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are called, which, based on an image and an object that belongs to in some way , provide a corresponding object "withdrawn along " ; it is often referred to as.
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
- With
So the return transport is a smooth function .
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 .
The return transport of a vector bundle
Let and be topological spaces , a vector bundle over and a continuous mapping . Then the withdrawn vector bundle is defined by
along with the projection . This vector bundle is usually noted by means of and is also called a pullback bundle from regarding .
If there is a cut in the vector bundle , the withdrawn cut is the one through
is given to all .
The withdrawn vector bundle is a special case of a fiber product .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.
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 .
Return transport of certain objects
In this section and are smooth manifolds and let be a smooth map .
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 .
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If there is a differentiable mapping and a k -form on , then the on is withdrawn differential form , which in the case of 1-forms by
given for tangential vectors at the point .
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
Individual evidence
- ↑ Allen Hatcher: Vector Bundles & K-Theory. Version 2.1, May 2009, p. 18 online (PDF; 1.11 MB) .
- ^ 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.