Differentiable Maps between Differentiable Manifolds

The class DiffMap implements differentiable maps from a differentiable manifold \(M\) to a differentiable manifold \(N\) over the same topological field \(K\) as \(M\) (in most applications, \(K = \RR\) or \(K = \CC\)):

\[\Phi: M \longrightarrow N\]

AUTHORS:

• Eric Gourgoulhon, Michal Bejger (2013-2015): initial version

• Marco Mancini (2018): pullback parallelization

REFERENCES:

• Chap. 1 of [KN1963]

• Chaps. 2 and 3 of [Lee2013]

class sage.manifolds.differentiable.diff_map.DiffMap(parent, coord_functions=None, name=None, latex_name=None, is_isomorphism=False, is_identity=False)

Bases: ContinuousMap

Differentiable map between two differentiable manifolds.

This class implements differentiable maps of the type

\[\Phi: M \longrightarrow N\]

where \(M\) and \(N\) are differentiable manifolds over the same topological field \(K\) (in most applications, \(K = \RR\) or \(K = \CC\)).

Differentiable maps are the morphisms of the category of differentiable manifolds. The set of all differentiable maps from \(M\) to \(N\) is therefore the homset between \(M\) and \(N\), which is denoted by \(\mathrm{Hom}(M,N)\).

The class DiffMap is a Sage element class, whose parent class is DifferentiableManifoldHomset. It inherits from the class ContinuousMap since a differentiable map is obviously a continuous one.

INPUT:

• parent – homset \(\mathrm{Hom}(M,N)\) to which the differentiable map belongs

• coord_functions – (default: None) if not None, must be a dictionary of the coordinate expressions (as lists (or tuples) of the coordinates of the image expressed in terms of the coordinates of the considered point) with the pairs of charts (chart1, chart2) as keys (chart1 being a chart on \(M\) and chart2 a chart on \(N\)). If the dimension of the map’s codomain is 1, a single coordinate expression can be passed instead of a tuple with a single element

• name – (default: None) name given to the differentiable map

• latex_name – (default: None) LaTeX symbol to denote the differentiable map; if None, the LaTeX symbol is set to name

• is_isomorphism – (default: False) determines whether the constructed object is a isomorphism (i.e. a diffeomorphism); if set to True, then the manifolds \(M\) and \(N\) must have the same dimension.

• is_identity – (default: False) determines whether the constructed object is the identity map; if set to True, then \(N\) must be \(M\) and the entry coord_functions is not used.

Note

If the information passed by means of the argument coord_functions is not sufficient to fully specify the differentiable map, further coordinate expressions, in other charts, can be subsequently added by means of the method add_expr()

EXAMPLES:

The standard embedding of the sphere \(S^2\) into \(\RR^3\):

sage: M = Manifold(2, 'S^2') # the 2-dimensional sphere S^2
sage: U = M.open_subset('U') # complement of the North pole
sage: c_xy.<x,y> = U.chart() # stereographic coordinates from the North pole
sage: V = M.open_subset('V') # complement of the South pole
sage: c_uv.<u,v> = V.chart() # stereographic coordinates from the South pole
sage: M.declare_union(U,V)   # S^2 is the union of U and V
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: N = Manifold(3, 'R^3', r'\RR^3')  # R^3
sage: c_cart.<X,Y,Z> = N.chart()  # Cartesian coordinates on R^3
sage: Phi = M.diff_map(N,
....: {(c_xy, c_cart): [2*x/(1+x^2+y^2), 2*y/(1+x^2+y^2), (x^2+y^2-1)/(1+x^2+y^2)],
....:  (c_uv, c_cart): [2*u/(1+u^2+v^2), 2*v/(1+u^2+v^2), (1-u^2-v^2)/(1+u^2+v^2)]},
....: name='Phi', latex_name=r'\Phi')
sage: Phi
Differentiable map Phi from the 2-dimensional differentiable manifold
 S^2 to the 3-dimensional differentiable manifold R^3
sage: Phi.parent()
Set of Morphisms from 2-dimensional differentiable manifold S^2 to
 3-dimensional differentiable manifold R^3 in Category of smooth
 manifolds over Real Field with 53 bits of precision
sage: Phi.parent() is Hom(M, N)
True
sage: type(Phi)
<class 'sage.manifolds.differentiable.manifold_homset.DifferentiableManifoldHomset_with_category.element_class'>
sage: Phi.display()
Phi: S^2 → R^3
on U: (x, y) ↦ (X, Y, Z) = (2*x/(x^2 + y^2 + 1), 2*y/(x^2 + y^2 + 1),
 (x^2 + y^2 - 1)/(x^2 + y^2 + 1))
on V: (u, v) ↦ (X, Y, Z) = (2*u/(u^2 + v^2 + 1), 2*v/(u^2 + v^2 + 1),
 -(u^2 + v^2 - 1)/(u^2 + v^2 + 1))

It is possible to create the map via the method diff_map() only in a single pair of charts: the argument coord_functions is then a mere list of coordinate expressions (and not a dictionary) and the arguments chart1 and chart2 have to be provided if the charts differ from the default ones on the domain and/or the codomain:

sage: Phi1 = M.diff_map(N, [2*x/(1+x^2+y^2), 2*y/(1+x^2+y^2),
....:                       (x^2+y^2-1)/(1+x^2+y^2)],
....:                   chart1=c_xy, chart2=c_cart, name='Phi',
....:                   latex_name=r'\Phi')

Since c_xy and c_cart are the default charts on respectively M and N, they can be omitted, so that the above declaration is equivalent to:

sage: Phi1 = M.diff_map(N, [2*x/(1+x^2+y^2), 2*y/(1+x^2+y^2),
....:                       (x^2+y^2-1)/(1+x^2+y^2)],
....:                   name='Phi', latex_name=r'\Phi')

With such a declaration, the differentiable map is only partially defined on the manifold \(S^2\), being known in only one chart:

sage: Phi1.display()
Phi: S^2 → R^3
on U: (x, y) ↦ (X, Y, Z) = (2*x/(x^2 + y^2 + 1), 2*y/(x^2 + y^2 + 1),
 (x^2 + y^2 - 1)/(x^2 + y^2 + 1))

The definition can be completed by means of the method add_expr():

sage: Phi1.add_expr(c_uv, c_cart, [2*u/(1+u^2+v^2), 2*v/(1+u^2+v^2),
....:                              (1-u^2-v^2)/(1+u^2+v^2)])
sage: Phi1.display()
Phi: S^2 → R^3
on U: (x, y) ↦ (X, Y, Z) = (2*x/(x^2 + y^2 + 1), 2*y/(x^2 + y^2 + 1),
 (x^2 + y^2 - 1)/(x^2 + y^2 + 1))
on V: (u, v) ↦ (X, Y, Z) = (2*u/(u^2 + v^2 + 1), 2*v/(u^2 + v^2 + 1),
 -(u^2 + v^2 - 1)/(u^2 + v^2 + 1))

At this stage, Phi1 and Phi are fully equivalent:

sage: Phi1 == Phi
True

The test suite is passed:

sage: TestSuite(Phi).run()
sage: TestSuite(Phi1).run()

The map acts on points:

sage: np = M.point((0,0), chart=c_uv, name='N')  # the North pole
sage: Phi(np)
Point Phi(N) on the 3-dimensional differentiable manifold R^3
sage: Phi(np).coord() # Cartesian coordinates
(0, 0, 1)
sage: sp = M.point((0,0), chart=c_xy, name='S')  # the South pole
sage: Phi(sp).coord() # Cartesian coordinates
(0, 0, -1)

The differential \(\mathrm{d}\Phi\) of the map \(\Phi\) at the North pole and at the South pole:

sage: Phi.differential(np)
Generic morphism:
  From: Tangent space at Point N on the 2-dimensional differentiable manifold S^2
  To:   Tangent space at Point Phi(N) on the 3-dimensional differentiable manifold R^3
sage: Phi.differential(sp)
Generic morphism:
  From: Tangent space at Point S on the 2-dimensional differentiable manifold S^2
  To:   Tangent space at Point Phi(S) on the 3-dimensional differentiable manifold R^3

The matrix of the linear map \(\mathrm{d}\Phi_N\) with respect to the default bases of \(T_N S^2\) and \(T_{\Phi(N)} \RR^3\):

sage: Phi.differential(np).matrix()
[2 0]
[0 2]
[0 0]

the default bases being:

sage: Phi.differential(np).domain().default_basis()
Basis (∂/∂u,∂/∂v) on the Tangent space at Point N on the 2-dimensional
 differentiable manifold S^2
sage: Phi.differential(np).codomain().default_basis()
Basis (∂/∂X,∂/∂Y,∂/∂Z) on the Tangent space at Point Phi(N) on the
 3-dimensional differentiable manifold R^3

Differentiable maps can be composed by means of the operator *: let us introduce the map \(\RR^3\rightarrow \RR^2\) corresponding to the projection from the point \((X,Y,Z)=(0,0,1)\) onto the equatorial plane \(Z=0\):

sage: P = Manifold(2, 'R^2', r'\RR^2') # R^2 (equatorial plane)
sage: cP.<xP, yP> = P.chart()
sage: Psi = N.diff_map(P, (X/(1-Z), Y/(1-Z)), name='Psi',
....:                  latex_name=r'\Psi')
sage: Psi
Differentiable map Psi from the 3-dimensional differentiable manifold
 R^3 to the 2-dimensional differentiable manifold R^2
sage: Psi.display()
Psi: R^3 → R^2
   (X, Y, Z) ↦ (xP, yP) = (-X/(Z - 1), -Y/(Z - 1))

Then we compose Psi with Phi, thereby getting a map \(S^2\rightarrow \RR^2\):

sage: ster = Psi*Phi ; ster
Differentiable map from the 2-dimensional differentiable manifold S^2
 to the 2-dimensional differentiable manifold R^2

Let us test on the South pole (sp) that ster is indeed the composite of Psi and Phi:

sage: ster(sp) == Psi(Phi(sp))
True

Actually ster is the stereographic projection from the North pole, as its coordinate expression reveals:

sage: ster.display()
S^2 → R^2
on U: (x, y) ↦ (xP, yP) = (x, y)
on V: (u, v) ↦ (xP, yP) = (u/(u^2 + v^2), v/(u^2 + v^2))

If its codomain is 1-dimensional, a differentiable map must be defined by a single symbolic expression for each pair of charts, and not by a list/tuple with a single element:

sage: N = Manifold(1, 'N')
sage: c_N = N.chart('X')
sage: Phi = M.diff_map(N, {(c_xy, c_N): x^2+y^2,
....: (c_uv, c_N): 1/(u^2+v^2)})  # not ...[1/(u^2+v^2)] or (1/(u^2+v^2),)

An example of differentiable map \(\RR \rightarrow \RR^2\):

sage: R = Manifold(1, 'R')  # field R
sage: T.<t> = R.chart()  # canonical chart on R
sage: R2 = Manifold(2, 'R^2')  # R^2
sage: c_xy.<x,y> = R2.chart() # Cartesian coordinates on R^2
sage: Phi = R.diff_map(R2, [cos(t), sin(t)], name='Phi') ; Phi
Differentiable map Phi from the 1-dimensional differentiable manifold R
 to the 2-dimensional differentiable manifold R^2
sage: Phi.parent()
Set of Morphisms from 1-dimensional differentiable manifold R to
 2-dimensional differentiable manifold R^2 in Category of smooth
 manifolds over Real Field with 53 bits of precision
sage: Phi.parent() is Hom(R, R2)
True
sage: Phi.display()
Phi: R → R^2
   t ↦ (x, y) = (cos(t), sin(t))

An example of diffeomorphism between the unit open disk and the Euclidean plane \(\RR^2\):

sage: D = R2.open_subset('D', coord_def={c_xy: x^2+y^2<1}) # the open unit disk
sage: Phi = D.diffeomorphism(R2, [x/sqrt(1-x^2-y^2), y/sqrt(1-x^2-y^2)],
....:                        name='Phi', latex_name=r'\Phi')
sage: Phi
Diffeomorphism Phi from the Open subset D of the 2-dimensional
 differentiable manifold R^2 to the 2-dimensional differentiable
 manifold R^2
sage: Phi.parent()
Set of Morphisms from Open subset D of the 2-dimensional differentiable
 manifold R^2 to 2-dimensional differentiable manifold R^2 in Category
 of smooth manifolds over Real Field with 53 bits of precision
sage: Phi.parent() is Hom(D, R2)
True
sage: Phi.display()
Phi: D → R^2
   (x, y) ↦ (x, y) = (x/sqrt(-x^2 - y^2 + 1), y/sqrt(-x^2 - y^2 + 1))

The image of a point:

sage: p = D.point((1/2,0))
sage: q = Phi(p) ; q
Point on the 2-dimensional differentiable manifold R^2
sage: q.coord()
(1/3*sqrt(3), 0)

The inverse diffeomorphism is computed by means of the method inverse():

sage: Phi.inverse()
Diffeomorphism Phi^(-1) from the 2-dimensional differentiable manifold R^2
 to the Open subset D of the 2-dimensional differentiable manifold R^2
sage: Phi.inverse().display()
Phi^(-1): R^2 → D
   (x, y) ↦ (x, y) = (x/sqrt(x^2 + y^2 + 1), y/sqrt(x^2 + y^2 + 1))

Equivalently, one may use the notations ^(-1) or ~ to get the inverse:

sage: Phi^(-1) is Phi.inverse()
True
sage: ~Phi is Phi.inverse()
True

Check that ~Phi is indeed the inverse of Phi:

sage: (~Phi)(q) == p
True
sage: Phi * ~Phi == R2.identity_map()
True
sage: ~Phi * Phi == D.identity_map()
True

The coordinate expression of the inverse diffeomorphism:

sage: (~Phi).display()
Phi^(-1): R^2 → D
   (x, y) ↦ (x, y) = (x/sqrt(x^2 + y^2 + 1), y/sqrt(x^2 + y^2 + 1))

A special case of diffeomorphism: the identity map of the open unit disk:

sage: id = D.identity_map() ; id
Identity map Id_D of the Open subset D of the 2-dimensional
 differentiable manifold R^2
sage: latex(id)
\mathrm{Id}_{D}
sage: id.parent()
Set of Morphisms from Open subset D of the 2-dimensional differentiable
 manifold R^2 to Open subset D of the 2-dimensional differentiable
 manifold R^2 in Join of Category of subobjects of sets and Category of
 smooth manifolds over Real Field with 53 bits of precision
sage: id.parent() is Hom(D, D)
True
sage: id is Hom(D,D).one()  # the identity element of the monoid Hom(D,D)
True

The identity map acting on a point:

sage: id(p)
Point on the 2-dimensional differentiable manifold R^2
sage: id(p) == p
True
sage: id(p) is p
True

The coordinate expression of the identity map:

sage: id.display()
Id_D: D → D
   (x, y) ↦ (x, y)

The identity map is its own inverse:

sage: id^(-1) is id
True
sage: ~id is id
True

differential(point)

Return the differential of self at a given point.

If the differentiable map self is

\[\Phi: M \longrightarrow N,\]

where \(M\) and \(N\) are differentiable manifolds, the differential of \(\Phi\) at a point \(p \in M\) is the tangent space linear map:

\[\mathrm{d}\Phi_p: T_p M \longrightarrow T_{\Phi(p)} N\]

defined by

\[\begin{split}\begin{array}{rccc} \forall v\in T_p M,\quad \mathrm{d}\Phi_p(v) : & C^k(N) & \longrightarrow & \mathbb{R} \\ & f & \longmapsto & v(f\circ \Phi) \end{array}\end{split}\]

INPUT:

• point – point \(p\) in the domain \(M\) of the differentiable map \(\Phi\)

OUTPUT:

• \(\mathrm{d}\Phi_p\), the differential of \(\Phi\) at \(p\), as a FiniteRankFreeModuleMorphism

EXAMPLES:

Differential of a differentiable map between a 2-dimensional manifold and a 3-dimensional one:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N')
sage: Y.<u,v,w> = N.chart()
sage: Phi = M.diff_map(N, {(X,Y): (x-2*y, x*y, x^2-y^3)}, name='Phi',
....:                  latex_name = r'\Phi')
sage: p = M.point((2,-1), name='p')
sage: dPhip = Phi.differential(p) ; dPhip
Generic morphism:
  From: Tangent space at Point p on the 2-dimensional differentiable manifold M
  To:   Tangent space at Point Phi(p) on the 3-dimensional differentiable manifold N
sage: latex(dPhip)
{\mathrm{d}\Phi}_{p}
sage: dPhip.parent()
Set of Morphisms from Tangent space at Point p on the 2-dimensional
 differentiable manifold M to Tangent space at Point Phi(p) on the
 3-dimensional differentiable manifold N in Category of finite
 dimensional vector spaces over Symbolic Ring

The matrix of \(\mathrm{d}\Phi_p\) w.r.t. to the default bases of \(T_p M\) and \(T_{\Phi(p)} N\):

sage: dPhip.matrix()
[ 1 -2]
[-1  2]
[ 4 -3]

differential_functions(chart1=None, chart2=None)

Return the coordinate expression of the differential of the differentiable map with respect to a pair of charts.

If the differentiable map is

\[\Phi: M \longrightarrow N,\]

where \(M\) and \(N\) are differentiable manifolds, the differential of \(\Phi\) at a point \(p \in M\) is the tangent space linear map:

\[\mathrm{d}\Phi_p: T_p M \longrightarrow T_{\Phi(p)} N\]

defined by

\[\begin{split}\begin{array}{rccc} \forall v\in T_p M,\quad \mathrm{d}\Phi_p(v) : & C^k(N) & \longrightarrow & \mathbb{R}, \\ & f & \longmapsto & v(f\circ \Phi). \end{array}\end{split}\]

If the coordinate expression of \(\Phi\) is

\[y^i = Y^i(x^1, \ldots, x^n), \quad 1 \leq i \leq m,\]

where \((x^1, \ldots, x^n)\) are coordinates of a chart on \(M\) and \((y^1, \ldots, y^m)\) are coordinates of a chart on \(\Phi(M)\), the expression of the differential of \(\Phi\) with respect to these coordinates is

\[J_{ij} = \frac{\partial Y^i}{\partial x^j} \quad 1\leq i \leq m, \qquad 1 \leq j \leq n.\]

\(\left. J_{ij} \right|_p\) is then the matrix of the linear map \(\mathrm{d}\Phi_p\) with respect to the bases of \(T_p M\) and \(T_{\Phi(p)} N\) associated to the above charts:

\[\mathrm{d}\Phi_p\left( \left. \frac{\partial}{\partial x^j} \right|_p \right) = \left. J_{ij} \right|_p \; \left. \frac{\partial}{\partial y^i} \right| _{\Phi(p)}.\]

INPUT:

• chart1 – (default: None) chart on the domain \(M\) of \(\Phi\) (coordinates denoted by \((x^j)\) above); if None, the domain’s default chart is assumed

• chart2 – (default: None) chart on the codomain of \(\Phi\) (coordinates denoted by \((y^i)\) above); if None, the codomain’s default chart is assumed

OUTPUT:

• the functions \(J_{ij}\) as a double array, \(J_{ij}\) being the element [i][j] represented by a ChartFunction

To get symbolic expressions, use the method jacobian_matrix() instead.

EXAMPLES:

Differential functions of a map between a 2-dimensional manifold and a 3-dimensional one:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N')
sage: Y.<u,v,w> = N.chart()
sage: Phi = M.diff_map(N, {(X,Y): (x-2*y, x*y, x^2-y^3)}, name='Phi',
....:                  latex_name = r'\Phi')
sage: J = Phi.differential_functions(X, Y) ; J
[     1     -2]
[     y      x]
[   2*x -3*y^2]

The result is cached:

sage: Phi.differential_functions(X, Y) is J
True

The elements of J are functions of the coordinates of the chart X:

sage: J[2][0]
2*x
sage: type(J[2][0])
<class 'sage.manifolds.chart_func.ChartFunctionRing_with_category.element_class'>

sage: J[2][0].display()
(x, y) ↦ 2*x

In contrast, the method jacobian_matrix() leads directly to symbolic expressions:

sage: JJ = Phi.jacobian_matrix(X,Y) ; JJ
[     1     -2]
[     y      x]
[   2*x -3*y^2]
sage: JJ[2,0]
2*x
sage: type(JJ[2,0])
<class 'sage.symbolic.expression.Expression'>
sage: bool( JJ[2,0] == J[2][0].expr() )
True

jacobian_matrix(chart1=None, chart2=None)

Return the Jacobian matrix resulting from the coordinate expression of the differentiable map with respect to a pair of charts.

If \(\Phi\) is the current differentiable map and its coordinate expression is

\[y^i = Y^i(x^1, \ldots, x^n), \quad 1 \leq i \leq m,\]

where \((x^1, \ldots, x^n)\) are coordinates of a chart \(X\) on the domain of \(\Phi\) and \((y^1, \ldots, y^m)\) are coordinates of a chart \(Y\) on the codomain of \(\Phi\), the Jacobian matrix of the differentiable map \(\Phi\) w.r.t. to charts \(X\) and \(Y\) is

\[\begin{split}J = \left( \frac{\partial Y^i}{\partial x^j} \right)_{\substack{1 \leq i \leq m \\ 1 \leq j \leq n}},\end{split}\]

where \(i\) is the row index and \(j\) the column one.

INPUT:

• chart1 – (default: None) chart \(X\) on the domain of \(\Phi\); if none is provided, the domain’s default chart is assumed

• chart2 – (default: None) chart \(Y\) on the codomain of \(\Phi\); if none is provided, the codomain’s default chart is assumed

OUTPUT:

• the matrix \(J\) defined above

EXAMPLES:

Jacobian matrix of a map between a 2-dimensional manifold and a 3-dimensional one:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: N = Manifold(3, 'N')
sage: Y.<u,v,w> = N.chart()
sage: Phi = M.diff_map(N, {(X,Y): (x-2*y, x*y, x^2-y^3)}, name='Phi',
....:                  latex_name = r'\Phi')
sage: Phi.display()
Phi: M → N
   (x, y) ↦ (u, v, w) = (x - 2*y, x*y, -y^3 + x^2)
sage: J = Phi.jacobian_matrix(X, Y) ; J
[     1     -2]
[     y      x]
[   2*x -3*y^2]
sage: J.parent()
Full MatrixSpace of 3 by 2 dense matrices over Symbolic Ring

pullback(tensor_or_codomain_subset, name=None, latex_name=None)

Pullback operator associated with self.

In what follows, let \(\Phi\) denote a differentiable map self, \(M\) its domain and \(N\) its codomain.

INPUT:

One of the following:

• tensor_or_codomain_subset – one of the following:

  • a TensorField; a fully covariant tensor field \(T\) on \(N\), i.e. a tensor field of type \((0, p)\), with \(p\) a positive or zero integer; the case \(p = 0\) corresponds to a scalar field

  • a ManifoldSubset \(S\)

OUTPUT:

• (if the input is a tensor field \(T\)) a TensorField representing a fully covariant tensor field on \(M\) that is the pullback of \(T\) by \(\Phi\)

• (if the input is a manifold subset \(S\)) a ManifoldSubset that is the preimage \(\Phi^{-1}(S)\); same as preimage()

EXAMPLES:

Pullback on \(S^2\) of a scalar field defined on \(R^3\):

sage: M = Manifold(2, 'S^2', start_index=1)
sage: U = M.open_subset('U') # the complement of a meridian (domain of spherical coordinates)
sage: c_spher.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi') # spherical coord. on U
sage: N = Manifold(3, 'R^3', r'\RR^3', start_index=1)
sage: c_cart.<x,y,z> = N.chart() # Cartesian coord. on R^3
sage: Phi = U.diff_map(N, (sin(th)*cos(ph), sin(th)*sin(ph), cos(th)),
....:                  name='Phi', latex_name=r'\Phi')
sage: f = N.scalar_field(x*y*z, name='f') ; f
Scalar field f on the 3-dimensional differentiable manifold R^3
sage: f.display()
f: R^3 → ℝ
   (x, y, z) ↦ x*y*z
sage: pf = Phi.pullback(f) ; pf
Scalar field Phi^*(f) on the Open subset U of the 2-dimensional
 differentiable manifold S^2
sage: pf.display()
Phi^*(f): U → ℝ
   (th, ph) ↦ cos(ph)*cos(th)*sin(ph)*sin(th)^2

Pullback on \(S^2\) of the standard Euclidean metric on \(R^3\):

sage: g = N.sym_bilin_form_field(name='g')
sage: g[1,1], g[2,2], g[3,3] = 1, 1, 1
sage: g.display()
g = dx⊗dx + dy⊗dy + dz⊗dz
sage: pg = Phi.pullback(g) ; pg
Field of symmetric bilinear forms Phi^*(g) on the Open subset U of
 the 2-dimensional differentiable manifold S^2
sage: pg.display()
Phi^*(g) = dth⊗dth + sin(th)^2 dph⊗dph

Parallel computation:

sage: Parallelism().set('tensor', nproc=2)
sage: pg = Phi.pullback(g) ; pg
Field of symmetric bilinear forms Phi^*(g) on the Open subset U of
 the 2-dimensional differentiable manifold S^2
sage: pg.display()
Phi^*(g) = dth⊗dth + sin(th)^2 dph⊗dph
sage: Parallelism().set('tensor', nproc=1)  # switch off parallelization

Pullback on \(S^2\) of a 3-form on \(R^3\):

sage: a = N.diff_form(3, name='A')
sage: a[1,2,3] = f
sage: a.display()
A = x*y*z dx∧dy∧dz
sage: pa = Phi.pullback(a) ; pa
3-form Phi^*(A) on the Open subset U of the 2-dimensional
 differentiable manifold S^2
sage: pa.display() # should be zero (as any 3-form on a 2-dimensional manifold)
Phi^*(A) = 0

pushforward(tensor)

Pushforward operator associated with self.

In what follows, let \(\Phi\) denote the differentiable map, \(M\) its domain and \(N\) its codomain.

INPUT:

• tensor – TensorField; a fully contrariant tensor field \(T\) on \(M\), i.e. a tensor field of type \((p, 0)\), with \(p\) a positive integer

OUTPUT:

• a TensorField representing a fully contravariant tensor field along \(M\) with values in \(N\), which is the pushforward of \(T\) by \(\Phi\)

EXAMPLES:

Pushforward of a vector field on the 2-sphere \(S^2\) to the Euclidean 3-space \(\RR^3\), via the standard embedding of \(S^2\):

sage: S2 = Manifold(2, 'S^2', start_index=1)
sage: U = S2.open_subset('U')  # domain of spherical coordinates
sage: spher.<th,ph> = U.chart(r'th:(0,pi):\theta ph:(0,2*pi):\phi')
sage: R3 = Manifold(3, 'R^3', start_index=1)
sage: cart.<x,y,z> = R3.chart()
sage: Phi = U.diff_map(R3, {(spher, cart): [sin(th)*cos(ph),
....:   sin(th)*sin(ph), cos(th)]}, name='Phi', latex_name=r'\Phi')
sage: v = U.vector_field(name='v')
sage: v[:] = 0, 1
sage: v.display()
v = ∂/∂ph
sage: pv = Phi.pushforward(v); pv
Vector field Phi_*(v) along the Open subset U of the 2-dimensional
 differentiable manifold S^2 with values on the 3-dimensional
 differentiable manifold R^3
sage: pv.display()
Phi_*(v) = -sin(ph)*sin(th) ∂/∂x + cos(ph)*sin(th) ∂/∂y

Pushforward of a vector field on the real line to the \(\RR^3\), via a helix embedding:

sage: R.<t> = manifolds.RealLine()
sage: Psi = R.diff_map(R3, [cos(t), sin(t), t], name='Psi',
....:                  latex_name=r'\Psi')
sage: u = R.vector_field(name='u')
sage: u[0] = 1
sage: u.display()
u = ∂/∂t
sage: pu = Psi.pushforward(u); pu
Vector field Psi_*(u) along the Real number line ℝ with values on
 the 3-dimensional differentiable manifold R^3
sage: pu.display()
Psi_*(u) = -sin(t) ∂/∂x + cos(t) ∂/∂y + ∂/∂z