Section Modules¶

The set of sections over a vector bundle \(E \to M\) of class \(C^k\) on a domain \(U \in M\) is a module over the algebra \(C^k(U)\) of scalar fields on \(U\).

Depending on the domain, there are two classes of section modules:

SectionModule for local sections over a non-trivial part of a topological vector bundle
SectionFreeModule for local sections over a trivial part of a topological vector bundle

AUTHORS:

Michael Jung (2019): initial version

class sage.manifolds.section_module.SectionFreeModule(vbundle, domain)[source]¶

Bases: FiniteRankFreeModule

Free module of sections over a vector bundle \(E \to M\) of class \(C^k\) on a domain \(U \in M\) which admits a trivialization or local frame.

The section module \(C^k(U;E)\) is the set of all \(C^k\)-maps, called sections, of type

\[s: U \longrightarrow E\]

such that

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

where \(E_p\) is the vector bundle fiber of \(E\) at the point \(p\).

Since the domain \(U\) admits a local frame, the corresponding vector bundle \(E|_U \to U\) is trivial and \(C^k(U;E)\) is a free module over \(C^k(U)\).

Note

If \(E|_U\) is not trivial, the class SectionModule should be used instead, for \(C^k(U;E)\) is no longer a free module.

INPUT:

vbundle – vector bundle \(E\) on which the sections takes its values
domain – (default: None) subdomain \(U\) of the base space on which the sections are defined

EXAMPLES:

Module of sections on the 2-rank trivial vector bundle over the Euclidean plane \(\RR^2\):

Sage

sage: M = Manifold(2, 'R^2', structure='top')
sage: c_cart.<x,y> = M.chart()
sage: E = M.vector_bundle(2, 'E')
sage: e = E.local_frame('e') # Trivializes the vector bundle
sage: C0 = E.section_module(); C0
Free module C^0(R^2;E) of sections on the 2-dimensional topological
 manifold R^2 with values in the real vector bundle E of rank 2
sage: C0.category()
Category of finite dimensional modules over Algebra of scalar fields on
 the 2-dimensional topological manifold R^2
sage: C0.base_ring() is M.scalar_field_algebra()
True

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'R^2', structure='top')
>>> c_cart = M.chart(names=('x', 'y',)); (x, y,) = c_cart._first_ngens(2)
>>> E = M.vector_bundle(Integer(2), 'E')
>>> e = E.local_frame('e') # Trivializes the vector bundle
>>> C0 = E.section_module(); C0
Free module C^0(R^2;E) of sections on the 2-dimensional topological
 manifold R^2 with values in the real vector bundle E of rank 2
>>> C0.category()
Category of finite dimensional modules over Algebra of scalar fields on
 the 2-dimensional topological manifold R^2
>>> C0.base_ring() is M.scalar_field_algebra()
True

Sage Live

M = Manifold(2, 'R^2', structure='top')
c_cart.<x,y> = M.chart()
E = M.vector_bundle(2, 'E')
e = E.local_frame('e') # Trivializes the vector bundle
C0 = E.section_module(); C0
C0.category()
C0.base_ring() is M.scalar_field_algebra()

The vector bundle admits a global frame and is therefore trivial:

Sage

sage: E.is_manifestly_trivial()
True

Python

>>> from sage.all import *
>>> E.is_manifestly_trivial()
True

Sage Live

E.is_manifestly_trivial()

Since the vector bundle is trivial, its section module of global sections is a free module:

Sage

sage: isinstance(C0, FiniteRankFreeModule)
True

Python

>>> from sage.all import *
>>> isinstance(C0, FiniteRankFreeModule)
True

Sage Live

isinstance(C0, FiniteRankFreeModule)

Some elements are:

Sage

sage: C0.an_element().display()
2 e_0 + 2 e_1
sage: C0.zero().display()
zero = 0
sage: s = C0([-y,x]); s
Section on the 2-dimensional topological manifold R^2 with values in the
 real vector bundle E of rank 2
sage: s.display()
-y e_0 + x e_1

Python

>>> from sage.all import *
>>> C0.an_element().display()
2 e_0 + 2 e_1
>>> C0.zero().display()
zero = 0
>>> s = C0([-y,x]); s
Section on the 2-dimensional topological manifold R^2 with values in the
 real vector bundle E of rank 2
>>> s.display()
-y e_0 + x e_1

Sage Live

C0.an_element().display()
C0.zero().display()
s = C0([-y,x]); s
s.display()

The rank of the free module equals the rank of the vector bundle:

Sage

sage: C0.rank()
2

Python

>>> from sage.all import *
>>> C0.rank()
2

Sage Live

C0.rank()

The basis is given by the definition above:

Sage

sage: C0.bases()
[Local frame (E|_R^2, (e_0,e_1))]

Python

>>> from sage.all import *
>>> C0.bases()
[Local frame (E|_R^2, (e_0,e_1))]

Sage Live

C0.bases()

The test suite is passed as well:

Sage

sage: TestSuite(C0).run()

Python

>>> from sage.all import *
>>> TestSuite(C0).run()

Sage Live

TestSuite(C0).run()

Element[source]¶: alias of TrivialSection

base_space()[source]¶

Return the base space of the sections in this module.

EXAMPLES:

Sage

sage: M = Manifold(3, 'M', structure='top')
sage: U = M.open_subset('U')
sage: E = U.vector_bundle(2, 'E')
sage: C0 = E.section_module(force_free=True); C0
Free module C^0(U;E) of sections on the Open subset U of the
 3-dimensional topological manifold M with values in the real
 vector bundle E of rank 2
sage: C0.base_space()
Open subset U of the 3-dimensional topological manifold M

Python

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='top')
>>> U = M.open_subset('U')
>>> E = U.vector_bundle(Integer(2), 'E')
>>> C0 = E.section_module(force_free=True); C0
Free module C^0(U;E) of sections on the Open subset U of the
 3-dimensional topological manifold M with values in the real
 vector bundle E of rank 2
>>> C0.base_space()
Open subset U of the 3-dimensional topological manifold M

Sage Live

M = Manifold(3, 'M', structure='top')
U = M.open_subset('U')
E = U.vector_bundle(2, 'E')
C0 = E.section_module(force_free=True); C0
C0.base_space()

basis(symbol=None, latex_symbol=None, from_frame=None, indices=None, latex_indices=None, symbol_dual=None, latex_symbol_dual=None)[source]¶

Define a basis of self.

A basis of the section module is actually a local frame on the differentiable manifold \(U\) over which the section module is defined.

If the basis specified by the given symbol already exists, it is simply returned. If no argument is provided the module’s default basis is returned.

INPUT:

symbol – (default: None) either a string, to be used as a common base for the symbols of the elements of the basis, or a tuple of strings, representing the individual symbols of the elements of the basis
latex_symbol – (default: None) either a string, to be used as a common base for the LaTeX symbols of the elements of the basis, or a tuple of strings, representing the individual LaTeX symbols of the elements of the basis; if None, symbol is used in place of latex_symbol
indices – (default: None; used only if symbol is a single string) tuple of strings representing the indices labelling the elements of the basis; if None, the indices will be generated as integers within the range declared on self
latex_indices – (default: None) tuple of strings representing the indices for the LaTeX symbols of the elements of the basis; if None, indices is used instead
symbol_dual – (default: None) same as symbol but for the dual basis; if None, symbol must be a string and is used for the common base of the symbols of the elements of the dual basis
latex_symbol_dual – (default: None) same as latex_symbol but for the dual basis

OUTPUT:

a LocalFrame representing a basis on self

EXAMPLES:

Sage

sage: M = Manifold(2, 'M', structure='top')
sage: E = M.vector_bundle(2, 'E')
sage: C0 = E.section_module(force_free=True)
sage: e = C0.basis('e'); e
Local frame (E|_M, (e_0,e_1))

Python

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M', structure='top')
>>> E = M.vector_bundle(Integer(2), 'E')
>>> C0 = E.section_module(force_free=True)
>>> e = C0.basis('e'); e
Local frame (E|_M, (e_0,e_1))

Sage Live

M = Manifold(2, 'M', structure='top')
E = M.vector_bundle(2, 'E')
C0 = E.section_module(force_free=True)
e = C0.basis('e'); e

See LocalFrame for more examples and documentation.

default_frame()[source]¶: alias of default_basis().

domain()[source]¶

Return the domain of the section module.

EXAMPLES:

Sage

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); C0_U
Free module C^0(U;E) of sections on the Open subset U of the
 3-dimensional topological manifold M with values in the real vector
 bundle E of rank 2
sage: C0_U.domain()
Open subset U of the 3-dimensional topological manifold M

Python

>>> 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); C0_U
Free module C^0(U;E) of sections on the Open subset U of the
 3-dimensional topological manifold M with values in the real vector
 bundle E of rank 2
>>> C0_U.domain()
Open subset U of the 3-dimensional topological manifold M

Sage Live

M = Manifold(3, 'M', structure='top')
U = M.open_subset('U')
E = M.vector_bundle(2, 'E')
C0_U = E.section_module(domain=U, force_free=True); C0_U
C0_U.domain()

set_default_frame(basis)[source]¶: alias of set_default_basis().

vector_bundle()[source]¶

Return the overlying vector bundle on which the section module is defined.

EXAMPLES:

Sage

sage: M = Manifold(3, 'M', structure='top')
sage: E = M.vector_bundle(2, 'E')
sage: C0 = E.section_module(force_free=True); C0
Free module C^0(M;E) of sections on the 3-dimensional topological
 manifold M with values in the real vector bundle E of rank 2
sage: C0.vector_bundle()
Topological real vector bundle E -> M of rank 2 over the base space
 3-dimensional topological manifold M
sage: E is C0.vector_bundle()
True

Python

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='top')
>>> E = M.vector_bundle(Integer(2), 'E')
>>> C0 = E.section_module(force_free=True); C0
Free module C^0(M;E) of sections on the 3-dimensional topological
 manifold M with values in the real vector bundle E of rank 2
>>> C0.vector_bundle()
Topological real vector bundle E -> M of rank 2 over the base space
 3-dimensional topological manifold M
>>> E is C0.vector_bundle()
True

Sage Live

M = Manifold(3, 'M', structure='top')
E = M.vector_bundle(2, 'E')
C0 = E.section_module(force_free=True); C0
C0.vector_bundle()
E is C0.vector_bundle()

class sage.manifolds.section_module.SectionModule(vbundle, domain)[source]¶

Bases: UniqueRepresentation, Parent

Module of sections over a vector bundle \(E \to M\) of class \(C^k\) on a domain \(U \in M\).

The section module \(C^k(U;E)\) is the set of all \(C^k\)-maps, called sections, of type

\[s: U \longrightarrow E\]

such that

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

where \(E_p\) is the vector bundle fiber of \(E\) at the point \(p\).

\(C^k(U;E)\) is a module over \(C^k(U)\), the algebra of \(C^k\) scalar fields on \(U\).

INPUT:

vbundle – vector bundle \(E\) on which the sections takes its values
domain – (default: None) subdomain \(U\) of the base space on which the sections are defined

EXAMPLES:

Module of sections on the Möbius bundle:

Sage

sage: M = Manifold(1, 'RP^1', structure='top', start_index=1)
sage: U = M.open_subset('U')  # the complement of one point
sage: c_u.<u> =  U.chart() # [1:u] in homogeneous coord.
sage: V = M.open_subset('V') # the complement of the point u=0
sage: M.declare_union(U,V)   # [v:1] in homogeneous coord.
sage: c_v.<v> = V.chart()
sage: u_to_v = c_u.transition_map(c_v, (1/u),
....:                             intersection_name='W',
....:                             restrictions1 = u!=0,
....:                             restrictions2 = v!=0)
sage: v_to_u = u_to_v.inverse()
sage: W = U.intersection(V)
sage: E = M.vector_bundle(1, 'E')
sage: phi_U = E.trivialization('phi_U', latex_name=r'\varphi_U',
....:                          domain=U)
sage: phi_V = E.trivialization('phi_V', latex_name=r'\varphi_V',
....:                          domain=V)
sage: transf = phi_U.transition_map(phi_V, [[u]])
sage: C0 = E.section_module(); C0
Module C^0(RP^1;E) of sections on the 1-dimensional topological manifold
 RP^1 with values in the real vector bundle E of rank 1

Python

>>> from sage.all import *
>>> M = Manifold(Integer(1), 'RP^1', structure='top', start_index=Integer(1))
>>> U = M.open_subset('U')  # the complement of one point
>>> c_u = U.chart(names=('u',)); (u,) = c_u._first_ngens(1)# [1:u] in homogeneous coord.
>>> V = M.open_subset('V') # the complement of the point u=0
>>> M.declare_union(U,V)   # [v:1] in homogeneous coord.
>>> c_v = V.chart(names=('v',)); (v,) = c_v._first_ngens(1)
>>> u_to_v = c_u.transition_map(c_v, (Integer(1)/u),
...                             intersection_name='W',
...                             restrictions1 = u!=Integer(0),
...                             restrictions2 = v!=Integer(0))
>>> v_to_u = u_to_v.inverse()
>>> W = U.intersection(V)
>>> E = M.vector_bundle(Integer(1), 'E')
>>> phi_U = E.trivialization('phi_U', latex_name=r'\varphi_U',
...                          domain=U)
>>> phi_V = E.trivialization('phi_V', latex_name=r'\varphi_V',
...                          domain=V)
>>> transf = phi_U.transition_map(phi_V, [[u]])
>>> C0 = E.section_module(); C0
Module C^0(RP^1;E) of sections on the 1-dimensional topological manifold
 RP^1 with values in the real vector bundle E of rank 1

Sage Live

M = Manifold(1, 'RP^1', structure='top', start_index=1)
U = M.open_subset('U')  # the complement of one point
c_u.<u> =  U.chart() # [1:u] in homogeneous coord.
V = M.open_subset('V') # the complement of the point u=0
M.declare_union(U,V)   # [v:1] in homogeneous coord.
c_v.<v> = V.chart()
u_to_v = c_u.transition_map(c_v, (1/u),
                            intersection_name='W',
                            restrictions1 = u!=0,
                            restrictions2 = v!=0)
v_to_u = u_to_v.inverse()
W = U.intersection(V)
E = M.vector_bundle(1, 'E')
phi_U = E.trivialization('phi_U', latex_name=r'\varphi_U',
                         domain=U)
phi_V = E.trivialization('phi_V', latex_name=r'\varphi_V',
                         domain=V)
transf = phi_U.transition_map(phi_V, [[u]])
C0 = E.section_module(); C0

\(C^0(\RR P^1;E)\) is a module over the algebra \(C^0(\RR P^1)\):

Sage

sage: C0.category()
Category of modules over Algebra of scalar fields on the 1-dimensional
 topological manifold RP^1
sage: C0.base_ring() is M.scalar_field_algebra()
True

Python

>>> from sage.all import *
>>> C0.category()
Category of modules over Algebra of scalar fields on the 1-dimensional
 topological manifold RP^1
>>> C0.base_ring() is M.scalar_field_algebra()
True

Sage Live

C0.category()
C0.base_ring() is M.scalar_field_algebra()

However, \(C^0(\RR P^1;E)\) is not a free module:

Sage

sage: isinstance(C0, FiniteRankFreeModule)
False

Python

>>> from sage.all import *
>>> isinstance(C0, FiniteRankFreeModule)
False

Sage Live

isinstance(C0, FiniteRankFreeModule)

since the Möbius bundle is not trivial:

Sage

sage: E.is_manifestly_trivial()
False

Python

>>> from sage.all import *
>>> E.is_manifestly_trivial()
False

Sage Live

E.is_manifestly_trivial()

The section module over \(U\), on the other hand, is a free module since \(E|_U\) admits a trivialization and therefore has a local frame:

Sage

sage: C0_U = E.section_module(domain=U)
sage: isinstance(C0_U, FiniteRankFreeModule)
True

Python

>>> from sage.all import *
>>> C0_U = E.section_module(domain=U)
>>> isinstance(C0_U, FiniteRankFreeModule)
True

Sage Live

C0_U = E.section_module(domain=U)
isinstance(C0_U, FiniteRankFreeModule)

The zero element of the module:

Sage

sage: z = C0.zero() ; z
Section zero on the 1-dimensional topological manifold RP^1 with values
 in the real vector bundle E of rank 1
sage: z.display(phi_U.frame())
zero = 0
sage: z.display(phi_V.frame())
zero = 0

Python

>>> from sage.all import *
>>> z = C0.zero() ; z
Section zero on the 1-dimensional topological manifold RP^1 with values
 in the real vector bundle E of rank 1
>>> z.display(phi_U.frame())
zero = 0
>>> z.display(phi_V.frame())
zero = 0

Sage Live

z = C0.zero() ; z
z.display(phi_U.frame())
z.display(phi_V.frame())

The module \(C^0(M;E)\) coerces to any module of sections defined on a subdomain of \(M\), for instance \(C^0(U;E)\):

Sage

sage: C0_U.has_coerce_map_from(C0)
True
sage: C0_U.coerce_map_from(C0)
Coercion map:
  From: Module C^0(RP^1;E) of sections on the 1-dimensional topological
   manifold RP^1 with values in the real vector bundle E of rank 1
  To:   Free module C^0(U;E) of sections on the Open subset U of the
   1-dimensional topological manifold RP^1 with values in the real vector
   bundle E of rank 1

Python

>>> from sage.all import *
>>> C0_U.has_coerce_map_from(C0)
True
>>> C0_U.coerce_map_from(C0)
Coercion map:
  From: Module C^0(RP^1;E) of sections on the 1-dimensional topological
   manifold RP^1 with values in the real vector bundle E of rank 1
  To:   Free module C^0(U;E) of sections on the Open subset U of the
   1-dimensional topological manifold RP^1 with values in the real vector
   bundle E of rank 1

Sage Live

C0_U.has_coerce_map_from(C0)
C0_U.coerce_map_from(C0)

The conversion map is actually the restriction of sections defined on \(M\) to \(U\).

Element[source]¶: alias of Section

base_space()[source]¶

Return the base space of the sections in this module.

EXAMPLES:

Sage

sage: M = Manifold(3, 'M', structure='top')
sage: U = M.open_subset('U')
sage: E = U.vector_bundle(2, 'E')
sage: C0 = E.section_module(); C0
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
sage: C0.base_space()
Open subset U of the 3-dimensional topological manifold M

Python

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='top')
>>> U = M.open_subset('U')
>>> E = U.vector_bundle(Integer(2), 'E')
>>> C0 = E.section_module(); C0
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
>>> C0.base_space()
Open subset U of the 3-dimensional topological manifold M

Sage Live

M = Manifold(3, 'M', structure='top')
U = M.open_subset('U')
E = U.vector_bundle(2, 'E')
C0 = E.section_module(); C0
C0.base_space()

default_frame()[source]¶

Return the default frame defined on self.

EXAMPLES:

Get the default local frame of a non-trivial section module:

Sage

sage: M = Manifold(3, 'M', structure='top')
sage: U = M.open_subset('U')
sage: V = M.open_subset('V')
sage: M.declare_union(U, V)
sage: E = M.vector_bundle(2, 'E')
sage: C0 = E.section_module()
sage: e = E.local_frame('e', domain=U)
sage: C0.default_frame()
Local frame (E|_U, (e_0,e_1))

Python

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='top')
>>> U = M.open_subset('U')
>>> V = M.open_subset('V')
>>> M.declare_union(U, V)
>>> E = M.vector_bundle(Integer(2), 'E')
>>> C0 = E.section_module()
>>> e = E.local_frame('e', domain=U)
>>> C0.default_frame()
Local frame (E|_U, (e_0,e_1))

Sage Live

M = Manifold(3, 'M', structure='top')
U = M.open_subset('U')
V = M.open_subset('V')
M.declare_union(U, V)
E = M.vector_bundle(2, 'E')
C0 = E.section_module()
e = E.local_frame('e', domain=U)
C0.default_frame()

The local frame is indeed the same, and not a copy:

Sage

sage: e is C0.default_frame()
True

Python

>>> from sage.all import *
>>> e is C0.default_frame()
True

Sage Live

e is C0.default_frame()

domain()[source]¶

Return the domain of the section module.

EXAMPLES:

Sage

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); C0_U
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
sage: C0_U.domain()
Open subset U of the 3-dimensional topological manifold M

Python

>>> 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); C0_U
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
>>> C0_U.domain()
Open subset U of the 3-dimensional topological manifold M

Sage Live

M = Manifold(3, 'M', structure='top')
U = M.open_subset('U')
E = M.vector_bundle(2, 'E')
C0_U = E.section_module(domain=U); C0_U
C0_U.domain()

set_default_frame(basis)[source]¶

Set the default local frame on self.

EXAMPLES:

Set a default frame of a non-trivial section module:

Sage

sage: M = Manifold(3, 'M', structure='top')
sage: U = M.open_subset('U')
sage: V = M.open_subset('V')
sage: M.declare_union(U, V)
sage: E = M.vector_bundle(2, 'E')
sage: C0 = E.section_module(); C0
Module C^0(M;E) of sections on the 3-dimensional topological
 manifold M with values in the real vector bundle E of rank 2
sage: e = E.local_frame('e', domain=U)
sage: C0.set_default_frame(e)
sage: C0.default_frame()
Local frame (E|_U, (e_0,e_1))

Python

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='top')
>>> U = M.open_subset('U')
>>> V = M.open_subset('V')
>>> M.declare_union(U, V)
>>> E = M.vector_bundle(Integer(2), 'E')
>>> C0 = E.section_module(); C0
Module C^0(M;E) of sections on the 3-dimensional topological
 manifold M with values in the real vector bundle E of rank 2
>>> e = E.local_frame('e', domain=U)
>>> C0.set_default_frame(e)
>>> C0.default_frame()
Local frame (E|_U, (e_0,e_1))

Sage Live

M = Manifold(3, 'M', structure='top')
U = M.open_subset('U')
V = M.open_subset('V')
M.declare_union(U, V)
E = M.vector_bundle(2, 'E')
C0 = E.section_module(); C0
e = E.local_frame('e', domain=U)
C0.set_default_frame(e)
C0.default_frame()

The local frame is indeed the same, and not a copy:

Sage

sage: e is C0.default_frame()
True

Python

>>> from sage.all import *
>>> e is C0.default_frame()
True

Sage Live

e is C0.default_frame()

Notice, that the local frame is defined on a subset and is not part of the section module \(C^k(M;E)\):

Sage

sage: C0.default_frame().domain()
Open subset U of the 3-dimensional topological manifold M

Python

>>> from sage.all import *
>>> C0.default_frame().domain()
Open subset U of the 3-dimensional topological manifold M

Sage Live

C0.default_frame().domain()

vector_bundle()[source]¶

Return the overlying vector bundle on which the section module is defined.

EXAMPLES:

Sage

sage: M = Manifold(3, 'M', structure='top')
sage: E = M.vector_bundle(2, 'E')
sage: C0 = E.section_module(); C0
Module C^0(M;E) of sections on the 3-dimensional topological
 manifold M with values in the real vector bundle E of rank 2
sage: C0.vector_bundle()
Topological real vector bundle E -> M of rank 2 over the base space
 3-dimensional topological manifold M
sage: E is C0.vector_bundle()
True

Python

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', structure='top')
>>> E = M.vector_bundle(Integer(2), 'E')
>>> C0 = E.section_module(); C0
Module C^0(M;E) of sections on the 3-dimensional topological
 manifold M with values in the real vector bundle E of rank 2
>>> C0.vector_bundle()
Topological real vector bundle E -> M of rank 2 over the base space
 3-dimensional topological manifold M
>>> E is C0.vector_bundle()
True

Sage Live

M = Manifold(3, 'M', structure='top')
E = M.vector_bundle(2, 'E')
C0 = E.section_module(); C0
C0.vector_bundle()
E is C0.vector_bundle()

zero()[source]¶

Return the zero of self.

EXAMPLES:

Sage

sage: M = Manifold(2, 'M', structure='top')
sage: X.<x,y> = M.chart()
sage: E = M.vector_bundle(2, 'E')
sage: C0 = E.section_module()
sage: z = C0.zero(); z
Section zero on the 2-dimensional topological manifold M with values
 in the real vector bundle E of rank 2
sage: z == 0
True

Python

>>> 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')
>>> C0 = E.section_module()
>>> z = C0.zero(); z
Section zero on the 2-dimensional topological manifold M with values
 in the real vector bundle E of rank 2
>>> z == Integer(0)
True

Sage Live

M = Manifold(2, 'M', structure='top')
X.<x,y> = M.chart()
E = M.vector_bundle(2, 'E')
C0 = E.section_module()
z = C0.zero(); z
z == 0