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{{Short description|Linear approximation of smooth maps on tangent spaces}}

In [[mathematics]], the '''push forward''' (or '''pushforward''') of a [[smooth map]] ''F'' : ''M'' → ''N'' between [[smooth manifold]]s at a point ''p'' is, in some sense, the best linear approximation of ''F'' near ''p''. It can be viewed as generalization of the [[total derivative]] of ordinary calculus. Explicitly, it is a [[linear map]] from the [[tangent space]] of ''M'' at ''p'' to the tangent space of ''N'' at ''F''(''p'').

{{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''.]]

The push forward of a map ''F'' is also called, by various authors, the '''derivative''', '''total derivative''', or '''differential''' of ''F''.

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>F:U\to V</math> be a [[Smooth_function#Smooth_maps_of_manifolds|smooth map]] from an [[Open_subset#Euclidean_space|open subset]], <math>U</math>, of <math>\mathbb R^n</math> to an open subset, <math>V</math>, of <math>\mathbb R^m</math>. Let <math>(x^1,\ldots,x^n)</math> be the [[Coordinate system|coordinates]] in <math>U</math> and <math>(y^1,\ldots,y^m)</math> those in <math>V</math>. For any <math>p\in U</math>, the [[Jacobian]] of <math>F</math> is the [[matrix]] representation of the total derivative

== Motivation ==

:<math>DF(p):\mathbb R^n\to\mathbb R^m</math>.

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>

We wish to generalize this to the case that <math>F</math> is a smooth function between ''any'' [[Manifold#Differentiable_manifolds|smooth manifolds]] <math>M</math> and <math>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>.

==Definition==

Let <math>F:M\to N</math> be a smooth map of smooth manifolds. Given some <math>p\in M</math>, the ''push forward'' is a linear map

== The differential of a smooth map ==

:<math>F_*:T_pM\to T_{F(p)}N\,</math>

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 ''M'' at ''p'' to the tangent space of ''N'' at ''F''(''p''). The exact definition depends on the definition one uses for tangent vectors (for the various definitions see [[tangent space]]).

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 one defines tangent vectors as equivalence classes of curves through ''p'' then the push forward is given by

:<math>F_{*}(\gamma'(0)) = (F \circ \gamma)'(0)</math>

Here <math>\gamma</math> is a curve is ''M'' with <math>\gamma(0) = p</math>. The push forward is just the tangent vector to the curve <math>F\circ \gamma</math> at 0.

Alternatively, ifIf tangent vectors are defined as [[derivationequivalence (abstractclasses algebra)|derivations]]of actingthe oncurves smooth<math>\gamma</math> real-valuedfor functionswhich the<math> push\gamma(0) = x, </math> then the forwarddifferential is given by

:<math>F_{*}(X)(f) = X(f \circ F)</math>

Here ''X'' is a derivation on ''M'' and ''f'' is a smooth real-valued function on ''N''. One can show that <math>F_*(X)</math> is a indeed a derivation.

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

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

:<math>dF_p,\;DF_p,\;F'(p)</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>

==Properties==

Alternatively, if tangent vectors are defined as [[derivation (abstract algebra)|derivations]] acting on smooth real-valued functions, then the differential is given by

One can show that push forward of a [[function composition|composition]] is the composition of push forwards (i.e., [[functor]]ial behaviour), and the push forward of a [[local diffeomorphism]] is an [[isomorphism]] of tangent spaces.

:<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>.

Returning to the motivating example, it can be shown that the push forward of <math>F:U\to V</math>, in the given standard coordinates, is the matrix <math>J</math> whose entries are <math>J_{ij}=\partial F^{i}/\partial x^j(p)</math>. This is the Jacobian of <math>F</math>. More generally, given a smooth map <math>F:M\to N</math> the push forward of ''F'' written in local coordinates will always be given by the Jacobian of ''F'' in those coordinates.

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

The push forward of ''F'' induces in an obvious manner a [[vector bundle morphism]] from the [[tangent bundle]] of ''M'' to the tangent bundle of ''N'':

[[Image:SmoothPushforward-01.png|center]]

:<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>

==Push forwards of vector fields==

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.

Although one can always push forward tangent vectors, the push forward of a ''[[vector field]]'' does not always make sense. For example, if the map ''F'' is not surjective how should one define the vector outside the range of ''F''? Conversely, if ''F'' is not injective there may be more than one choice of the push forward of the field at a given point.

Extending by linearity gives the following matrix

There is one special situation where one can push forward vector fields, namely if the map ''F'' is a [[diffeomorphism]]. In this case, suppose ''X'' is a vector field on ''M'', the push forward defines a vector field ''Y'' on ''N'', given by <math>Y=F_*X</math> with

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

:<math>Y_p=F_*(X_{F^{-1}(p)})</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>

Here, <math>F^{-1}(p)</math> maps the point ''p'' back from the manifold ''N'' to the manifold ''M''. Then <math>X_{F^{-1}(p)}</math> is the vector field at the point <math>F^{-1}(p)</math> on ''M''.

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>\operatorname{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''^∗''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''^∗''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''^∗''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''_∗''X'', which is a vector field along ''φ'', i.e., a section of ''φ''^∗''TN'' over ''M''.

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

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''_''x''(''X_x'') is independent of the choice of ''x'' in ''φ''⁻¹({''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 }}

*[[pullback]]

* {{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}}

==References==

* John M. Lee, ''Introduction to Smooth Manifolds'', (2003) Springer Graduate Texts in Mathematics 218.

* Jurgen Jost, ''Riemannian Geometry and Geometric Analysis'', (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 ''See section 1.6''.

* [[Ralph Abraham]] and Jarrold E. Marsden, ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X ''See section 1.7 and 2.3''.

[[Category:Generalizations of the derivative]]

[[Category:Differential geometry]]

[[Category:Smooth functions]]