Pushforward (differential)#The differential on the tangent bundle

Let $\backslash varphi:\; U\; \backslash to\; V$ be a smooth map from an open subset $U$ of $\backslash R^m$ to an open subset $V$ of $\backslash R^n$. For any point $x$ in $U$, the Jacobian of $\backslash varphi$ at $x$ (with respect to the standard coordinates) is the matrix representation of the total derivative of $\backslash varphi$ at $x$, which is a linear map

:$d\backslash varphi\_x:T\_x\backslash R^m\backslash to\; T\_\{\backslash varphi(x)\}\backslash R^n$

between their tangent spaces. Note the tangent spaces $T\_x\backslash R^m,T\_\{\backslash varphi(x)\}\backslash R^n$ are isomorphic to $\backslash mathbb\{R\}^m$ and $\backslash mathbb\{R\}^n$, respectively. The pushforward generalizes this construction to the case that $\backslash varphi$ is a smooth function between any smooth manifolds $M$ and $N$.

Let $\backslash varphi\; \backslash colon\; M\; \backslash to\; N$ be a smooth map of smooth manifolds. Given $x\; \backslash in\; M,$ the differential of $\backslash varphi$ at $x$ is a linear map

:$d\backslash varphi\_x\; \backslash colon\backslash \; T\_xM\backslash to\; T\_\{\backslash varphi(x)\}N\backslash ,$

from the tangent space of $M$ at $x$ to the tangent space of $N$ at $\backslash varphi(x).$ The image $d\backslash varphi\_x\; X$ of a tangent vector $X\; \backslash in\; T\_x\; M$ under $d\backslash varphi\_x$ is sometimes called the pushforward of $X$ by $\backslash varphi.$ 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 $\backslash gamma$ for which $\backslash gamma(0)\; =\; x,$ then the differential is given by

:$d\backslash varphi\_x(\backslash gamma\text{'}(0))\; =\; (\backslash varphi\; \backslash circ\; \backslash gamma)\text{'}(0).$

Here, $\backslash gamma$ is a curve in $M$ with $\backslash gamma(0)\; =\; x,$ and $\backslash gamma\text{'}(0)$ is tangent vector to the curve $\backslash gamma$ at $0.$ In other words, the pushforward of the tangent vector to the curve $\backslash gamma$ at $0$ is the tangent vector to the curve $\backslash varphi\; \backslash circ\; \backslash gamma$ at $0.$

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by

:$d\backslash varphi\_x(X)(f)\; =\; X(f\; \backslash circ\; \backslash varphi),$

for an arbitrary function $f\; \backslash in\; C^\backslash infty(N)$ and an arbitrary derivation $X\; \backslash in\; T\_xM$ at point $x\; \backslash in\; M$ (a derivation is defined as a linear map $X\; \backslash colon\; C^\backslash infty(M)\; \backslash to\; \backslash R$ that satisfies the Leibniz rule, see: definition of tangent space via derivations). By definition, the pushforward of $X$ is in $T\_\{\backslash varphi(x)\}N$ and therefore itself is a derivation, $d\backslash varphi\_x(X)\; \backslash colon\; C^\backslash infty(N)\; \backslash to\; \backslash R$.

After choosing two charts around $x$ and around $\backslash varphi(x),$ $\backslash varphi$ is locally determined by a smooth map $\backslash widehat\{\backslash varphi\}\; \backslash colon\; U\; \backslash to\; V$ between open sets of $\backslash R^m$ and $\backslash R^n$, and

:$d\backslash varphi\_x\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; u^a\}\backslash right)\; =\; \backslash frac\{\backslash partial\{\backslash widehat\{\backslash varphi\}\}^b\}\{\backslash partial\; u^a\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; v^b\},$

in the Einstein summation notation, where the partial derivatives are evaluated at the point in $U$ corresponding to $x$ in the given chart.

Extending by linearity gives the following matrix

:$\backslash left(d\backslash varphi\_x\backslash right)\_a^\{\backslash ;b\}\; =\; \backslash frac\{\backslash partial\{\backslash widehat\{\backslash varphi\}\}^b\}\{\backslash partial\; u^a\}.$

Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map $\backslash varphi$ at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from $\backslash R^m$ to $\backslash R^n$. In general, the differential need not be invertible. However, if $\backslash varphi$ is a local diffeomorphism, then $d\backslash varphi\_x$ is invertible, and the inverse gives the pullback of $T\_\{\backslash varphi(x)\}\; N.$

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

:$D\backslash varphi\_x,\backslash left(\backslash varphi\_*\backslash right)\_x,\; \backslash varphi\text{'}(x),T\_x\backslash varphi.$

It follows from the definition that the differential of a composite is the composite of the differentials (i.e., functorial 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 of a smooth map $\backslash varphi$ induces, in an obvious manner, a bundle map (in fact a vector bundle homomorphism) from the tangent bundle of $M$ to the tangent bundle of $N$, denoted by $d\backslash varphi$, which fits into the following commutative diagram:

center

where $\backslash pi\_M$ and $\backslash pi\_N$ denote the bundle projections of the tangent bundles of $M$ and $N$ respectively.

$\backslash operatorname\{d\}\backslash !\backslash varphi$ induces a bundle map from $TM$ to the pullback bundle φ^∗TN over $M$ via

:$(m,v\_m)\; \backslash mapsto\; (\backslash varphi(m),\backslash operatorname\{d\}\backslash !\backslash varphi\; (m,v\_m)),$

where $m\; \backslash in\; M$ and $v\_m\; \backslash in\; T\_mM.$ The latter map may in turn be viewed as a section of the vector bundle {{nowrap|Hom(TM, φ^∗TN)}} over M. The bundle map $\backslash operatorname\{d\}\backslash !\backslash varphi$ is also denoted by $T\backslash varphi$ and called the tangent map. In this way, $T$ is a functor.

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 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, $\backslash operatorname\{d\}\backslash !\backslash phi\; \backslash circ\; X$ 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 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

:$Y\_y\; =\; \backslash phi\_*\backslash left(X\_\{\backslash phi^\{-1\}(y)\}\backslash right).$

A more general situation arises when φ is surjective (for example the 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.

Given a Lie group $G$, we can use the multiplication map $m(-,-)\; :\; G\backslash times\; G\; \backslash to\; G$ to get left multiplication $L\_g\; =\; m(g,-)$ and right multiplication $R\_g\; =\; m(-,g)$ maps $G\; \backslash to\; G$. These maps can be used to construct left or right invariant vector fields on $G$ from its tangent space at the origin $\backslash mathfrak\{g\}\; =\; T\_e\; G$ (which is its associated Lie algebra). For example, given $X\; \backslash in\; \backslash mathfrak\{g\}$ we get an associated vector field $\backslash mathfrak\{X\}$ on $G$ defined by

$$\backslash mathfrak\{X\}\_g\; =\; (L\_g)\_*(X)\; \backslash in\; T\_g\; G$$

for every $g\; \backslash in\; G$. This can be readily computed using the curves definition of pushforward maps. If we have a curve

$$\backslash gamma:\; (-1,1)\; \backslash to\; G$$

where

$$\backslash gamma(0)\; =\; e\; \backslash ,\; ,\; \backslash quad\; \backslash gamma\text{'}(0)\; =\; X$$

we get

$$\backslash begin\{align\}$$

(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}

since $L\_g$ is constant with respect to $\backslash gamma$. This implies we can interpret the tangent spaces $T\_g\; G$ as $T\_g\; G\; =\; g\backslash cdot\; T\_e\; G\; =\; g\backslash cdot\; \backslash mathfrak\{g\}$.

For example, if $G$ is the Heisenberg group given by matrices

$$H\; =\; \backslash left\backslash \{$$

\begin{bmatrix}

1 & a & b \\

0 & 1 & c \\

0 & 0 & 1

\end{bmatrix} : a,b,c \in \mathbb{R}

\right\}

it has Lie algebra given by the set of matrices

$$\backslash mathfrak\{h\}\; =\; \backslash left\backslash \{$$

\begin{bmatrix}

0 & a & b \\

0 & 0 & c \\

0 & 0 & 0

\end{bmatrix} : a,b,c \in \mathbb{R}

\right\}

since we can find a path $\backslash gamma:(-1,1)\; \backslash to\; H$ giving any real number in one of the upper matrix entries with (i-th row and j-th column). Then, for

$$g\; =\; \backslash begin\{bmatrix\}$$

1 & 2 & 3 \\

0 & 1 & 4 \\

0 & 0 & 1

\end{bmatrix}

we have

$$T\_gH\; =\; g\backslash cdot\; \backslash mathfrak\{h\}\; =$$

\left\{

\begin{bmatrix}

0 & a & b + 2c \\

0 & 0 & c \\

0 & 0 & 0

\end{bmatrix} : a,b,c \in \mathbb{R}

\right\}

which is equal to the original set of matrices. This is not always the case, for example, in the group

$$G\; =\; \backslash left\backslash \{$$

\begin{bmatrix}

a & b \\

0 & 1/a

\end{bmatrix} : a,b \in \mathbb{R}, a \neq 0

\right\}

we have its Lie algebra as the set of matrices

$$\backslash mathfrak\{g\}\; =\; \backslash left\backslash \{$$

\begin{bmatrix}

a & b \\

0 & -a

\end{bmatrix} : a,b \in \mathbb{R}

\right\}

hence for some matrix

$$g\; =\; \backslash begin\{bmatrix\}$$

2 & 3 \\

0 & 1/2

\end{bmatrix}

we have

$$T\_gG\; =\; \backslash left\backslash \{$$

\begin{bmatrix}

2a & 2b - 3a \\

0 & -a/2

\end{bmatrix} : a,b\in \mathbb{R}

\right\}

which is not the same set of matrices.

• Pullback (differential geometry)
• Flow-based generative model

• {{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