Multivector Field Modules#

The set \(A^p(U, \Phi)\) of \(p\)-vector fields along a differentiable manifold \(U\) with values on a differentiable manifold \(M\) via a differentiable map \(\Phi:\ U \rightarrow M\) (possibly \(U = M\) and \(\Phi = \mathrm{Id}_M\)) is a module over the algebra \(C^k(U)\) of differentiable scalar fields on \(U\). It is a free module if and only if \(M\) is parallelizable. Accordingly, two classes implement \(A^p(U,\Phi)\):

MultivectorModule for \(p\)-vector fields with values on a generic (in practice, not parallelizable) differentiable manifold \(M\)
MultivectorFreeModule for \(p\)-vector fields with values on a parallelizable manifold \(M\)

AUTHORS:

Eric Gourgoulhon (2017): initial version

REFERENCES:

R. L. Bishop and S. L. Goldberg (1980) [BG1980]
C.-M. Marle (1997) [Mar1997]

class sage.manifolds.differentiable.multivector_module.MultivectorFreeModule(vector_field_module, degree)#

Bases: ExtPowerFreeModule

Free module of multivector fields of a given degree \(p\) (\(p\)-vector fields) along a differentiable manifold \(U\) with values on a parallelizable manifold \(M\).

Given a differentiable manifold \(U\) and a differentiable map \(\Phi:\; U \rightarrow M\) to a parallelizable manifold \(M\) of dimension \(n\), the set \(A^p(U, \Phi)\) of \(p\)-vector fields (i.e. alternating tensor fields of type \((p,0)\)) along \(U\) with values on \(M\) is a free module of rank \(\binom{n}{p}\) over \(C^k(U)\), the commutative algebra of differentiable scalar fields on \(U\) (see DiffScalarFieldAlgebra). The standard case of \(p\)-vector fields on a differentiable manifold \(M\) corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).

Note

This class implements \(A^p(U, \Phi)\) in the case where \(M\) is parallelizable; \(A^p(U, \Phi)\) is then a free module. If \(M\) is not parallelizable, the class MultivectorModule must be used instead.

INPUT:

vector_field_module – free module \(\mathfrak{X}(U,\Phi)\) of vector fields along \(U\) associated with the map \(\Phi: U \rightarrow V\)
degree – positive integer; the degree \(p\) of the multivector fields

EXAMPLES:

Free module of 2-vector fields on a parallelizable 3-dimensional manifold:

sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: XM = M.vector_field_module() ; XM
Free module X(M) of vector fields on the 3-dimensional
 differentiable manifold M
sage: A = M.multivector_module(2) ; A
Free module A^2(M) of 2-vector fields on the 3-dimensional
 differentiable manifold M
sage: latex(A)
A^{2}\left(M\right)

A is nothing but the second exterior power of XM, i.e. we have \(A^{2}(M) = \Lambda^2(\mathfrak{X}(M))\) (see ExtPowerFreeModule):

sage: A is XM.exterior_power(2)
True

\(A^{2}(M)\) is a module over the algebra \(C^k(M)\) of (differentiable) scalar fields on \(M\):

sage: A.category()
Category of finite dimensional modules over Algebra of
 differentiable scalar fields on the 3-dimensional
 differentiable manifold M
sage: CM = M.scalar_field_algebra() ; CM
Algebra of differentiable scalar fields on the 3-dimensional
 differentiable manifold M
sage: A in Modules(CM)
True
sage: A.base_ring()
Algebra of differentiable scalar fields on
 the 3-dimensional differentiable manifold M
sage: A.base_module()
Free module X(M) of vector fields on
 the 3-dimensional differentiable manifold M
sage: A.base_module() is XM
True
sage: A.rank()
3

Elements can be constructed from \(A\). In particular, 0 yields the zero element of \(A\):

sage: A(0)
2-vector field zero on the 3-dimensional differentiable
 manifold M
sage: A(0) is A.zero()
True

while non-zero elements are constructed by providing their components in a given vector frame:

sage: comp = [[0,3*x,-z],[-3*x,0,4],[z,-4,0]]
sage: a = A(comp, frame=X.frame(), name='a') ; a
2-vector field a on the 3-dimensional differentiable manifold M
sage: a.display()
a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z

An alternative is to construct the 2-vector field from an empty list of components and to set the nonzero nonredundant components afterwards:

sage: a = A([], name='a')
sage: a[0,1] = 3*x  # component in the manifold's default frame
sage: a[0,2] = -z
sage: a[1,2] = 4
sage: a.display()
a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z

The module \(A^1(M)\) is nothing but \(\mathfrak{X}(M)\) (the free module of vector fields on \(M\)):

sage: A1 = M.multivector_module(1) ; A1
Free module X(M) of vector fields on the 3-dimensional
 differentiable manifold M
sage: A1 is XM
True

There is a coercion map \(A^p(M) \rightarrow T^{(p,0)}(M)\):

sage: T20 = M.tensor_field_module((2,0)); T20
Free module T^(2,0)(M) of type-(2,0) tensors fields on the
 3-dimensional differentiable manifold M
sage: T20.has_coerce_map_from(A)
True

but of course not in the reverse direction, since not all contravariant tensor field is alternating:

sage: A.has_coerce_map_from(T20)
False

The coercion map \(A^2(M) \rightarrow T^{(2,0)}(M)\) in action:

sage: T20 = M.tensor_field_module((2,0)) ; T20
Free module T^(2,0)(M) of type-(2,0) tensors fields on the
 3-dimensional differentiable manifold M
sage: ta = T20(a) ; ta
Tensor field a of type (2,0) on the 3-dimensional differentiable
 manifold M
sage: ta.display()
a = 3*x ∂/∂x⊗∂/∂y - z ∂/∂x⊗∂/∂z - 3*x ∂/∂y⊗∂/∂x + 4 ∂/∂y⊗∂/∂z
 + z ∂/∂z⊗∂/∂x - 4 ∂/∂z⊗∂/∂y
sage: a.display()
a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z
sage: ta.symmetries()  # the antisymmetry is preserved
no symmetry;  antisymmetry: (0, 1)

There is also coercion to subdomains, which is nothing but the restriction of the multivector field to some subset of its domain:

sage: U = M.open_subset('U', coord_def={X: x^2+y^2<1})
sage: B = U.multivector_module(2) ; B
Free module A^2(U) of 2-vector fields on the Open subset U of the
 3-dimensional differentiable manifold M
sage: B.has_coerce_map_from(A)
True
sage: a_U = B(a) ; a_U
2-vector field a on the Open subset U of the 3-dimensional
 differentiable manifold M
sage: a_U.display()
a = 3*x ∂/∂x∧∂/∂y - z ∂/∂x∧∂/∂z + 4 ∂/∂y∧∂/∂z

Element#: alias of MultivectorFieldParal

class sage.manifolds.differentiable.multivector_module.MultivectorModule(vector_field_module, degree)#

Bases: UniqueRepresentation, Parent

Module of multivector fields of a given degree \(p\) (\(p\)-vector fields) along a differentiable manifold \(U\) with values on a differentiable manifold \(M\).

Given a differentiable manifold \(U\) and a differentiable map \(\Phi: U \rightarrow M\) to a differentiable manifold \(M\), the set \(A^p(U, \Phi)\) of \(p\)-vector fields (i.e. alternating tensor fields of type \((p,0)\)) along \(U\) with values on \(M\) is a module over \(C^k(U)\), the commutative algebra of differentiable scalar fields on \(U\) (see DiffScalarFieldAlgebra). The standard case of \(p\)-vector fields on a differentiable manifold \(M\) corresponds to \(U = M\) and \(\Phi = \mathrm{Id}_M\). Other common cases are \(\Phi\) being an immersion and \(\Phi\) being a curve in \(M\) (\(U\) is then an open interval of \(\RR\)).

Note

This class implements \(A^p(U,\Phi)\) in the case where \(M\) is not assumed to be parallelizable; the module \(A^p(U, \Phi)\) is then not necessarily free. If \(M\) is parallelizable, the class MultivectorFreeModule must be used instead.

INPUT:

vector_field_module – module \(\mathfrak{X}(U, \Phi)\) of vector fields along \(U\) with values on \(M\) via the map \(\Phi: U \rightarrow M\)
degree – positive integer; the degree \(p\) of the multivector fields

EXAMPLES:

Module of 2-vector fields on a non-parallelizable 2-dimensional manifold:

sage: M = Manifold(2, 'M')
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V)   # M is the union of U and V
sage: c_xy.<x,y> = U.chart() ; c_uv.<u,v> = V.chart()
sage: transf = c_xy.transition_map(c_uv, (x+y, x-y),
....:  intersection_name='W', restrictions1= x>0,
....:  restrictions2= u+v>0)
sage: inv = transf.inverse()
sage: W = U.intersection(V)
sage: eU = c_xy.frame() ; eV = c_uv.frame()
sage: XM = M.vector_field_module() ; XM
Module X(M) of vector fields on the 2-dimensional differentiable
 manifold M
sage: A = M.multivector_module(2) ; A
Module A^2(M) of 2-vector fields on the 2-dimensional
 differentiable manifold M
sage: latex(A)
A^{2}\left(M\right)

A is nothing but the second exterior power of XM, i.e. we have \(A^{2}(M) = \Lambda^2(\mathfrak{X}(M))\):

sage: A is XM.exterior_power(2)
True

Modules of multivector fields are unique:

sage: A is M.multivector_module(2)
True

\(A^2(M)\) is a module over the algebra \(C^k(M)\) of (differentiable) scalar fields on \(M\):

sage: A.category()
Category of modules over Algebra of differentiable scalar fields
 on the 2-dimensional differentiable manifold M
sage: CM = M.scalar_field_algebra() ; CM
Algebra of differentiable scalar fields on the 2-dimensional
 differentiable manifold M
sage: A in Modules(CM)
True
sage: A.base_ring() is CM
True
sage: A.base_module()
Module X(M) of vector fields on the 2-dimensional differentiable
 manifold M
sage: A.base_module() is XM
True

Elements can be constructed from A(). In particular, 0 yields the zero element of A:

sage: z = A(0) ; z
2-vector field zero on the 2-dimensional differentiable
 manifold M
sage: z.display(eU)
zero = 0
sage: z.display(eV)
zero = 0
sage: z is A.zero()
True

while non-zero elements are constructed by providing their components in a given vector frame:

sage: a = A([[0,3*x],[-3*x,0]], frame=eU, name='a') ; a
2-vector field a on the 2-dimensional differentiable manifold M
sage: a.add_comp_by_continuation(eV, W, c_uv) # finishes initializ. of a
sage: a.display(eU)
a = 3*x ∂/∂x∧∂/∂y
sage: a.display(eV)
a = (-3*u - 3*v) ∂/∂u∧∂/∂v

An alternative is to construct the 2-vector field from an empty list of components and to set the nonzero nonredundant components afterwards:

sage: a = A([], name='a')
sage: a[eU,0,1] = 3*x
sage: a.add_comp_by_continuation(eV, W, c_uv)
sage: a.display(eU)
a = 3*x ∂/∂x∧∂/∂y
sage: a.display(eV)
a = (-3*u - 3*v) ∂/∂u∧∂/∂v

The module \(A^1(M)\) is nothing but the dual of \(\mathfrak{X}(M)\) (the module of vector fields on \(M\)):

sage: A1 = M.multivector_module(1) ; A1
Module X(M) of vector fields on the 2-dimensional differentiable
 manifold M
sage: A1 is XM
True

There is a coercion map \(A^p(M)\rightarrow T^{(p,0)}(M)\):

sage: T20 = M.tensor_field_module((2,0)) ; T20
Module T^(2,0)(M) of type-(2,0) tensors fields on the
 2-dimensional differentiable manifold M
sage: T20.has_coerce_map_from(A)
True

but of course not in the reverse direction, since not all contravariant tensor field is alternating:

sage: A.has_coerce_map_from(T20)
False

The coercion map \(A^2(M) \rightarrow T^{(2,0)}(M)\) in action:

sage: ta = T20(a) ; ta
Tensor field a of type (2,0) on the 2-dimensional differentiable
 manifold M
sage: ta.display(eU)
a = 3*x ∂/∂x⊗∂/∂y - 3*x ∂/∂y⊗∂/∂x
sage: a.display(eU)
a = 3*x ∂/∂x∧∂/∂y
sage: ta.display(eV)
a = (-3*u - 3*v) ∂/∂u⊗∂/∂v + (3*u + 3*v) ∂/∂v⊗∂/∂u
sage: a.display(eV)
a = (-3*u - 3*v) ∂/∂u∧∂/∂v

There is also coercion to subdomains, which is nothing but the restriction of the multivector field to some subset of its domain:

sage: A2U = U.multivector_module(2) ; A2U
Free module A^2(U) of 2-vector fields on the Open subset U of
 the 2-dimensional differentiable manifold M
sage: A2U.has_coerce_map_from(A)
True
sage: a_U = A2U(a) ; a_U
2-vector field a on the Open subset U of the 2-dimensional
 differentiable manifold M
sage: a_U.display(eU)
a = 3*x ∂/∂x∧∂/∂y

Element#: alias of MultivectorField

base_module()#

Return the vector field module on which the multivector field module self is constructed.

OUTPUT:

a VectorFieldModule representing the module on which self is defined

EXAMPLES:

sage: M = Manifold(3, 'M')
sage: A2 = M.multivector_module(2) ; A2
Module A^2(M) of 2-vector fields on the 3-dimensional
 differentiable manifold M
sage: A2.base_module()
Module X(M) of vector fields on the 3-dimensional
 differentiable manifold M
sage: A2.base_module() is M.vector_field_module()
True
sage: U = M.open_subset('U')
sage: A2U = U.multivector_module(2) ; A2U
Module A^2(U) of 2-vector fields on the Open subset U of the
 3-dimensional differentiable manifold M
sage: A2U.base_module()
Module X(U) of vector fields on the Open subset U of the
 3-dimensional differentiable manifold M

degree()#

Return the degree of the multivector fields in self.

OUTPUT:

integer \(p\) such that self is a set of \(p\)-vector fields

EXAMPLES:

sage: M = Manifold(3, 'M')
sage: M.multivector_module(2).degree()
2
sage: M.multivector_module(3).degree()
3

zero()#

Return the zero of self.

EXAMPLES:

sage: M = Manifold(3, 'M')
sage: A2 = M.multivector_module(2)
sage: A2.zero()
2-vector field zero on the 3-dimensional differentiable
 manifold M