Graded Algebra of Mixed Differential Forms

Let \(M\) and \(N\) be differentiable manifolds and \(\varphi: M \to N\) a differentiable map. The space of mixed differential forms along \(\varphi\), denoted by \(\Omega^*(M,\varphi)\), is given by the direct sum \(\bigoplus^n_{j=0} \Omega^j(M,\varphi)\) of differential form modules, where \(n=\dim(N)\). With the wedge product, \(\Omega^*(M,\varphi)\) inherits the structure of a graded algebra.

AUTHORS:

  • Michael Jung (2019) : initial version
class sage.manifolds.differentiable.mixed_form_algebra.MixedFormAlgebra(vector_field_module)

Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

An instance of this class represents the graded algebra of mixed form. That is, if \(\varphi: M \to N\) is a differentiable map between two differentiable manifolds \(M\) and \(N\), the graded algebra of mixed forms \(\Omega^*(M,\varphi)\) along \(\varphi\) is defined via the direct sum \(\bigoplus^{n}_{j=0} \Omega^j(M,\varphi)\) consisting of differential form modules (cf. DiffFormModule), where \(n\) is the dimension of \(N\). Hence, \(\Omega^*(M,\varphi)\) is a module over \(C^k(M)\) and a vector space over \(\RR\) or \(\CC\). Furthermore notice, that

\[\Omega^*(M,\varphi) \cong C^k \left( \bigoplus^n_{j=0} \Lambda^j(\varphi^*T^*N) \right),\]

where \(C^k\) denotes the global section functor for differentiable sections of order \(k\) here.

The wedge product induces a multiplication on \(\Omega^*(M,\varphi)\) and gives it the structure of a graded algebra since

\[\Omega^k(M,\varphi) \wedge \Omega^l(M,\varphi) \subset \Omega^{k+l}(M,\varphi).\]

INPUT:

  • vector_field_module – module \(\mathfrak{X}(M,\varphi)\) of vector fields along \(M\) associated with the map \(\varphi: M \rightarrow N\)

EXAMPLES:

Graded algebra of mixed forms on a 3-dimensional manifold:

sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: Omega = M.mixed_form_algebra(); Omega
Graded algebra Omega^*(M) of mixed differential forms on the
 3-dimensional differentiable manifold M
sage: Omega.category()
Category of graded algebras over Symbolic Ring
sage: Omega.base_ring()
Symbolic Ring
sage: Omega.vector_field_module()
Free module X(M) of vector fields on the 3-dimensional differentiable
 manifold M

Elements can be created from scratch:

sage: A = Omega(0); A
Mixed differential form zero on the 3-dimensional differentiable
 manifold M
sage: A is Omega.zero()
True
sage: B = Omega(1); B
Mixed differential form one on the 3-dimensional differentiable
 manifold M
sage: B is Omega.one()
True
sage: C = Omega([2,0,0,0]); C
Mixed differential form on the 3-dimensional differentiable manifold M

There are some important coercions implemented:

sage: Omega0 = M.scalar_field_algebra(); Omega0
Algebra of differentiable scalar fields on the 3-dimensional
 differentiable manifold M
sage: Omega.has_coerce_map_from(Omega0)
True
sage: Omega2 = M.diff_form_module(2); Omega2
Free module Omega^2(M) of 2-forms on the 3-dimensional differentiable
 manifold M
sage: Omega.has_coerce_map_from(Omega2)
True

Restrictions induce coercions as well:

sage: U = M.open_subset('U'); U
Open subset U of the 3-dimensional differentiable manifold M
sage: OmegaU = U.mixed_form_algebra(); OmegaU
Graded algebra Omega^*(U) of mixed differential forms on the Open subset
 U of the 3-dimensional differentiable manifold M
sage: OmegaU.has_coerce_map_from(Omega)
True
Element

alias of sage.manifolds.differentiable.mixed_form.MixedForm

irange(start=None)

Single index generator.

INPUT:

  • start – (default: None) initial value \(i_0\) of the index between 0 and \(n\), where \(n\) is the manifold’s dimension; if none is provided, the value 0 is assumed

OUTPUT:

  • an iterable index, starting from \(i_0\) and ending at \(n\), where \(n\) is the manifold’s dimension

EXAMPLES:

sage: M = Manifold(3, 'M')
sage: A = M.mixed_form_algebra()
sage: list(A.irange())
[0, 1, 2, 3]
sage: list(A.irange(2))
[2, 3]
one()

Return the one of self.

EXAMPLES:

sage: M = Manifold(3, 'M')
sage: A = M.mixed_form_algebra()
sage: A.one()
Mixed differential form one on the 3-dimensional differentiable
 manifold M
vector_field_module()

Return the underlying vector field module.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: N = Manifold(3, 'N')
sage: Phi = M.diff_map(N, name='Phi'); Phi
Differentiable map Phi from the 2-dimensional differentiable
 manifold M to the 3-dimensional differentiable manifold N
sage: A = M.mixed_form_algebra(Phi); A
Graded algebra Omega^*(M,Phi) of mixed differential forms along the
 2-dimensional differentiable manifold M mapped into the
 3-dimensional differentiable manifold N via Phi
sage: A.vector_field_module()
Module X(M,Phi) of vector fields along the 2-dimensional
 differentiable manifold M mapped into the 3-dimensional
 differentiable manifold N
zero()

Return the zero of self.

EXAMPLES:

sage: M = Manifold(3, 'M')
sage: A = M.mixed_form_algebra()
sage: A.zero()
Mixed differential form zero on the 3-dimensional differentiable
 manifold M