Return transport

In various sub-areas of mathematics , as pullback or return transport (also: retraction , retreat ) constructions are called, which, based on an image and an object that belongs to in some way , provide a corresponding object "withdrawn along " ; it is often referred to as. ${\displaystyle f\colon X\to Y}$ ${\displaystyle E}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle f^{*}E}$

The dual concept is usually called push forward .

In category theory, pullback is another name for the fiber product . The dual concept is called pushout , cocartesian square or fiber sum .

Motivation: The return of a smooth function

Let be a diffeomorphism between smooth manifolds and be a smooth function on . Then the return transport from with respect to is defined by ${\displaystyle f\colon M\to N}$ ${\displaystyle \psi \colon N\to \mathbb {R} }$ ${\displaystyle N}$ ${\displaystyle \psi }$ ${\displaystyle f}$

${\displaystyle f^{*}\colon C^{\infty }(N)\to C^{\infty }(M)}$ With ${\displaystyle (f^{*}(\psi ))(x)=\psi (f(x))\,.}$

So the return transport is a smooth function . ${\displaystyle f^{*}\psi }$ ${\displaystyle M\to \mathbb {R} }$

If the function is restricted to an open subset , a smooth function is also obtained . The return transport is thus a morphism between the sheaves of the smooth functions of and . ${\displaystyle \psi }$ ${\displaystyle U\subset N}$ ${\displaystyle f^{-1}(U)\subset M}$ ${\displaystyle N}$ ${\displaystyle M}$

The return transport of a vector bundle

Scheme of a pullback using the example of the cotangent bundle

Let and be topological spaces , a vector bundle over and a continuous mapping . Then the withdrawn vector bundle is defined by ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle \pi \colon E\to N}$ ${\displaystyle N}$ ${\displaystyle f\colon M\to N}$ ${\displaystyle \pi _{E'}\colon E'\to M}$

${\displaystyle E':=\{(x,e)\in M\times E|f(x)=\pi (e)\}}$

along with the projection . This vector bundle is usually noted by means of and is also called a pullback bundle from regarding . ${\displaystyle \pi _{E'}(x,e):=x}$ ${\displaystyle f^{*}E}$ ${\displaystyle E}$ ${\displaystyle f}$

If there is a cut in the vector bundle , the withdrawn cut is the one through ${\displaystyle \omega \in \Gamma (N,E)}$ ${\displaystyle E}$ ${\displaystyle f^{*}\omega \in \Gamma (M,f^{*}E)}$

${\displaystyle (f^{*}\omega )_{p}=\omega _{f(p)}}$

is given to all . ${\displaystyle p\in M}$

The withdrawn vector bundle is a special case of a fiber product . In the area of differential geometry , smooth manifolds are usually considered instead of arbitrary topological spaces and . Then it is also required that the mapping and the vector bundle are differentiable. ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle f\colon M\to N}$

Dual operator

Let and be two vector bundles and a continuous mapping, so that the corresponding return transport is. The dual operator of the return transport is the push forward of . ${\displaystyle \pi _{E}\colon E\to N}$ ${\displaystyle \pi _{F}\colon F\to M}$ ${\displaystyle \phi \colon M\to N}$ ${\displaystyle \phi ^{*}\colon E\to F}$ ${\displaystyle \phi _{*}\colon F\to E}$ ${\displaystyle \phi }$

Scheme of a push forward, TN is the tangent bundle to the manifold N

Return transport of certain objects

In this section and are smooth manifolds and let be a smooth map . ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle f\colon M\to N}$

Smooth functions

The set of smooth functions can naturally be interpreted using the vector space of the smooth cuts in the vector bundle . Correspondingly, the return transport of a smooth function can also be understood as the return transport of a smooth section of the vector bundle . ${\displaystyle C^{\infty }(M)}$ ${\displaystyle \psi \colon N\to \mathbb {R} }$ ${\displaystyle \Gamma ^{\infty }(M,M\times \mathbb {R} )}$ ${\displaystyle \textstyle \coprod _{p\in M}\mathbb {R} \cong M\times \mathbb {R} }$ ${\displaystyle (f^{*}(\psi ))(x)=\psi (f(x))}$ ${\displaystyle M\times \mathbb {R} }$

Differential forms

Since the set of differential forms forms a vector bundle, the return transport of a differential form can be examined.

If there is a differentiable mapping and a k -form on , then the on is withdrawn differential form , which in the case of 1-forms by ${\displaystyle f\colon M\to N}$ ${\displaystyle \omega \in \Gamma (N,\Lambda ^{k}(T^{*}N))}$ ${\displaystyle N}$ ${\displaystyle M}$ ${\displaystyle f^{*}\omega }$

${\displaystyle (f^{*}\omega )_{p}(X)=\omega _{f(p)}(f_{*p}X)}$

given for tangential vectors at the point . ${\displaystyle X\in T_{p}M}$ ${\displaystyle p\in M}$

literature

• Otto Forster : Riemann surfaces (= Heidelberg pocket books 184). Springer, Berlin et al. 1977, ISBN 3-540-08034-1 (English: Lectures on Riemann Surfaces (= Graduate Texts in Mathematics 81). Corrected 2nd printing. Ibid 1991, ISBN 3-540-90617-7 ).
• R. Abraham, Jerrold E. Marsden , T. Ratiu: Manifolds, tensor analysis, and applications (= Applied mathematical sciences 75). 2nd Edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 .

Web links

• Pullback of a differential form, nLab

Individual evidence

1. ↑ Allen Hatcher: Vector Bundles & K-Theory. Version 2.1, May 2009, p. 18 online (PDF; 1.11 MB) .
2. ^ John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 , p. 111.