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.
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Motivation
Let <math>F:U\to V</math> be a smooth map from an open subset, <math>U</math>, of <math>\mathbb R^n</math> to an open subset, <math>V</math>, of <math>\mathbb R^m</math>. Let <math>(x^1,\ldots,x^n)</math> be the coordinates in <math>U</math> and <math>(y^1,\ldots,y^m)</math> those in <math>V</math>. For any <math>p\in U</math>, the Jacobian of <math>F</math> is the matrix representation of the total derivative
- <math>DF(p):\mathbb R^n\to\mathbb R^m</math>.
We wish to generalize this to the case that <math>F</math> is a smooth function between any smooth manifolds <math>M</math> and <math>N</math>.
Definition
Let <math>F:M\to N</math> be a smooth map of smooth manifolds. Given some <math>p\in M</math>, the push forward is a linear map
- <math>F_*:T_pM\to T_{F(p)}N\,</math>
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
- <math>F_{*}(\gamma'(0)) = (F \circ \gamma)'(0)</math>
Here <math>\gamma</math> is a curve in M with <math>\gamma(0) = p</math>. The push forward is just the tangent vector to the curve <math>F\circ \gamma</math> at 0.
Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions the push forward is given by
- <math>F_{*}(X)(f) = X(f \circ F)</math>
Here X is a derivation on M and f is a smooth real-valued function on N. One can show that <math>F_*(X)</math> is a indeed a derivation.
The push forward is frequently expressed using a variety of other notations such as
- <math>dF_p,\;DF_p,\;F'(p)</math>
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 <math>F:U\to V</math>, in the given standard coordinates, is the matrix <math>J</math> whose entries are <math>J_{ij}=\partial F^{i}/\partial x^j(p)</math>. This is the Jacobian of <math>F</math>. More generally, given a smooth map <math>F:M\to N</math> 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 <math>Y=F_*X</math> with
- <math>Y_p=F_*(X_{F^{-1}(p)})</math>
Here, <math>F^{-1}(p)</math> maps the point p back from the manifold N to the manifold M. Then <math>X_{F^{-1}(p)}</math> is the vector field at the point <math>F^{-1}(p)</math> on M.
See also
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.
