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Push forward

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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.

Contents

1 Motivation

2 Definition

3 Properties

4 Push forwards of vector fields

5 See also

6 References

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Motivation

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

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

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

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Definition

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

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

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

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

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

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

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

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

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

$dF_p,\;DF_p,\;F'(p)$

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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 $F:U\to V$ , in the given standard coordinates, is the matrix $J$ whose entries are $J_{ij}=\partial F^{i}/\partial x^j(p)$ . This is the Jacobian of $F$ . More generally, given a smooth map $F:M\to N$ 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:

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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 $Y = F * X$ with

$Y_p=F_*(X_{F^{-1}(p)})$

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

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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.

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Categories: Differential geometry