# 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)

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


The parent of $$s$$ is not a free module, since $$E$$ is not trivial:

sage: 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)


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


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)


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)


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.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)


and multiplied by scalar fields:

sage: f = M.scalar_field(y^2-x^2, name='f')
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)


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)


Since $$E|_U$$ is trivial, cU now belongs to the free module:

sage: 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


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

add_comp(basis=None)

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')
1-index components w.r.t. Trivialization frame (E|_V, ((phi_V^*e_1),(phi_V^*e_2)))
sage: 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)
1-index components w.r.t. Local frame (E|_V, (e_1,e_2))
sage: 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)

add_comp_by_continuation(frame, subdomain, chart=None)

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')


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)


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)


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)


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)


and $$a$$ is defined on the entire manifold $$S^2$$.

add_expr_from_subdomain(frame, subdomain)

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


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]


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)


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)


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)


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)


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)

at(point)

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:

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.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)

base_module()

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

comp(basis=None, from_basis=None)

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:

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]


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

copy(name=None, latex_name=None)

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: 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


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

disp(frame=None, chart=None)

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.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)


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)


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)


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)


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)


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


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)

display(frame=None, chart=None)

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.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)


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)


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)


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)


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)


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


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)

display_comp(frame=None, chart=None, only_nonzero=True)

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


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

domain()

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

restrict(subdomain)

Return the restriction of self to some subdomain.

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

INPUT:

OUTPUT:

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


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)


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

set_comp(basis=None)

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)


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


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)))

set_name(name=None, latex_name=None)

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

set_restriction(rst)

Define a restriction of self to some subdomain.

INPUT:

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

class sage.manifolds.section.TrivialSection(section_module, name=None, latex_name=None)

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


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

add_comp(basis=None)

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')
1-index components w.r.t. Local frame (E|_M, (e_0,e_1))
sage: s.display(e)
s = 2 e_0


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

sage: f = E.local_frame('f')
1-index components w.r.t. Local frame (E|_M, (f_0,f_1))
sage: 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


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)
1-index components w.r.t. Local frame (E|_U, (g_0,g_1))
sage: 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

at(point)

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:

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

comp(basis=None, from_basis=None)

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:

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]

display_comp(frame=None, chart=None, only_nonzero=False)

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


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


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


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


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

restrict(subdomain)

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


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

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


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

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


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

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


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

sage: s.restrict(M) is s
True

set_comp(basis=None)

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


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


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))


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))