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)#

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: 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

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

sage: E.is_manifestly_trivial()
True

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

sage: isinstance(C0, FiniteRankFreeModule)
True

Some elements are:

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

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

sage: C0.rank()
2

The basis is given by the definition above:

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

The test suite is passed as well:

sage: TestSuite(C0).run()

Element#: alias of TrivialSection

base_space()#

Return the base space of the sections in this module.

EXAMPLES:

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

basis(symbol=None, latex_symbol=None, from_frame=None, indices=None, latex_indices=None, symbol_dual=None, latex_symbol_dual=None)#

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: 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))

See LocalFrame for more examples and documentation.

default_frame()#

Return the default basis of the free module self.

The default basis is simply a basis whose name can be skipped in methods requiring a basis as an argument. By default, it is the first basis introduced on the module. It can be changed by the method set_default_basis().

OUTPUT:

instance of FreeModuleBasis

EXAMPLES:

At the module construction, no default basis is assumed:

sage: M = FiniteRankFreeModule(ZZ, 2, name='M', start_index=1)
sage: M.default_basis()
No default basis has been defined on the
 Rank-2 free module M over the Integer Ring

The first defined basis becomes the default one:

sage: e = M.basis('e') ; e
Basis (e_1,e_2) on the Rank-2 free module M over the Integer Ring
sage: M.default_basis()
Basis (e_1,e_2) on the Rank-2 free module M over the Integer Ring
sage: f =  M.basis('f') ; f
Basis (f_1,f_2) on the Rank-2 free module M over the Integer Ring
sage: M.default_basis()
Basis (e_1,e_2) on the Rank-2 free module M over the Integer Ring

domain()#

Return the domain of the section module.

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); 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

set_default_frame(basis)#

Sets the default basis of self.

The default basis is simply a basis whose name can be skipped in methods requiring a basis as an argument. By default, it is the first basis introduced on the module.

INPUT:

basis – instance of FreeModuleBasis representing a basis on self

EXAMPLES:

Changing the default basis on a rank-3 free module:

sage: M = FiniteRankFreeModule(ZZ, 3, name='M', start_index=1)
sage: e = M.basis('e') ; e
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
sage: f =  M.basis('f') ; f
Basis (f_1,f_2,f_3) on the Rank-3 free module M over the Integer Ring
sage: M.default_basis()
Basis (e_1,e_2,e_3) on the Rank-3 free module M over the Integer Ring
sage: M.set_default_basis(f)
sage: M.default_basis()
Basis (f_1,f_2,f_3) on the Rank-3 free module M over the Integer Ring

vector_bundle()#

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

EXAMPLES:

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

class sage.manifolds.section_module.SectionModule(vbundle, domain)#

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: 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

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

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

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

sage: isinstance(C0, FiniteRankFreeModule)
False

since the Möbius bundle is not trivial:

sage: E.is_manifestly_trivial()
False

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

The zero element of the module:

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

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: 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

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

Element#: alias of Section

base_space()#

Return the base space of the sections in this module.

EXAMPLES:

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

default_frame()#

Return the default frame defined on self.

EXAMPLES:

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

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))

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

sage: e is C0.default_frame()
True

domain()#

Return the domain of the section module.

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); 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

set_default_frame(basis)#

Set the default local frame on self.

EXAMPLES:

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

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))

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

sage: e is C0.default_frame()
True

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

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

vector_bundle()#

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

EXAMPLES:

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

zero()#

Return the zero of self.

EXAMPLES:

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