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 \(K = \CC\)):
AUTHORS:
Eric Gourgoulhon, Michal Bejger (2013-2015): initial version
Marco Mancini (2018): pullback parallelization
REFERENCES:
- 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 isDifferentiableManifoldHomset
. It inherits from the classContinuousMap
since a differentiable map is obviously a continuous one.INPUT:
parent
– homset \(\mathrm{Hom}(M,N)\) to which the differentiable map belongscoord_functions
– (default:None
) if notNone
, 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 elementname
– (default:None
) name given to the differentiable maplatex_name
– (default:None
) LaTeX symbol to denote the differentiable map; ifNone
, the LaTeX symbol is set toname
is_isomorphism
– (default:False
) determines whether the constructed object is a isomorphism (i.e. a diffeomorphism); if set toTrue
, 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 toTrue
, then \(N\) must be \(M\) and the entrycoord_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 methodadd_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 argumentcoord_functions
is then a mere list of coordinate expressions (and not a dictionary) and the argumentschart1
andchart2
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
andc_cart
are the default charts on respectivelyM
andN
, 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
andPhi
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
withPhi
, 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
) thatster
is indeed the composite ofPsi
andPhi
: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 ofPhi
: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); ifNone
, the domain’s default chart is assumedchart2
– (default:None
) chart on the codomain of \(\Phi\) (coordinates denoted by \((y^i)\) above); ifNone
, 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 aChartFunction
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 chartX
: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 assumedchart2
– (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 fielda
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 aspreimage()
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