# Ngô Quốc Anh

## April 16, 2011

### Pushforward (differential) of a smooth map

Filed under: Riemannian geometry — Ngô Quốc Anh @ 21:09

Suppose that $\varphi : M \to N$ is a smooth map between smooth manifolds; then the differential of $\varphi$ at a point $x$ is, in some sense, the best linear approximation of $\varphi$ near $x$. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of $M$ at $x$ to the tangent space of $N$ at $\varphi (x)$. Hence it can be used to push forward tangent vectors on $M$ to tangent vectors on $N$.

The differential of a map $\varphi$ is also called, by various authors, the derivative or total derivative of $\varphi$, and is sometimes itself called the pushforward.

If a map, $\varphi$, carries every point on manifold $M$ to manifold $N$ then the pushforward of $\varphi$ carries vectors in the tangent space at every point in $M$ to a tangent space at every point in $N$.

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

$\mathrm d \varphi_x:\mathbb R^m\to\mathbb R^n$

from $\mathbb R^m$ to $\mathbb R^n$. We wish to generalize this to the case that $\varphi$ is a smooth function between any smooth manifolds $M$ and $N$.

The differential of a smooth map. Let $\varphi : M \to N$ be a smooth map of smooth manifolds. Given some $x\in M$, the differential (or (total) derivative) of $\varphi$ at $x$ is a linear map

$\mathrm d \varphi_x:T_xM\to T_{\varphi(x)}N$

from the tangent space of $M$ at $x$ to the tangent space of $N$ at $\varphi (x)$. The application of $d\varphi x$ to a tangent vector $X$ is sometimes called the pushforward of $X$ by $\varphi$. 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 $x$ then the differential is given by

$\mathrm d \varphi_x(\gamma'(0)) = (\varphi \circ \gamma)'(0).$

Here $\gamma$ is a curve in $M$ with $\gamma (0) = x$. In other words, the pushforward of the tangent vector to the curve $\gamma$ at $0$ is just the tangent vector to the curve $\varphi \circ \gamma$ at $0$.

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

$\mathrm d\varphi_x(X)(f) = X(f \circ \varphi).$

Here $X \in T_xM$, therefore $X$ is a derivation defined on $M$ and $f$ is a smooth real-valued function on $N$. By definition, the pushforward of $X$ at a given $x$ in $M$ is in $T_{\varphi (x)}N$ and therefore itself is a derivation.

After choosing charts around $x$ and $\varphi (x)$, $F$ is locally determined by a smooth map

${\hat \varphi} : U \rightarrow V$

between open sets of $\mathbb R^m$ and $\mathbb R^n$, and $d\varphi x$ has representation (at $x$)

$\displaystyle\mathrm d \varphi_x\Bigl(\frac{ \partial }{\partial u^a}\Bigr) = \frac{\partial {\hat \varphi}^b}{\partial u^a} \frac{ \partial }{\partial v^b},$

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

$\displaystyle (\mathrm d\varphi_x)_a^{\;b}= \frac{\partial {\hat\varphi}^b}{\partial u^a}.$

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

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

$D\varphi_x,\; (\varphi_*)_x, \;\varphi'(x).$

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.

Source: wikipedia

If we have a triple $(N, \varphi, \pi_N)$, where $\varphi: M\to N$ – a smooth mapping, $\pi_N: TN\to N$ – bundle projection of the tangent bundles of N, then the pullback of $(N, \varphi, \pi_N)$ is the triple $(TM,\pi_M, d\varphi)$, i.e. $TM$ is fiber product of $\varphi, \pi_N$.
On the other hand, $T$ can be interpreted as a functor $T: M \to TM$ such that $T(\phi)= d\phi$