Multivector Fields

Let \(U\) and \(M\) be two differentiable manifolds. Given a positive integer \(p\) and a differentiable map \(\Phi: U \rightarrow M\), a multivector field of degree \(p\), or \(p\)-vector field, along \(U\) with values on \(M\) is a field along \(U\) of alternating contravariant tensors of rank \(p\) in the tangent spaces to \(M\). The standard case of a multivector field on a differentiable manifold 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 \(\RR\)).

Two classes implement multivector fields, depending whether the manifold \(M\) is parallelizable:

AUTHORS:

  • Eric Gourgoulhon (2017): initial version

REFERENCES:

class sage.manifolds.differentiable.multivectorfield.MultivectorField(vector_field_module, degree, name=None, latex_name=None)

Bases: sage.manifolds.differentiable.tensorfield.TensorField

Multivector field with values on a generic (i.e. a priori not parallelizable) differentiable manifold.

Given a differentiable manifold \(U\), a differentiable map \(\Phi: U \rightarrow M\) to a differentiable manifold \(M\) and a positive integer \(p\), a multivector field of degree \(p\) (or \(p\)-vector field) along \(U\) with values on \(M\supset\Phi(U)\) is a differentiable map

\[a:\ U \longrightarrow T^{(p,0)}M\]

(\(T^{(p,0)}M\) being the tensor bundle of type \((p,0)\) over \(M\)) such that

\[\forall x \in U,\quad a(x) \in \Lambda^p(T_{\Phi(x)} M) ,\]

where \(T_{\Phi(x)} M\) is the vector space tangent to \(M\) at \(\Phi(x)\) and \(\Lambda^p\) stands for the exterior power of degree \(p\) (cf. ExtPowerFreeModule). In other words, \(a(x)\) is an alternating contravariant tensor of degree \(p\) of the tangent vector space \(T_{\Phi(x)} M\).

The standard case of a multivector field on a 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

If \(M\) is parallelizable, the class MultivectorFieldParal 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 \(\Phi\)
  • degree – the degree of the multivector field (i.e. its tensor rank)
  • name – (default: None) name given to the multivector field
  • latex_name – (default: None) LaTeX symbol to denote the multivector field; if none is provided, the LaTeX symbol is set to name

EXAMPLES:

Multivector field of degree 2 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: xy_to_uv = c_xy.transition_map(c_uv, (x+y, x-y),
....:                         intersection_name='W',
....:                         restrictions1= x>0, restrictions2= u+v>0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V)
sage: eU = c_xy.frame() ; eV = c_uv.frame()
sage: a = M.multivector_field(2, name='a') ; a
2-vector field a on the 2-dimensional differentiable manifold M
sage: a.parent()
Module A^2(M) of 2-vector fields on the 2-dimensional differentiable
 manifold M
sage: a.degree()
2

Setting the components of a:

sage: a[eU,0,1] = x*y^2 + 2*x
sage: a.add_comp_by_continuation(eV, W, c_uv)
sage: a.display(eU)
a = (x*y^2 + 2*x) d/dx/\d/dy
sage: a.display(eV)
a = (-1/4*u^3 + 1/4*u*v^2 - 1/4*v^3 + 1/4*(u^2 - 8)*v - 2*u) d/du/\d/dv

The exterior product of two vector fields is a 2-vector field:

sage: a = M.vector_field(name='a')
sage: a[eU,:] = [-y, x]
sage: a.add_comp_by_continuation(eV, W, c_uv)
sage: b = M.vector_field(name='b')
sage: b[eU,:] = [1+x*y, x^2]
sage: b.add_comp_by_continuation(eV, W, c_uv)
sage: s = a.wedge(b) ; s
2-vector field a/\b on the 2-dimensional differentiable manifold M
sage: s.display(eU)
a/\b = (-2*x^2*y - x) d/dx/\d/dy
sage: s.display(eV)
a/\b = (1/2*u^3 - 1/2*u*v^2 - 1/2*v^3 + 1/2*(u^2 + 2)*v + u) d/du/\d/dv

Multiplying a 2-vector field by a scalar field results in another 2-vector field:

sage: f = M.scalar_field({c_xy: (x+y)^2, c_uv: u^2}, name='f')
sage: s = f*s ; s
2-vector field on the 2-dimensional differentiable manifold M
sage: s.display(eU)
(-2*x^2*y^3 - x^3 - (4*x^3 + x)*y^2 - 2*(x^4 + x^2)*y) d/dx/\d/dy
sage: s.display(eV)
(1/2*u^5 - 1/2*u^3*v^2 - 1/2*u^2*v^3 + u^3 + 1/2*(u^4 + 2*u^2)*v)
  d/du/\d/dv
bracket(other)

Return the Schouten-Nijenhuis bracket of self with another multivector field.

The Schouten-Nijenhuis bracket extends the Lie bracket of vector fields (cf. bracket()) to multivector fields.

Denoting by \(A^p(M)\) the \(C^k(M)\)-module of \(p\)-vector fields on the \(C^k\)-differentiable manifold \(M\) over the field \(K\) (cf. MultivectorModule), the Schouten-Nijenhuis bracket is a \(K\)-bilinear map

\[\begin{split}\begin{array}{ccc} A^p(M)\times A^q(M) & \longrightarrow & A^{p+q-1}(M) \\ (a,b) & \longmapsto & [a,b] \end{array}\end{split}\]

which obeys the following properties:

  • if \(p=0\) and \(q=0\), (i.e. \(a\) and \(b\) are two scalar fields), \([a,b]=0\)
  • if \(p=0\) (i.e. \(a\) is a scalar field) and \(q\geq 1\), \([a,b] = - \iota_{\mathrm{d}a} b\) (minus the interior product of the differential of \(a\) by \(b\))
  • if \(p=1\) (i.e. \(a\) is a vector field), \([a,b] = \mathcal{L}_a b\) (the Lie derivative of \(b\) along \(a\))
  • \([a,b] = -(-1)^{(p-1)(q-1)} [b,a]\)
  • for any multivector field \(c\) and \((a,b) \in A^p(M)\times A^q(M)\), \([a,.]\) obeys the graded Leibniz rule
\[[a,b\wedge c] = [a,b]\wedge c + (-1)^{(p-1)q} b\wedge [a,c]\]
  • for \((a,b,c) \in A^p(M)\times A^q(M)\times A^r(M)\), the graded Jacobi identity holds:
\[(-1)^{(p-1)(r-1)}[a,[b,c]] + (-1)^{(q-1)(p-1)}[b,[c,a]] + (-1)^{(r-1)(q-1)}[c,[a,b]] = 0\]

Note

There are two definitions of the Schouten-Nijenhuis bracket in the literature, which differ from each other when \(p\) is even by an overall sign. The definition adopted here is that of [Mar1997], [Kos1985] and Wikipedia article Schouten-Nijenhuis_bracket. The other definition, adopted e.g. by [Nij1955], [Lic1977] and [Vai1994], is \([a,b]' = (-1)^{p+1} [a,b]\).

INPUT:

  • other – a multivector field

OUTPUT:

  • instance of MultivectorField (or of DiffScalarField if \(p=1\) and \(q=0\)) representing the Schouten-Nijenhuis bracket \([a,b]\), where \(a\) is self and \(b\) is other

EXAMPLES:

Bracket of two vector fields on the 2-sphere:

sage: M = Manifold(2, 'S^2', start_index=1) # the sphere S^2
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V)   # S^2 is the union of U and V
sage: c_xy.<x,y> = U.chart() # stereographic coord. North
sage: c_uv.<u,v> = V.chart() # stereographic coord. South
sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
....:                intersection_name='W', restrictions1= x^2+y^2!=0,
....:                restrictions2= u^2+v^2!=0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V) # The complement of the two poles
sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
sage: a = M.vector_field(name='a')
sage: a[e_xy,:] = y, x
sage: a.add_comp_by_continuation(e_uv, W, c_uv)
sage: b = M.vector_field(name='b')
sage: b[e_xy,:] = x*y, x-y
sage: b.add_comp_by_continuation(e_uv, W, c_uv)
sage: s = a.bracket(b); s
Vector field [a,b] on the 2-dimensional differentiable manifold S^2
sage: s.display(e_xy)
[a,b] = (x^2 + y^2 - x + y) d/dx + (-(x - 1)*y - x) d/dy

For two vector fields, the bracket coincides with the Lie derivative:

sage: s == b.lie_derivative(a)
True

Schouten-Nijenhuis bracket of a 2-vector field and a 1-vector field:

sage: c = a.wedge(b); c
2-vector field a/\b on the 2-dimensional differentiable
 manifold S^2
sage: s = c.bracket(a); s
2-vector field [a/\b,a] on the 2-dimensional differentiable
 manifold S^2
sage: s.display(e_xy)
[a/\b,a] = (x^3 + (2*x - 1)*y^2 - x^2 + 2*x*y) d/dx/\d/dy

Since \(a\) is a vector field, we have in this case:

sage: s == - c.lie_derivative(a)
True

See also

MultivectorFieldParal.bracket() for more examples and check of standards identities involving the Schouten-Nijenhuis bracket

degree()

Return the degree of self.

OUTPUT:

  • integer \(p\) such that self is a \(p\)-vector field

EXAMPLES:

sage: M = Manifold(3, 'M')
sage: a = M.multivector_field(2); a
2-vector field on the 3-dimensional differentiable manifold M
sage: a.degree()
2
sage: b = M.vector_field(); b
Vector field on the 3-dimensional differentiable manifold M
sage: b.degree()
1
interior_product(form)

Interior product with a differential form.

If self is a multivector field \(A\) of degree \(p\) and \(B\) is a differential form of degree \(q\geq p\) on the same manifold as \(A\), the interior product of \(A\) by \(B\) is the differential form \(\iota_A B\) of degree \(q-p\) defined by

\[(\iota_A B)_{i_1\ldots i_{q-p}} = A^{k_1\ldots k_p} B_{k_1\ldots k_p i_1\ldots i_{q-p}}\]

Note

A.interior_product(B) yields the same result as A.contract(0,..., p-1, B, 0,..., p-1) (cf. contract()), but interior_product is more efficient, the alternating character of \(A\) being not used to reduce the computation in contract()

INPUT:

  • form – differential form \(B\) (instance of DiffForm); the degree of \(B\) must be at least equal to the degree of self

OUTPUT:

  • scalar field (case \(p=q\)) or DiffForm (case \(p<q\)) representing the interior product \(\iota_A B\), where \(A\) is self

See also

interior_product() for the interior product of a differential form with a multivector field

EXAMPLES:

Interior product of a vector field (\(p=1\)) with a 2-form (\(q=2\)) on the 2-sphere:

sage: M = Manifold(2, 'S^2', start_index=1) # the sphere S^2
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V)   # S^2 is the union of U and V
sage: c_xy.<x,y> = U.chart() # stereographic coord. North
sage: c_uv.<u,v> = V.chart() # stereographic coord. South
sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
....:                intersection_name='W', restrictions1= x^2+y^2!=0,
....:                restrictions2= u^2+v^2!=0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V) # The complement of the two poles
sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
sage: a = M.vector_field(name='a')
sage: a[e_xy,:] = -y, x
sage: a.add_comp_by_continuation(e_uv, W, c_uv)
sage: b = M.diff_form(2, name='b')
sage: b[e_xy,1,2] = 4/(x^2+y^2+1)^2   # the standard area 2-form
sage: b.add_comp_by_continuation(e_uv, W, c_uv)
sage: b.display(e_xy)
b = 4/(x^2 + y^2 + 1)^2 dx/\dy
sage: b.display(e_uv)
b = -4/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) du/\dv
sage: s = a.interior_product(b); s
1-form i_a b on the 2-dimensional differentiable manifold S^2
sage: s.display(e_xy)
i_a b = -4*x/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dx
 - 4*y/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1) dy
sage: s.display(e_uv)
 i_a b = 4*u/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) du
  + 4*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1) dv
sage: s == a.contract(b)
True

Example with \(p=2\) and \(q=2\):

sage: a = M.multivector_field(2, name='a')
sage: a[e_xy,1,2] = x*y
sage: a.add_comp_by_continuation(e_uv, W, c_uv)
sage: a.display(e_xy)
a = x*y d/dx/\d/dy
sage: a.display(e_uv)
a = -u*v d/du/\d/dv
sage: s = a.interior_product(b); s
Scalar field i_a b on the 2-dimensional differentiable manifold S^2
sage: s.display()
i_a b: S^2 --> R
on U: (x, y) |--> 8*x*y/(x^4 + y^4 + 2*(x^2 + 1)*y^2 + 2*x^2 + 1)
on V: (u, v) |--> 8*u*v/(u^4 + v^4 + 2*(u^2 + 1)*v^2 + 2*u^2 + 1)

Some checks:

sage: s == a.contract(0, 1, b, 0, 1)
True
sage: s.restrict(U) == 2 * a[[e_xy,1,2]] * b[[e_xy,1,2]]
True
sage: s.restrict(V) == 2 * a[[e_uv,1,2]] * b[[e_uv,1,2]]
True
wedge(other)

Exterior product with another multivector field.

INPUT:

  • other – another multivector field (on the same manifold)

OUTPUT:

EXAMPLES:

Exterior product of two vector fields on the 2-sphere:

sage: M = Manifold(2, 'S^2', start_index=1) # the sphere S^2
sage: U = M.open_subset('U') ; V = M.open_subset('V')
sage: M.declare_union(U,V)   # S^2 is the union of U and V
sage: c_xy.<x,y> = U.chart() # stereographic coord. North
sage: c_uv.<u,v> = V.chart() # stereographic coord. South
sage: xy_to_uv = c_xy.transition_map(c_uv, (x/(x^2+y^2), y/(x^2+y^2)),
....:                intersection_name='W', restrictions1= x^2+y^2!=0,
....:                restrictions2= u^2+v^2!=0)
sage: uv_to_xy = xy_to_uv.inverse()
sage: W = U.intersection(V) # The complement of the two poles
sage: e_xy = c_xy.frame() ; e_uv = c_uv.frame()
sage: a = M.vector_field(name='a')
sage: a[e_xy,:] = y, x
sage: a.add_comp_by_continuation(e_uv, W, c_uv)
sage: b = M.vector_field(name='b')
sage: b[e_xy,:] = x^2 + y^2, y
sage: b.add_comp_by_continuation(e_uv, W, c_uv)
sage: c = a.wedge(b); c
2-vector field a/\b on the 2-dimensional differentiable
 manifold S^2
sage: c.display(e_xy)
a/\b = (-x^3 - (x - 1)*y^2) d/dx/\d/dy
sage: c.display(e_uv)
a/\b = (-v^2 + u) d/du/\d/dv
class sage.manifolds.differentiable.multivectorfield.MultivectorFieldParal(vector_field_module, degree, name=None, latex_name=None)

Bases: sage.tensor.modules.alternating_contr_tensor.AlternatingContrTensor, sage.manifolds.differentiable.tensorfield_paral.TensorFieldParal

Multivector field with values on a parallelizable manifold.

Given a differentiable manifold \(U\), a differentiable map \(\Phi: U \rightarrow M\) to a parallelizable manifold \(M\) and a positive integer \(p\), a multivector field of degree \(p\) (or \(p\)-vector field) along \(U\) with values on \(M\supset\Phi(U)\) is a differentiable map

\[a:\ U \longrightarrow T^{(p,0)}M\]

(\(T^{(p,0)}M\) being the tensor bundle of type \((p,0)\) over \(M\)) such that

\[\forall x \in U,\quad a(x) \in \Lambda^p(T_{\Phi(x)} M) ,\]

where \(T_{\Phi(x)} M\) is the vector space tangent to \(M\) at \(\Phi(x)\) and \(\Lambda^p\) stands for the exterior power of degree \(p\) (cf. ExtPowerFreeModule). In other words, \(a(x)\) is an alternating contravariant tensor of degree \(p\) of the tangent vector space \(T_{\Phi(x)} M\).

The standard case of a multivector field on a 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

If \(M\) is not parallelizable, the class MultivectorField must be used instead.

INPUT:

  • vector_field_module – free module \(\mathfrak{X}(U,\Phi)\) of vector fields along \(U\) with values on \(M\) via the map \(\Phi\)
  • degree – the degree of the multivector field (i.e. its tensor rank)
  • name – (default: None) name given to the multivector field
  • latex_name – (default: None) LaTeX symbol to denote the multivector field; if none is provided, the LaTeX symbol is set to name

EXAMPLES:

A 2-vector field on a 4-dimensional manifold:

sage: M = Manifold(4, 'M')
sage: c_txyz.<t,x,y,z> = M.chart()
sage: a = M.multivector_field(2, 'a') ; a
2-vector field a on the 4-dimensional differentiable manifold M
sage: a.parent()
Free module A^2(M) of 2-vector fields on the 4-dimensional
 differentiable manifold M

A multivector field is a tensor field of purely contravariant type:

sage: a.tensor_type()
(2, 0)

It is antisymmetric, its components being CompFullyAntiSym:

sage: a.symmetries()
no symmetry;  antisymmetry: (0, 1)
sage: a[0,1] = 2*x
sage: a[1,0]
-2*x
sage: a.comp()
Fully antisymmetric 2-indices components w.r.t. Coordinate frame
 (M, (d/dt,d/dx,d/dy,d/dz))
sage: type(a.comp())
<class 'sage.tensor.modules.comp.CompFullyAntiSym'>

Setting a component with repeated indices to a non-zero value results in an error:

sage: a[1,1] = 3
Traceback (most recent call last):
...
ValueError: by antisymmetry, the component cannot have a nonzero value
 for the indices (1, 1)
sage: a[1,1] = 0  # OK, albeit useless
sage: a[1,2] = 3  # OK

The expansion of a multivector field with respect to a given frame is displayed via the method display():

sage: a.display() # expansion w.r.t. the default frame
a = 2*x d/dt/\d/dx + 3 d/dx/\d/dy
sage: latex(a.display()) # output for the notebook
a = 2 \, x \frac{\partial}{\partial t }\wedge \frac{\partial}{\partial x }
 + 3 \frac{\partial}{\partial x }\wedge \frac{\partial}{\partial y }

Multivector fields can be added or subtracted:

sage: b = M.multivector_field(2)
sage: b[0,1], b[0,2], b[0,3] = y, 2, x+z
sage: s = a + b ; s
2-vector field on the 4-dimensional differentiable manifold M
sage: s.display()
(2*x + y) d/dt/\d/dx + 2 d/dt/\d/dy + (x + z) d/dt/\d/dz + 3 d/dx/\d/dy
sage: s = a - b ; s
2-vector field on the 4-dimensional differentiable manifold M
sage: s.display()
(2*x - y) d/dt/\d/dx - 2 d/dt/\d/dy + (-x - z) d/dt/\d/dz + 3 d/dx/\d/dy

An example of 3-vector field in \(\RR^3\) with Cartesian coordinates:

sage: M = Manifold(3, 'R3', '\RR^3', start_index=1)
sage: c_cart.<x,y,z> = M.chart()
sage: a = M.multivector_field(3, name='a')
sage: a[1,2,3] = x^2+y^2+z^2  # the only independent component
sage: a[:] # all the components are set from the previous line:
[[[0, 0, 0], [0, 0, x^2 + y^2 + z^2], [0, -x^2 - y^2 - z^2, 0]],
 [[0, 0, -x^2 - y^2 - z^2], [0, 0, 0], [x^2 + y^2 + z^2, 0, 0]],
 [[0, x^2 + y^2 + z^2, 0], [-x^2 - y^2 - z^2, 0, 0], [0, 0, 0]]]
sage: a.display()
a = (x^2 + y^2 + z^2) d/dx/\d/dy/\d/dz

Spherical components from the tensorial change-of-frame formula:

sage: c_spher.<r,th,ph> = M.chart(r'r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi')
sage: spher_to_cart = c_spher.transition_map(c_cart,
....:                [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)])
sage: cart_to_spher = spher_to_cart.set_inverse(sqrt(x^2+y^2+z^2),
....:                              atan2(sqrt(x^2+y^2),z), atan2(y, x))
sage: a.comp(c_spher.frame()) # computation of components w.r.t. spherical frame
Fully antisymmetric 3-indices components w.r.t. Coordinate frame
 (R3, (d/dr,d/dth,d/dph))
sage: a.comp(c_spher.frame())[1,2,3, c_spher]
1/sin(th)
sage: a.display(c_spher.frame())
a = sqrt(x^2 + y^2 + z^2)/sqrt(x^2 + y^2) d/dr/\d/dth/\d/dph
sage: a.display(c_spher.frame(), c_spher)
a = 1/sin(th) d/dr/\d/dth/\d/dph

The exterior product of two multivector fields is performed via the method wedge():

sage: a = M.vector_field(name='A')
sage: a[:] = (x*y, -z*x, y)
sage: b = M.vector_field(name='B')
sage: b[:] = (y, z+y, x^2-z^2)
sage: ab = a.wedge(b) ; ab
2-vector field A/\B on the 3-dimensional differentiable manifold R3
sage: ab.display()
A/\B = (x*y^2 + 2*x*y*z) d/dx/\d/dy + (x^3*y - x*y*z^2 - y^2) d/dx/\d/dz
 + (x*z^3 - y^2 - (x^3 + y)*z) d/dy/\d/dz

Let us check the formula relating the exterior product to the tensor product for vector fields:

sage: a.wedge(b) == a*b - b*a
True

The tensor product of a vector field and a 2-vector field is not a 3-vector field but a tensor field of type \((3,0)\) with less symmetries:

sage: c = a*ab ; c
Tensor field A*(A/\B) of type (3,0) on the 3-dimensional differentiable
 manifold R3
sage: c.symmetries()  # the antisymmetry is only w.r.t. the last 2 arguments:
no symmetry;  antisymmetry: (1, 2)

The Lie derivative of a 2-vector field is a 2-vector field:

sage: ab.lie_der(a)
2-vector field on the 3-dimensional differentiable manifold R3
bracket(other)

Return the Schouten-Nijenhuis bracket of self with another multivector field.

The Schouten-Nijenhuis bracket extends the Lie bracket of vector fields (cf. bracket()) to multivector fields.

Denoting by \(A^p(M)\) the \(C^k(M)\)-module of \(p\)-vector fields on the \(C^k\)-differentiable manifold \(M\) over the field \(K\) (cf. MultivectorModule), the Schouten-Nijenhuis bracket is a \(K\)-bilinear map

\[\begin{split}\begin{array}{ccc} A^p(M)\times A^q(M) & \longrightarrow & A^{p+q-1}(M) \\ (a,b) & \longmapsto & [a,b] \end{array}\end{split}\]

which obeys the following properties:

  • if \(p=0\) and \(q=0\), (i.e. \(a\) and \(b\) are two scalar fields), \([a,b]=0\)
  • if \(p=0\) (i.e. \(a\) is a scalar field) and \(q\geq 1\), \([a,b] = - \iota_{\mathrm{d}a} b\) (minus the interior product of the differential of \(a\) by \(b\))
  • if \(p=1\) (i.e. \(a\) is a vector field), \([a,b] = \mathcal{L}_a b\) (the Lie derivative of \(b\) along \(a\))
  • \([a,b] = -(-1)^{(p-1)(q-1)} [b,a]\)
  • for any multivector field \(c\) and \((a,b) \in A^p(M)\times A^q(M)\), \([a,.]\) obeys the graded Leibniz rule
\[[a,b\wedge c] = [a,b]\wedge c + (-1)^{(p-1)q} b\wedge [a,c]\]
  • for \((a,b,c) \in A^p(M)\times A^q(M)\times A^r(M)\), the graded Jacobi identity holds:
\[(-1)^{(p-1)(r-1)}[a,[b,c]] + (-1)^{(q-1)(p-1)}[b,[c,a]] + (-1)^{(r-1)(q-1)}[c,[a,b]] = 0\]

Note

There are two definitions of the Schouten-Nijenhuis bracket in the literature, which differ from each other when \(p\) is even by an overall sign. The definition adopted here is that of [Mar1997], [Kos1985] and Wikipedia article Schouten-Nijenhuis_bracket. The other definition, adopted e.g. by [Nij1955], [Lic1977] and [Vai1994], is \([a,b]' = (-1)^{p+1} [a,b]\).

INPUT:

  • other – a multivector field

OUTPUT:

EXAMPLES:

Let us consider two vector fields on a 3-dimensional manifold:

sage: M = Manifold(3, 'M')
sage: X.<x,y,z> = M.chart()
sage: a = M.vector_field(name='a')
sage: a[:] = x*y+z, x+y-z, z-2*x+y
sage: b = M.vector_field(name='b')
sage: b[:] = y+2*z-x, x^2-y+z, z-x

and form their Schouten-Nijenhuis bracket:

sage: s = a.bracket(b); s
Vector field [a,b] on the 3-dimensional differentiable manifold M
sage: s.display()
[a,b] = (-x^3 + (x + 3)*y - y^2 - (x + 2*y + 1)*z - 2*x) d/dx
 + (2*x^2*y - x^2 + 2*x*z - 3*x) d/dy
 + (-x^2 - (x - 4)*y - 3*x + 2*z) d/dz

Check that \([a,b]\) is actually the Lie bracket:

sage: f = M.scalar_field({X: x+y*z}, name='f')
sage: s(f) == a(b(f)) - b(a(f))
True

Check that \([a,b]\) coincides with the Lie derivative of \(b\) along \(a\):

sage: s == b.lie_derivative(a)
True

Schouten-Nijenhuis bracket for \(p=0\) and \(q=1\):

sage: s = f.bracket(a); s
Scalar field -i_df a on the 3-dimensional differentiable manifold M
sage: s.display()
-i_df a: M --> R
   (x, y, z) |--> x*y - y^2 - (x + 2*y + 1)*z + z^2

Check that \([f,a] = - \iota_{\mathrm{d}f} a = - \mathrm{d}f(a)\):

sage: s == - f.differential()(a)
True

Schouten-Nijenhuis bracket for \(p=0\) and \(q=2\):

sage: c = M.multivector_field(2, name='c')
sage: c[0,1], c[0,2], c[1,2] = x+z+1, x*y+z, x-y
sage: s = f.bracket(c); s
Vector field -i_df c on the 3-dimensional differentiable manifold M
sage: s.display()
-i_df c = (x*y^2 + (x + y + 1)*z + z^2) d/dx
 + (x*y - y^2 - x - z - 1) d/dy + (-x*y - (x - y + 1)*z) d/dz

Check that \([f,c] = - \iota_{\mathrm{d}f} c\):

sage: s == - f.differential().interior_product(c)
True

Schouten-Nijenhuis bracket for \(p=1\) and \(q=2\):

sage: s = a.bracket(c); s
2-vector field [a,c] on the 3-dimensional differentiable manifold M
sage: s.display()
[a,c] = ((x - 1)*y - (y - 2)*z - 2*x - 1) d/dx/\d/dy
 + ((x + 1)*y - (x + 1)*z - 3*x - 1) d/dx/\d/dz
 + (-5*x + y - z - 2) d/dy/\d/dz

Again, since \(a\) is a vector field, the Schouten-Nijenhuis bracket coincides with the Lie derivative:

sage: s == c.lie_derivative(a)
True

Schouten-Nijenhuis bracket for \(p=2\) and \(q=2\):

sage: d = M.multivector_field(2, name='d')
sage: d[0,1], d[0,2], d[1,2] = x-y^2, x+z, z-x-1
sage: s = c.bracket(d); s
3-vector field [c,d] on the 3-dimensional differentiable manifold M
sage: s.display()
[c,d] = (-y^3 + (3*x + 1)*y - y^2 - x - z + 2) d/dx/\d/dy/\d/dz

Let us check the component formula (with respect to the manifold’s default coordinate chart, i.e. X) for \(p=q=2\), taking into account the tensor antisymmetries:

sage: s[0,1,2] == - sum(c[i,0]*d[1,2].diff(i)
....:                 + c[i,1]*d[2,0].diff(i) + c[i,2]*d[0,1].diff(i)
....:                 + d[i,0]*c[1,2].diff(i) + d[i,1]*c[2,0].diff(i)
....:                 + d[i,2]*c[0,1].diff(i) for i in M.irange())
True

Schouten-Nijenhuis bracket for \(p=1\) and \(q=3\):

sage: e = M.multivector_field(3, name='e')
sage: e[0,1,2] = x+y*z+1
sage: s = a.bracket(e); s
3-vector field [a,e] on the 3-dimensional differentiable manifold M
sage: s.display()
[a,e] = (-(2*x + 1)*y + y^2 - (y^2 - x - 1)*z - z^2
 - 2*x - 2) d/dx/\d/dy/\d/dz

Again, since \(p=1\), the bracket coincides with the Lie derivative:

sage: s == e.lie_derivative(a)
True

Schouten-Nijenhuis bracket for \(p=2\) and \(q=3\):

sage: s = c.bracket(e); s
4-vector field [c,e] on the 3-dimensional differentiable manifold M

Since on a 3-dimensional manifold, any 4-vector field is zero, we have:

sage: s.display()
[c,e] = 0

Let us check the graded commutation law \([a,b] = -(-1)^{(p-1)(q-1)} [b,a]\) for various values of \(p\) and \(q\):

sage: f.bracket(a) == - a.bracket(f)  # p=0 and q=1
True
sage: f.bracket(c) == c.bracket(f)    # p=0 and q=2
True
sage: a.bracket(b) == - b.bracket(a)  # p=1 and q=1
True
sage: a.bracket(c) == - c.bracket(a)  # p=1 and q=2
True
sage: c.bracket(d) == d.bracket(c)    # p=2 and q=2
True

Let us check the graded Leibniz rule for \(p=1\) and \(q=1\):

sage: a.bracket(b.wedge(c)) == a.bracket(b).wedge(c) + b.wedge(a.bracket(c))
True

as well as for \(p=2\) and \(q=1\):

sage: c.bracket(a.wedge(b)) == c.bracket(a).wedge(b) - a.wedge(c.bracket(b))
True

Finally let us check the graded Jacobi identity for \(p=1\), \(q=1\) and \(r=2\):

sage: a.bracket(b.bracket(c)) + b.bracket(c.bracket(a)) \
....: + c.bracket(a.bracket(b)) == 0
True

as well as for \(p=1\), \(q=2\) and \(r=2\):

sage: a.bracket(c.bracket(d)) + c.bracket(d.bracket(a)) \
....: - d.bracket(a.bracket(c)) == 0
True
interior_product(form)

Interior product with a differential form.

If self is a multivector field \(A\) of degree \(p\) and \(B\) is a differential form of degree \(q\geq p\) on the same manifold as \(A\), the interior product of \(A\) by \(B\) is the differential form \(\iota_A B\) of degree \(q-p\) defined by

\[(\iota_A B)_{i_1\ldots i_{q-p}} = A^{k_1\ldots k_p} B_{k_1\ldots k_p i_1\ldots i_{q-p}}\]

Note

A.interior_product(B) yields the same result as A.contract(0,..., p-1, B, 0,..., p-1) (cf. contract()), but interior_product is more efficient, the alternating character of \(A\) being not used to reduce the computation in contract()

INPUT:

  • form – differential form \(B\) (instance of DiffFormParal); the degree of \(B\) must be at least equal to the degree of self

OUTPUT:

  • scalar field (case \(p=q\)) or DiffFormParal (case \(p<q\)) representing the interior product \(\iota_A B\), where \(A\) is self

See also

interior_product() for the interior product of a differential form with a multivector field

EXAMPLES:

Interior product with \(p=1\) and \(q=1\) on 4-dimensional manifold:

sage: M = Manifold(4, 'M')
sage: X.<t,x,y,z> = M.chart()
sage: a = M.vector_field(name='a')
sage: a[:] = [x, 1+t^2, x*z, y-3]
sage: b = M.one_form(name='b')
sage: b[:] = [-z^2, 2, x, x-y]
sage: s = a.interior_product(b); s
Scalar field i_a b on the 4-dimensional differentiable manifold M
sage: s.display()
i_a b: M --> R
   (t, x, y, z) |--> x^2*z - x*z^2 + 2*t^2 + (x + 3)*y - y^2
    - 3*x + 2

In this case, we have \(\iota_a b = a^i b_i = a(b) = b(a)\):

sage: all([s == a.contract(b), s == a(b), s == b(a)])
True

Case \(p=1\) and \(q=3\):

sage: c = M.diff_form(3, name='c')
sage: c[0,1,2], c[0,1,3] = x*y - z, -3*t
sage: c[0,2,3], c[1,2,3] = t+x, y
sage: s = a.interior_product(c); s
2-form i_a c on the 4-dimensional differentiable manifold M
sage: s.display()
i_a c = (x^2*y*z - x*z^2 - 3*t*y + 9*t) dt/\dx
 + (-(t^2*x - t)*y + (t^2 + 1)*z - 3*t - 3*x) dt/\dy
 + (3*t^3 - (t*x + x^2)*z + 3*t) dt/\dz
 + ((x^2 - 3)*y + y^2 - x*z) dx/\dy
 + (-x*y*z - 3*t*x) dx/\dz + (t*x + x^2 + (t^2 + 1)*y) dy/\dz
sage: s == a.contract(c)
True

Case \(p=2\) and \(q=3\):

sage: d = M.multivector_field(2, name='d')
sage: d[0,1], d[0,2], d[0,3] = t-x, 2*z, y-1
sage: d[1,2], d[1,3], d[2,3] = z, y+t, 4
sage: s = d.interior_product(c); s
1-form i_d c on the 4-dimensional differentiable manifold M
sage: s.display()
i_d c = (2*x*y*z - 6*t^2 - 6*t*y - 2*z^2 + 8*t + 8*x) dt
 + (-4*x*y*z + 2*(3*t + 4)*y + 4*z^2 - 6*t) dx
 + (2*((t - 1)*x - x^2 - 2*t)*y - 2*y^2 - 2*(t - x)*z + 2*t
 + 2*x) dy + (-6*t^2 + 6*t*x + 2*(2*t + 2*x + y)*z) dz
sage: s == d.contract(0, 1, c, 0, 1)
True
wedge(other)

Exterior product of self with another multivector field.

INPUT:

  • other – another multivector field

OUTPUT:

EXAMPLES:

Exterior product of a vector field and a 2-vector field on a 3-dimensional manifold:

sage: M = Manifold(3, 'M', start_index=1)
sage: X.<x,y,z> = M.chart()
sage: a = M.vector_field(name='a')
sage: a[:] = [2, 1+x, y*z]
sage: b = M.multivector_field(2, name='b')
sage: b[1,2], b[1,3], b[2,3] = y^2, z+x, z^2
sage: a.display()
a = 2 d/dx + (x + 1) d/dy + y*z d/dz
sage: b.display()
b = y^2 d/dx/\d/dy + (x + z) d/dx/\d/dz + z^2 d/dy/\d/dz
sage: s = a.wedge(b); s
3-vector field a/\b on the 3-dimensional differentiable
 manifold M
sage: s.display()
a/\b = (-x^2 + (y^3 - x - 1)*z + 2*z^2 - x) d/dx/\d/dy/\d/dz

Check:

sage: s[1,2,3] == a[1]*b[2,3] + a[2]*b[3,1] + a[3]*b[1,2]
True