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)[source]

Bases: Parent, UniqueRepresentation

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

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M')
>>> X = M.chart(names=('x', 'y', 'z',)); (x, y, z,) = X._first_ngens(3)
>>> Omega = M.mixed_form_algebra(); Omega
Graded algebra Omega^*(M) of mixed differential forms on the
 3-dimensional differentiable manifold M
>>> Omega.category()
Join of Category of graded algebras over Symbolic Ring and Category of
 chain complexes over Symbolic Ring
>>> Omega.base_ring()
Symbolic Ring
>>> 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

>>> from sage.all import *
>>> A = Omega(Integer(0)); A
Mixed differential form zero on the 3-dimensional differentiable
 manifold M
>>> A is Omega.zero()
True
>>> B = Omega(Integer(1)); B
Mixed differential form one on the 3-dimensional differentiable
 manifold M
>>> B is Omega.one()
True
>>> C = Omega([Integer(2),Integer(0),Integer(0),Integer(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

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

>>> from sage.all import *
>>> U = M.open_subset('U'); U
Open subset U of the 3-dimensional differentiable manifold M
>>> 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
>>> OmegaU.has_coerce_map_from(Omega)
True
Element[source]

alias of MixedForm

cohomology(*args, **kwargs)[source]

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 also

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

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', latex_name=r'\mathcal{M}')
>>> A = M.mixed_form_algebra()
>>> A.cohomology()
De Rham cohomology ring on the 3-dimensional differentiable
 manifold M
differential(degree=None)[source]

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

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> C = M.de_rham_complex()
>>> d = C.differential(); d
Generic endomorphism of Graded algebra Omega^*(M) of mixed
 differential forms on the 2-dimensional differentiable manifold M
>>> d0 = C.differential(Integer(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
>>> f = M.scalar_field(x, name='f'); f.display()
f: M → ℝ
   (x, y) ↦ x
>>> d0(f).display()
df = dx
homology(*args, **kwargs)[source]

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 also

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

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M', latex_name=r'\mathcal{M}')
>>> A = M.mixed_form_algebra()
>>> A.cohomology()
De Rham cohomology ring on the 3-dimensional differentiable
 manifold M
irange(start=None)[source]

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]

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M')
>>> A = M.mixed_form_algebra()
>>> list(A.irange())
[0, 1, 2, 3]
>>> list(A.irange(Integer(2)))
[2, 3]
lift_from_homology(x)[source]

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

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M')
>>> X = M.chart(names=('x', 'y',)); (x, y,) = X._first_ngens(2)
>>> C = M.de_rham_complex()
>>> H = C.cohomology()
>>> alpha = M.diff_form(Integer(1), [Integer(1),Integer(1)], name='alpha')
>>> alpha.display()
alpha = dx + dy
>>> a = H(alpha); a
[alpha]
>>> C.lift_from_homology(a)
Mixed differential form alpha on the 2-dimensional differentiable
 manifold M
one()[source]

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

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M')
>>> A = M.mixed_form_algebra()
>>> A.one()
Mixed differential form one on the 3-dimensional differentiable
 manifold M
vector_field_module()[source]

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

>>> from sage.all import *
>>> M = Manifold(Integer(2), 'M')
>>> N = Manifold(Integer(3), 'N')
>>> 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
>>> 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
>>> 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()[source]

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

>>> from sage.all import *
>>> M = Manifold(Integer(3), 'M')
>>> A = M.mixed_form_algebra()
>>> A.zero()
Mixed differential form zero on the 3-dimensional differentiable
 manifold M