Editing Pushforward (differential)

{{Short description|Linear approximation of smooth maps on tangent spaces}}
{{other uses|Pushforward (disambiguation){{!}}Pushforward}}
{{Use American English|date = March 2019}}

[[File:pushforward.svg|thumb|upright=1.5|alt="If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N."|If a map, ''φ'', carries every point on manifold ''M'' to manifold ''N'' then the pushforward of ''φ'' carries vectors in the tangent space at every point in ''M'' to a tangent space at every point in ''N''.]]

In [[differential geometry]], '''pushforward''' is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that <math>\varphi\colon M\to N</math> is a [[smooth map]] between [[smooth manifold]]s; then the '''differential''' of <math>\varphi</math> at a point <math>x</math>, denoted <math>\mathrm d\varphi_x</math>, is, in some sense, the best [[linear approximation]] of <math>\varphi</math> near <math>x</math>. It can be viewed as a generalization of the [[total derivative]] of ordinary calculus. Explicitly, the differential is a [[linear map]] from the [[tangent space]] of <math>M</math> at <math>x</math> to the tangent space of <math>N</math> at <math>\varphi(x)</math>, <math>\mathrm d\varphi_x\colon T_xM \to T_{\varphi(x)}N</math>. Hence it can be used to ''push'' tangent vectors on <math>M</math> ''forward'' to tangent vectors on <math>N</math>. The differential of a map <math>\varphi</math> is also called, by various authors, the '''derivative''' or '''total derivative''' of <math>\varphi</math>.

== Motivation ==
Let <math>\varphi: U \to V</math> be a [[Smooth function#Smooth functions on and between manifolds|smooth map]] from an [[Open subset#Euclidean space|open subset]] <math>U</math> of <math>\R^m</math> to an open subset <math>V</math> of <math>\R^n</math>. For any point <math>x</math> in <math>U</math>, the [[Jacobian matrix and determinant|Jacobian]] of <math>\varphi</math> at <math>x</math> (with respect to the standard coordinates) is the [[matrix (mathematics)|matrix]] representation of the [[total derivative]] of <math>\varphi</math> at <math>x</math>, which is a [[linear map]]

:<math>d\varphi_x:T_x\R^m\to T_{\varphi(x)}\R^n</math>

between their tangent spaces. Note the tangent spaces <math>T_x\R^m,T_{\varphi(x)}\R^n</math> are isomorphic to <math>\mathbb{R}^m</math> and <math>\mathbb{R}^n</math>, respectively. The pushforward generalizes this construction to the case that <math>\varphi</math> is a smooth function between ''any'' [[Manifold#Differentiable manifolds|smooth manifolds]] <math>M</math> and <math>N</math>.

== The differential of a smooth map ==
Let <math>\varphi \colon M \to N </math> be a smooth map of smooth manifolds. Given <math> x \in M, </math> the '''differential''' of <math> \varphi </math> at <math> x </math> is a linear map

:<math>d\varphi_x \colon\ T_xM\to T_{\varphi(x)}N\,</math>

from the [[tangent space]] of <math> M </math> at <math> x </math> to the tangent space of <math> N </math> at <math> \varphi(x). </math> The image <math> d\varphi_x X </math> of a tangent vector <math> X \in T_x M </math> under <math> d\varphi_x </math> is sometimes called the '''pushforward''' of <math> X </math> by <math> \varphi. </math> The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see [[tangent space]]).

If tangent vectors are defined as equivalence classes of the curves <math>\gamma</math> for which <math> \gamma(0) = x, </math> then the differential is given by

:<math>d\varphi_x(\gamma'(0)) = (\varphi \circ \gamma)'(0).</math>

Here, <math> \gamma </math> is a curve in <math> M </math> with <math> \gamma(0) = x, </math> and <math>\gamma'(0)</math> is tangent vector to the curve <math> \gamma </math> at <math> 0. </math> In other words, the pushforward of the tangent vector to the curve <math> \gamma </math> at <math> 0 </math> is the tangent vector to the curve <math>\varphi \circ \gamma</math> at <math> 0. </math>

Alternatively, if tangent vectors are defined as [[derivation (abstract algebra)|derivations]] acting on smooth real-valued functions, then the differential is given by
:<math>d\varphi_x(X)(f) = X(f \circ \varphi),</math>

for an arbitrary function <math>f \in C^\infty(N)</math> and an arbitrary derivation <math>X \in T_xM</math> at point <math>x \in M</math> (a [[Derivation (abstract algebra)|derivation]] is defined as a [[linear map]] <math>X \colon C^\infty(M) \to \R</math> that satisfies the [[Product rule|Leibniz rule]], see: [[Tangent space#Definition via derivations|definition of tangent space via derivations]]). By definition, the pushforward of <math>X</math> is in <math>T_{\varphi(x)}N</math> and therefore itself is a derivation, <math>d\varphi_x(X) \colon C^\infty(N) \to \R</math>.

After choosing two [[manifold (mathematics)|charts]] around <math> x </math> and around <math> \varphi(x), </math> <math> \varphi </math> is locally determined by a smooth map <math>\widehat{\varphi} \colon U \to V</math> between open sets of <math>\R^m</math> and <math>\R^n</math>, and

:<math>d\varphi_x\left(\frac{\partial}{\partial u^a}\right)  = \frac{\partial{\widehat{\varphi}}^b}{\partial u^a} \frac{\partial}{\partial v^b},</math>

in the [[Einstein summation notation]], where the partial derivatives are evaluated at the point in <math> U </math> corresponding to <math> x </math> in the given chart.

Extending by linearity gives the following matrix

:<math>\left(d\varphi_x\right)_a^{\;b} = \frac{\partial{\widehat{\varphi}}^b}{\partial u^a}.</math>

Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map <math> \varphi </math> at each point. Therefore, in some chosen local coordinates, it is represented by the [[Jacobian matrix]] of the corresponding smooth map from <math>\R^m</math> to <math>\R^n</math>. In general, the differential need not be invertible. However, if <math> \varphi </math> is a [[local diffeomorphism]], then <math> d\varphi_x </math> is invertible, and the inverse gives the [[pullback (differential geometry)|pullback]] of <math> T_{\varphi(x)} N.</math>

The differential is frequently expressed using a variety of other notations such as

:<math>D\varphi_x,\left(\varphi_*\right)_x, \varphi'(x),T_x\varphi.</math>

It follows from the definition that the differential of a [[function composition|composite]] is the composite of the differentials (i.e., [[functor]]ial behaviour). This is the ''chain rule'' for smooth maps.

Also, the differential of a [[local diffeomorphism]] is a [[linear isomorphism]] of tangent spaces.

==The differential on the tangent bundle ==
The differential of a smooth map <math>\varphi</math> induces, in an obvious manner, a [[bundle map]] (in fact a [[vector bundle homomorphism]]) from the [[tangent bundle]] of <math>M</math> to the tangent bundle of <math>N</math>, denoted by <math>d\varphi</math>, which fits into the following [[commutative diagram]]:
[[Image:SmoothPushforward-01.svg|center]]

where <math>\pi_M</math> and <math>\pi_N</math> denote the bundle projections of the tangent bundles of <math>M</math> and <math>N</math> respectively.

<math>\operatorname{d}\!\varphi</math> induces a [[bundle map]] from <math>TM</math> to the [[pullback bundle]] ''φ''<sup>∗</sup>''TN'' over <math>M</math> via

:<math>(m,v_m) \mapsto (\varphi(m),\operatorname{d}\!\varphi (m,v_m)),</math>

where <math>m \in M</math> and <math>v_m \in T_mM.</math> The latter map may in turn be viewed as a [[section (fiber bundle)|section]] of the [[vector bundle]] {{nowrap|Hom(''TM'', ''φ''<sup>∗</sup>''TN'')}} over ''M''. The bundle map <math>\operatorname{d}\!\varphi</math> is also denoted by <math>T\varphi</math> and called the '''tangent map'''. In this way, <math>T</math> is a [[functor]].

== Pushforward of vector fields ==
Given a smooth map {{nowrap|''φ'' : ''M'' → ''N''}} and a [[vector field]] ''X'' on ''M'', it is not usually possible to identify a pushforward of ''X'' by ''φ'' with some vector field ''Y'' on ''N''. For example, if the map ''φ'' is not surjective, there is no natural way to define such a pushforward outside of the image of ''φ''. Also, if ''φ'' is not injective there may be more than one choice of pushforward at a given point. Nevertheless, one can make this difficulty precise, using the notion of a vector field along a map.

A [[vector bundle|section]] of ''φ''<sup>∗</sup>''TN'' over ''M'' is called a '''vector field along ''φ'''''. For example, if ''M'' is a submanifold of ''N'' and ''φ'' is the inclusion, then a vector field along ''φ'' is just a section of the tangent bundle of ''N'' along ''M''; in particular, a vector field on ''M'' defines such a section via the inclusion of ''TM'' inside ''TN''. This idea generalizes to arbitrary smooth maps.

Suppose that ''X'' is a vector field on ''M'', i.e., a section of ''TM''. Then, <math>\operatorname{d}\!\phi \circ X</math> yields, in the above sense, the '''pushforward''' ''φ''<sub>∗</sub>''X'', which is a vector field along ''φ'', i.e., a section of ''φ''<sup>∗</sup>''TN'' over ''M''.

Any vector field ''Y'' on ''N'' defines a [[pullback bundle|pullback section]] ''φ''<sup>∗</sup>''Y'' of ''φ''<sup>∗</sup>''TN'' with {{nowrap|1=(''φ''<sup>∗</sup>''Y'')<sub>''x''</sub> = ''Y''<sub>''φ''(''x'')</sub>}}. A vector field ''X'' on ''M'' and a vector field ''Y'' on ''N'' are said to be '''''φ''-related''' if {{nowrap|1=''φ''<sub>∗</sub>''X'' = ''φ''<sup>∗</sup>''Y''}} as vector fields along ''φ''. In other words, for all ''x'' in ''M'', {{nowrap|1=''dφ''<sub>''x''</sub>(''X'') = ''Y''<sub>''φ''(''x'')</sub>}}.

In some situations, given a ''X'' vector field on ''M'', there is a unique vector field ''Y'' on ''N'' which is ''φ''-related to ''X''. This is true in particular when ''φ'' is a [[diffeomorphism]]. In this case, the pushforward defines a vector field ''Y'' on ''N'', given by
:<math>Y_y = \phi_*\left(X_{\phi^{-1}(y)}\right).</math>

A more general situation arises when ''φ'' is surjective (for example the [[fiber bundle|bundle projection]] of a fiber bundle). Then a vector field ''X'' on ''M'' is said to be '''projectable''' if for all ''y'' in ''N'', ''dφ''<sub>''x''</sub>(''X<sub>x</sub>'') is independent of the choice of ''x'' in ''φ''<sup>−1</sup>({''y''}). This is precisely the condition that guarantees that a pushforward of ''X'', as a vector field on ''N'', is well defined.

=== Examples ===

==== Pushforward from multiplication on Lie groups ====
Given a [[Lie group]] <math>G</math>, we can use the multiplication map <math>m(-,-) : G\times G \to G</math> to get left multiplication <math>L_g = m(g,-)</math> and right multiplication <math>R_g = m(-,g)</math> maps <math>G \to G</math>. These maps can be used to construct left or right invariant vector fields on <math>G</math> from its tangent space at the origin <math>\mathfrak{g} = T_e G</math> (which is its associated [[Lie algebra]]). For example, given <math>X \in \mathfrak{g}</math> we get an associated vector field <math>\mathfrak{X}</math> on <math>G</math> defined by
<math display="block">\mathfrak{X}_g = (L_g)_*(X) \in T_g G</math>
for every <math>g \in G</math>. This can be readily computed using the curves definition of pushforward maps. If we have a curve
<math display="block">\gamma: (-1,1) \to G</math>
where
<math display="block">\gamma(0) = e \, , \quad \gamma'(0) = X</math>
we get
<math display="block">\begin{align}
(L_g)_*(X) &= (L_g\circ \gamma)'(0) \\
&= (g\cdot \gamma(t))'(0) \\
&= \frac{dg}{dt}\gamma(0) + g\cdot \frac{d\gamma}{dt} (0) \\
&= g \cdot \gamma'(0)
\end{align}</math>
since <math>L_g</math> is constant with respect to <math>\gamma</math>. This implies we can interpret the tangent spaces <math>T_g G</math> as <math>T_g G = g\cdot T_e G = g\cdot \mathfrak{g}</math>.

==== Pushforward for some Lie groups ====
For example, if <math>G</math> is the Heisenberg group given by matrices
<math display="block">H = \left\{
\begin{bmatrix}
1 & a & b \\
0 & 1 & c \\
0 & 0 & 1
\end{bmatrix} : a,b,c \in \mathbb{R}
\right\}</math>
it has Lie algebra given by the set of matrices
<math display="block">\mathfrak{h} = \left\{
\begin{bmatrix}
0 & a & b \\
0 & 0 & c \\
0 & 0 & 0
\end{bmatrix} : a,b,c \in \mathbb{R}
\right\}</math>
since we can find a path <math>\gamma:(-1,1) \to H</math> giving any real number in one of the upper matrix entries with <math>i < j</math> (i-th row and j-th column). Then, for
<math display="block">g = \begin{bmatrix}
1 & 2 & 3 \\
0 & 1 & 4 \\
0 & 0 & 1
\end{bmatrix}</math>
we have
<math display="block">T_gH = g\cdot \mathfrak{h} = 
\left\{
\begin{bmatrix}
0 & a & b + 2c \\
0 & 0 & c \\
0 & 0 & 0
\end{bmatrix} : a,b,c \in \mathbb{R}
\right\}</math>
which is equal to the original set of matrices. This is not always the case, for example, in the group
<math display="block">G = \left\{
\begin{bmatrix}
a & b \\
0 & 1/a
\end{bmatrix} : a,b \in \mathbb{R}, a \neq 0
\right\}</math>
we have its Lie algebra as the set of matrices
<math display="block">\mathfrak{g} = \left\{
\begin{bmatrix}
a & b \\
0 & -a
\end{bmatrix} : a,b \in \mathbb{R}
\right\}</math>
hence for some matrix
<math display="block">g = \begin{bmatrix}
2 & 3 \\
0 & 1/2
\end{bmatrix}</math>
we have
<math display="block">T_gG = \left\{
\begin{bmatrix}
2a & 2b - 3a \\
0 & -a/2
\end{bmatrix} : a,b\in \mathbb{R}
\right\}</math>
which is not the same set of matrices.

==See also==
*[[Pullback (differential geometry)]]
*[[Flow-based generative model]]

== References ==

* {{cite book|first=John M. |last=Lee |title=Introduction to Smooth Manifolds |year=2003 |series=Springer Graduate Texts in Mathematics |volume=218 }}
* {{cite book|first=Jürgen |last=Jost |title=Riemannian Geometry and Geometric Analysis |year=2002 |publisher=Springer-Verlag |location=Berlin |isbn=3-540-42627-2 }} ''See section 1.6''.
* {{cite book |author-link1=Ralph Abraham (mathematician) |first1=Ralph |last1=Abraham |first2=Jerrold E. |last2=Marsden |author-link2=Jerrold E. Marsden |title=Foundations of Mechanics |year=1978 |publisher=Benjamin-Cummings |location=London |isbn=0-8053-0102-X }} ''See section 1.7 and 2.3''.

{{Manifolds}}

[[Category:Generalizations of the derivative]]
[[Category:Differential geometry]]
[[Category:Smooth functions]]