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 be a smooth map from an open subset, U, of to an open subset, V, of . Let be the coordinates in U and those in V. For any , the Jacobian of F is the matrix representation of the total derivative

.

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

Definition

Let be a smooth map of smooth manifolds. Given some , the push forward is a linear map

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

Here γ is a curve is M with γ(0) = p. The push forward is just the tangent vector to the curve at 0.

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

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

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 , in the given standard coordinates, is the matrix J whose entries are . This is the Jacobian of F. More generally, given a smooth map 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 Y = F _* X with

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