Pushforward (differential)

In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds 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).

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

Motivation

Let ${\displaystyle F:U\to V}$  be a smooth map from an open subset, ${\displaystyle U}$ , of ${\displaystyle \mathbb {R} ^{n}}$  to an open subset, ${\displaystyle V}$ , of ${\displaystyle \mathbb {R} ^{m}}$ . Let ${\displaystyle (x^{1},\ldots ,x^{n})}$  be the coordinates in ${\displaystyle U}$  and ${\displaystyle (y^{1},\ldots ,y^{m})}$  those in ${\displaystyle V}$ . For any ${\displaystyle p\in U}$ , the Jacobian of ${\displaystyle F}$  is the matrix representation of the total derivative

${\displaystyle DF(p):\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ .

We wish to generalize this to the case that ${\displaystyle F}$  is a smooth function between any smooth manifolds ${\displaystyle M}$  and ${\displaystyle N}$ .

Definition

Let ${\displaystyle F:M\to N}$  be a smooth map of smooth manifolds. Given some ${\displaystyle p\in M}$ , the push forward is a linear map

${\displaystyle F_{*}:T_{p}M\to T_{F(p)}N\,}$ 

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).

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

${\displaystyle F_{*}(\gamma '(0))=(F\circ \gamma )'(0)}$ 

Here ${\displaystyle \gamma }$  is a curve in M with ${\displaystyle \gamma (0)=p}$ . The push forward is just the tangent vector to the curve ${\displaystyle F\circ \gamma }$  at 0.

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

${\displaystyle F_{*}(X)(f)=X(f\circ F)}$ 

Here X is a derivation on M and f is a smooth real-valued function on N. One can show that ${\displaystyle F_{*}(X)}$  is a indeed a derivation.

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

${\displaystyle dF_{p},\;DF_{p},\;F'(p)}$ 

Properties

One can show that push forward of a composition is the composition of push forwards (i.e., functorial behaviour), and the push forward of a local diffeomorphism is an isomorphism of tangent spaces.

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

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:

 

Push forwards of vector fields

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.

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 ${\displaystyle Y=F_{*}X}$  with

${\displaystyle Y_{p}=F_{*}(X_{F^{-1}(p)})}$ 

Here, ${\displaystyle F^{-1}(p)}$  maps the point p back from the manifold N to the manifold M. Then ${\displaystyle X_{F^{-1}(p)}}$  is the vector field at the point ${\displaystyle F^{-1}(p)}$  on M.

See also

• pullback

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.