# 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. See MixedFormAlgebra for details.

This algebra is endowed with a natural chain complex structure induced by the exterior derivative. The corresponding homology is called de Rham cohomology. See DeRhamCohomologyRing for details.

AUTHORS:

• Michael Jung (2019) : initial version

class sage.manifolds.differentiable.mixed_form_algebra.MixedFormAlgebra(vector_field_module)#

An instance of this class represents the graded algebra of mixed forms. 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

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

Moreover, $$\Omega^*(M,\varphi)$$ inherits the structure of a chain complex, called de Rham complex, with the exterior derivative as boundary map, that is

$0 \rightarrow \Omega^0(M,\varphi) \xrightarrow{\mathrm{d}_0} \Omega^1(M,\varphi) \xrightarrow{\mathrm{d}_1} \dots \xrightarrow{\mathrm{d}_{n-1}} \Omega^n(M,\varphi) \xrightarrow{\mathrm{d}_{n}} 0.$

The induced cohomology is called de Rham cohomology, see cohomology() or DeRhamCohomologyRing respectively.

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()
Join of Category of graded algebras over Symbolic Ring and Category of
chain complexes 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#
cohomology(*args, **kwargs)#

Return the de Rham cohomology of the de Rham complex self.

The $$k$$-th de Rham cohomology is given by

$H^k_{\mathrm{dR}}(M, \varphi) = \left. \mathrm{ker}(\mathrm{d}_k) \middle/ \mathrm{im}(\mathrm{d}_{k-1}) \right. .$

The corresponding ring is given by

$H^*_{\mathrm{dR}}(M, \varphi) = \bigoplus^n_{k=0} H^k_{\mathrm{dR}}(M, \varphi),$

endowed with the cup product as multiplication induced by the wedge product.

See DeRhamCohomologyRing for details.

EXAMPLES:

sage: M = Manifold(3, 'M', latex_name=r'\mathcal{M}')
sage: A = M.mixed_form_algebra()
sage: A.cohomology()
De Rham cohomology ring on the 3-dimensional differentiable
manifold M
differential(degree=None)#

Return the differential of the de Rham complex self given by the exterior derivative.

INPUT:

• degree – (default: None) degree of the differential operator; if none is provided, the differential operator on self is returned.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: C = M.de_rham_complex()
sage: d = C.differential(); d
Generic endomorphism of Graded algebra Omega^*(M) of mixed
differential forms on the 2-dimensional differentiable manifold M
sage: d0 = C.differential(0); d0
Generic morphism:
From: Algebra of differentiable scalar fields on the
2-dimensional differentiable manifold M
To:   Free module Omega^1(M) of 1-forms on the 2-dimensional
differentiable manifold M
sage: f = M.scalar_field(x, name='f'); f.display()
f: M → ℝ
(x, y) ↦ x
sage: d0(f).display()
df = dx
homology(*args, **kwargs)#

Return the de Rham cohomology of the de Rham complex self.

The $$k$$-th de Rham cohomology is given by

$H^k_{\mathrm{dR}}(M, \varphi) = \left. \mathrm{ker}(\mathrm{d}_k) \middle/ \mathrm{im}(\mathrm{d}_{k-1}) \right. .$

The corresponding ring is given by

$H^*_{\mathrm{dR}}(M, \varphi) = \bigoplus^n_{k=0} H^k_{\mathrm{dR}}(M, \varphi),$

endowed with the cup product as multiplication induced by the wedge product.

See DeRhamCohomologyRing for details.

EXAMPLES:

sage: M = Manifold(3, 'M', latex_name=r'\mathcal{M}')
sage: A = M.mixed_form_algebra()
sage: A.cohomology()
De Rham cohomology ring on the 3-dimensional differentiable
manifold M
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]
lift_from_homology(x)#

Lift a cohomology class to the algebra of mixed differential forms.

EXAMPLES:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: C = M.de_rham_complex()
sage: H = C.cohomology()
sage: alpha = M.diff_form(1, [1,1], name='alpha')
sage: alpha.display()
alpha = dx + dy
sage: a = H(alpha); a
[alpha]
sage: C.lift_from_homology(a)
Mixed differential form alpha on the 2-dimensional differentiable
manifold M
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