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 3dimensional 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 3dimensional 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 3dimensional differentiable manifold M
Elements can be created from scratch:
sage: A = Omega(0); A Mixed differential form zero on the 3dimensional differentiable manifold M sage: A is Omega.zero() True sage: B = Omega(1); B Mixed differential form one on the 3dimensional differentiable manifold M sage: B is Omega.one() True sage: C = Omega([2,0,0,0]); C Mixed differential form on the 3dimensional differentiable manifold M
There are some important coercions implemented:
sage: Omega0 = M.scalar_field_algebra(); Omega0 Algebra of differentiable scalar fields on the 3dimensional 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 2forms on the 3dimensional 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 3dimensional 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 3dimensional differentiable manifold M sage: OmegaU.has_coerce_map_from(Omega) True

Element
¶

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