Pushforward (differential)

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Pushforward (differential)

Suppose that "&phi;" : "M" → "N" is a smooth map between smooth manifolds; then the differential of "&phi;" at a point "x" is, in some sense, the best linear approximation of "&phi;" near "x". 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 "x" to the tangent space of "N" at "&phi;"("x"). Hence it can be used to "push forward" tangent vectors on "M" to tangent vectors on "N".

The differential of a map "&phi;" is also called, by various authors, the derivative or total derivative of "&phi;", and is sometimes itself called the pushforward.

Motivation

Let "&phi;":"U"&rarr;"V" be a smooth map from an open subset "U" of Rm to an open subset "V" of Rn. For any point "x" in "U", the Jacobian of "&phi;" at "x" (with respect to the standard coordinates) is the matrix representation of the total derivative of "&phi;" at "x", which is a linear map:$mathrm d varphi_x:mathbb R^m omathbb R^n$from Rm to Rn.

We wish to generalize this to the case that "&phi;" is a smooth function between "any" smooth manifolds "M" and "N".

The differential of a smooth map

Let "&phi;" : "M" &rarr; "N" be a smooth map of smooth manifolds. Given some "x" &isin; "M", the differential (or (total) derivative) of "&phi;" at "x" is a linear
$mathrm d varphi_x:T_xM o T_\left\{varphi\left(x\right)\right\}N,$from the tangent space of "M" at "x" to the tangent space of "N" at "&phi;"("x"). The application of d"&phi;"x to a tangent vector "X" is sometimes called the pushforward of "X" by "&phi;". The exact definition of this pushforward 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 "x" then the differential is given by:$mathrm d varphi_x\left(gamma\text{'}\left(0\right)\right) = \left(varphi circ gamma\right)\text{'}\left(0\right).$Here "&gamma;" is a curve in "M" with "&gamma;"(0) = "x". In other words, the pushforward of the tangent vector to the curve "&gamma;" at 0 is just the tangent vector to the curve "&phi;"$circ$"&gamma;" at 0.

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by:$mathrm dvarphi_x\left(X\right)\left(f\right) = X\left(f circ varphi\right).$Here "X" &isin; "TxM", therefore "X" is a derivation defined on "M" and "f" is a smooth real-valued function on "N". By definition, the pushforward of "X" at a given "x" in "M" is in "T""&phi;"("x")"N" and therefore itself is a derivation.

After choosing charts around "x" and "&phi;"("x"), "F" is locally determined by a smooth map

:$\left\{hat varphi\right\} : U ightarrow V$

between open sets of R"m" and R"n", and d"&phi;"x has representation (at "x")

:$mathrm d varphi_xBigl\left(frac\left\{ partial \right\}\left\{partial u^a\right\}Bigr\right) = frac\left\{partial \left\{hat varphi\right\}^b\right\}\left\{partial u^a\right\} frac\left\{ partial \right\}\left\{partial v^b\right\},$

in the Einstein summation notation, where the partial derivatives are evaluated at the point in "U" corresponding to "x" in the given chart.

Extending by linearity gives the following matrix

:$\left(mathrm dvarphi_x\right)_a^\left\{;b\right\}= frac\left\{partial \left\{hatvarphi\right\}^b\right\}\left\{partial u^a\right\}.$

Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map "&phi;" at each point. Therefore, in some chosen local coordinates, it is represented by Jacobian of the corresponding smooth map from R"m" to R"n". In general the differential need not be invertible. If "&phi;" is a local diffeomorphism, then the pushforward at "x" is invertible and its inverse gives the pullback of "T""&phi;"("x")"N".

The differential is frequently expressed using a variety of other notations such as:$Dvarphi_x,; \left(varphi_*\right)_x, ;varphi\text{'}\left(x\right).$

It follows from the definition that the differential of a composite is the composite of the differentials (i.e., functorial behaviour). This is the "chain rule" for smooth maps.

Also, the differential of a local diffeomorphism is an linear isomorphism of tangent spaces.

The differential on the tangent bundle

The differential of a smooth map "&phi;" induces, in an obvious manner, a bundle map (in fact a vector bundle homomorphism) from the tangent bundle of "M" to the tangent bundle of "N", denoted by d"&phi;" or "&phi;"*, which fits into the following commutative diagram:where "πM" and "πN" denote the bundle projections of the tangent bundles of "M" and "N" respectively.

Equivalently (see bundle map), "&phi;"* = d"&phi;" is a bundle map from "TM" to the pullback bundle "&phi;*TN" over "M", which may in turn be viewed as a section of the vector bundle Hom("TM","&phi;"*"TN") over "M".

Pushforward of vector fields

Given a smooth map "&phi;":"M"&rarr;"N" and a vector field "X" on "M", it is not usually possible to define a pushforward of "X" by "&phi;" as a vector field on "N". For example, if the map "&phi;" is not surjective, there is no natural way to define such a pushforward outside of the image of "&phi;". Also, if "&phi;" is not injective there may be more than one choice of pushforward at a given point. Nevertheless, one can make this difficulty precise, using the notion of a vector field along a map.

A section of "&phi;*TN" over "M" is called a vector field along "&phi;". For example, if "M" is a submanifold of "N" and "&phi;" is the inclusion, then a vector field along "&phi;" is just a section of the tangent bundle of "N" along "M"; in particular, a vector field on "M" defines such a section via the inclusion of "TM" inside "TN". This idea generalizes to arbitrary smooth maps.

Suppose that "X" is a vector field on "M", i.e., a section of "TM". Then, applying the differential pointwise to "X" yields the pushforward "&phi;*X", which is a vector field along "&phi;", i.e., a section of "&phi;*TN" over "M".

Any vector field "Y" on "N" defines a pullback section "&phi;*Y" of "&phi;*TN" with ("&phi;*Y")"x" = "Y""&phi;"("x"). A vector field "X" on "M" and a vector field "Y" on "N" are said to be "&phi;"-related if "&phi;*X" = "&phi;*Y" as vector fields along "&phi;". In other words, for all "x" in "M", d"&phi;""x"("X")="Y""&phi;"("x").

In some situations, given a "X" vector field on "M", there is a unique vector field "Y" on "M" which is "&phi;"-related to "X". This is true in particular when "&phi;" is a diffeomorphism. In this case, the pushforward defines a vector field "Y" on "N", given by:$Y_x=varphi_*\left(X_\left\{varphi^\left\{-1\right\}\left(x\right)\right\}\right).$

A more general situation arises when "&phi;" is surjective (for example the bundle projection of a fiber bundle). Then a vector field "X" on "M" is said to be projectable if for all "y" in "N", d"&phi;""x"("X""x") is independent of the choice of "x" in "&phi;"-1({"y"}). This is precisely the condition that guarantees that a pushforward of "X", as a vector field on "N", is well defined.

ee 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 3-540-42627-2 "See section 1.6".
* Ralph Abraham and Jerrold 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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