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.

The notion of push forward also exists in measure theory as the measure induced on Y by a measurable function f : X \to Y, given a measure on X.

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[edit] 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.

[edit] 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 \in T_pM, therefore x is a derivative defined on M and f is a smooth real-valued function on N. By definition, the push-forward of X is in TF(p)N and therefore itself is a derivative. In component notation the definition becomes,

(F_*(X))^a\partial_a (f)=X^b\partial_b F^a\partial_a f

therefore by identifying the following relation,

(F_*)^a_{\;b}= \frac{\partial F^a}{\partial x^b}

one can understand push-forward roughly as a coordinate transformation. Note that the push-forward is not gauranteed to have an inverse. Therefore in general we cannot define pullback for vectors.

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

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

[edit] 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\colon 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:

[edit] 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.

[edit] Push forward in measure theory

Given two measurable spaces (X, \mathcal{A}) and (Y, \mathcal{B}), a function f : X \to Y is called measurable if the pre-image of any measurable set in Y is measurable in X:

B \in \mathcal{B} \implies f^{-1} (B) \in \mathcal{A}.

If the space (X, \mathcal{A}) is equipped with a measure \mu : \mathcal{A} \to [0, + \infty], the push forward of μ by a measurable function f : X \to Y is the measure f_{*} (\mu) : \mathcal{B} \to [0, + \infty] defined by

(f_{*} (\mu)) (B) := \mu \left( f^{-1} (B) \right).

[edit] See also

[edit] 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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