

OUTPUT:

  • components in the local frame basis, as an instance of the class Components

EXAMPLES:

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') # makes E trivial
sage: s = E.section(name='s')
sage: s[1] = x*y
sage: s.comp(e)
1-index components w.r.t. Local frame (E|_M, (e_1,e_2))
sage: s.comp()  # the default frame is e
1-index components w.r.t. Local frame (E|_M, (e_1,e_2))
sage: s.comp()[:]
[x*y, 0]
sage: f = E.local_frame('f')
sage: s[f, 1] = x-3
sage: s.comp(f)
1-index components w.r.t. Local frame (E|_M, (f_1,f_2))
sage: s.comp(f)[:]
[x - 3, 0]

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top', start_index=Integer(1))
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> E = M.vector_bundle(Integer(2), 'E')
>>> e = E.local_frame('e') # makes E trivial
>>> s = E.section(name='s')
>>> s[Integer(1)] = x*y
>>> s.comp(e)
1-index components w.r.t. Local frame (E|_M, (e_1,e_2))
>>> s.comp()  # the default frame is e
1-index components w.r.t. Local frame (E|_M, (e_1,e_2))
>>> s.comp()[:]
[x*y, 0]
>>> f = E.local_frame('f')
>>> s[f, Integer(1)] = x-Integer(3)
>>> s.comp(f)
1-index components w.r.t. Local frame (E|_M, (f_1,f_2))
>>> s.comp(f)[:]
[x - 3, 0]
display_comp(frame=None, chart=None, only_nonzero=False)[source]#

Display the section components with respect to a given frame, one per line.

The output is either text-formatted (console mode) or LaTeX-formatted (notebook mode).

INPUT:

  • frame – (default: None) local frame with respect to which the section components are defined; if None, then the default basis of the section module on which the section is defined is used

  • chart – (default: None) chart specifying the coordinate expression of the components; if None, the default chart of the section module domain is used

  • only_nonzero – (default: False) boolean; if True, only nonzero components are displayed

EXAMPLES:

Display of the components of a section on a rank 4 vector bundle over a 2-dimensional manifold:

sage: M = Manifold(2, 'M', structure='top')
sage: X.<x,y> = M.chart()
sage: E = M.vector_bundle(3, 'E')
sage: e = E.local_frame('e') # makes E trivial
sage: s = E.section(name='s')
sage: s[0], s[2] = x+y, x*y
sage: s.display_comp()
s^0 = x + y
s^1 = 0
s^2 = x*y

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> E = M.vector_bundle(Integer(3), 'E')
>>> e = E.local_frame('e') # makes E trivial
>>> s = E.section(name='s')
>>> s[Integer(0)], s[Integer(2)] = x+y, x*y
>>> s.display_comp()
s^0 = x + y
s^1 = 0
s^2 = x*y

By default, the vanishing components are displayed, too; to see only non-vanishing components, the argument only_nonzero must be set to True:

sage: s.display_comp(only_nonzero=True)
s^0 = x + y
s^2 = x*y

>>> from sage.all import *
>>> s.display_comp(only_nonzero=True)
s^0 = x + y
s^2 = x*y

Display in a frame different from the default one:

sage: a = E.section_module().automorphism()
sage: a[:] = [[1+y^2, 0, 0], [0, 2+x^2, 0], [0, 0, 1]]
sage: f = e.new_frame(a, 'f')
sage: s.display_comp(frame=f)
s^0 = (x + y)/(y^2 + 1)
s^1 = 0
s^2 = x*y

>>> from sage.all import *
>>> a = E.section_module().automorphism()
>>> a[:] = [[Integer(1)+y**Integer(2), Integer(0), Integer(0)], [Integer(0), Integer(2)+x**Integer(2), Integer(0)], [Integer(0), Integer(0), Integer(1)]]
>>> f = e.new_frame(a, 'f')
>>> s.display_comp(frame=f)
s^0 = (x + y)/(y^2 + 1)
s^1 = 0
s^2 = x*y

Display with respect to a chart different from the default one:

sage: Y.<u,v> = M.chart()
sage: X_to_Y = X.transition_map(Y, [x+y, x-y])
sage: Y_to_X = X_to_Y.inverse()
sage: s.display_comp(chart=Y)
s^0 = u
s^1 = 0
s^2 = 1/4*u^2 - 1/4*v^2

>>> from sage.all import *
>>> Y = M.chart(names=('u', 'v',)); (u, v,) = Y._first_ngens(2)
>>> X_to_Y = X.transition_map(Y, [x+y, x-y])
>>> Y_to_X = X_to_Y.inverse()
>>> s.display_comp(chart=Y)
s^0 = u
s^1 = 0
s^2 = 1/4*u^2 - 1/4*v^2

Display of the components with respect to a specific frame, expressed in terms of a specific chart:

sage: s.display_comp(frame=f, chart=Y)
s^0 = 4*u/(u^2 - 2*u*v + v^2 + 4)
s^1 = 0
s^2 = 1/4*u^2 - 1/4*v^2

>>> from sage.all import *
>>> s.display_comp(frame=f, chart=Y)
s^0 = 4*u/(u^2 - 2*u*v + v^2 + 4)
s^1 = 0
s^2 = 1/4*u^2 - 1/4*v^2
restrict(subdomain)[source]#

Return the restriction of self to some subdomain.

If the restriction has not been defined yet, it is constructed here.

INPUT:

OUTPUT:

EXAMPLES:

Restriction of a section defined over \(\RR^2\) to a disk:

sage: M = Manifold(2, 'R^2')
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') # makes E trivial
sage: s = E.section(x+y, -1+x^2, name='s')
sage: D = M.open_subset('D') # the unit open disc
sage: e_D = e.restrict(D)
sage: c_cart_D = c_cart.restrict(D, x^2+y^2<1)
sage: s_D = s.restrict(D) ; s_D
Section s on the Open subset D of the 2-dimensional differentiable
 manifold R^2 with values in the real vector bundle E of rank 2
sage: s_D.display(e_D)
s = (x + y) e_0 + (x^2 - 1) 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
>>> E = M.vector_bundle(Integer(2), 'E')
>>> e = E.local_frame('e') # makes E trivial
>>> s = E.section(x+y, -Integer(1)+x**Integer(2), name='s')
>>> D = M.open_subset('D') # the unit open disc
>>> e_D = e.restrict(D)
>>> c_cart_D = c_cart.restrict(D, x**Integer(2)+y**Integer(2)<Integer(1))
>>> s_D = s.restrict(D) ; s_D
Section s on the Open subset D of the 2-dimensional differentiable
 manifold R^2 with values in the real vector bundle E of rank 2
>>> s_D.display(e_D)
s = (x + y) e_0 + (x^2 - 1) e_1

The symbolic expressions of the components with respect to Cartesian coordinates are equal:

sage: bool( s_D[1].expr() == s[1].expr() )
True

>>> from sage.all import *
>>> bool( s_D[Integer(1)].expr() == s[Integer(1)].expr() )
True

but neither the chart functions representing the components (they are defined on different charts):

sage: s_D[1] == s[1]
False

>>> from sage.all import *
>>> s_D[Integer(1)] == s[Integer(1)]
False

nor the scalar fields representing the components (they are defined on different open subsets):

sage: s_D[[1]] == s[[1]]
False

>>> from sage.all import *
>>> s_D[[Integer(1)]] == s[[Integer(1)]]
False

The restriction of the section to its own domain is of course itself:

sage: s.restrict(M) is s
True

>>> from sage.all import *
>>> s.restrict(M) is s
True
set_comp(basis=None)[source]#

Return the components of the section in a given local frame for assignment.

The components with respect to other frames on the same domain are deleted, in order to avoid any inconsistency. To keep them, use the method add_comp() instead.

INPUT:

  • basis – (default: None) local frame in which the components are defined; if none is provided, the components are assumed to refer to the section module’s default frame

OUTPUT:

  • components in the given frame, as an instance of the class Components; if such components did not exist previously, they are created

EXAMPLES:

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') # makes E trivial
sage: s = E.section(name='s')
sage: s.set_comp(e)
1-index components w.r.t. Local frame (E|_M, (e_0,e_1))
sage: s.set_comp(e)[0] = 2
sage: s.display(e)
s = 2 e_0

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> E = M.vector_bundle(Integer(2), 'E')
>>> e = E.local_frame('e') # makes E trivial
>>> s = E.section(name='s')
>>> s.set_comp(e)
1-index components w.r.t. Local frame (E|_M, (e_0,e_1))
>>> s.set_comp(e)[Integer(0)] = Integer(2)
>>> s.display(e)
s = 2 e_0

Setting components in a new frame (f):

sage: f = E.local_frame('f')
sage: s.set_comp(f)
1-index components w.r.t. Local frame (E|_M, (f_0,f_1))
sage: s.set_comp(f)[0] = x
sage: s.display(f)
s = x f_0

>>> from sage.all import *
>>> f = E.local_frame('f')
>>> s.set_comp(f)
1-index components w.r.t. Local frame (E|_M, (f_0,f_1))
>>> s.set_comp(f)[Integer(0)] = x
>>> s.display(f)
s = x f_0

The components with respect to the frame e have be erased:

sage: s.display(e)
Traceback (most recent call last):
...
ValueError: no basis could be found for computing the components
 in the Local frame (E|_M, (e_0,e_1))

>>> from sage.all import *
>>> s.display(e)
Traceback (most recent call last):
...
ValueError: no basis could be found for computing the components
 in the Local frame (E|_M, (e_0,e_1))

Setting components in a frame defined on a subdomain deletes previously defined components as well:

sage: U = M.open_subset('U', coord_def={X: x>0})
sage: g = E.local_frame('g', domain=U)
sage: s.set_comp(g)
1-index components w.r.t. Local frame (E|_U, (g_0,g_1))
sage: s.set_comp(g)[0] = 1+y
sage: s.display(g)
s = (y + 1) g_0
sage: s.display(f)
Traceback (most recent call last):
...
ValueError: no basis could be found for computing the components
 in the Local frame (E|_M, (f_0,f_1))

>>> from sage.all import *
>>> U = M.open_subset('U', coord_def={X: x>Integer(0)})
>>> g = E.local_frame('g', domain=U)
>>> s.set_comp(g)
1-index components w.r.t. Local frame (E|_U, (g_0,g_1))
>>> s.set_comp(g)[Integer(0)] = Integer(1)+y
>>> s.display(g)
s = (y + 1) g_0
>>> s.display(f)
Traceback (most recent call last):
...
ValueError: no basis could be found for computing the components
 in the Local frame (E|_M, (f_0,f_1))