﻿ Pullback (differential geometry)

# Pullback (differential geometry)

In various areas of mathematics is called the pullback or repatriation (also: withdrawal, withdrawal) designs that provide the basis of a figure and an object that is to be heard in any way, a corresponding " along reclusive " object; it is often referred to.

The dual concept usually called Push Forward.

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

## Motivation: The return of a smooth function

Be a diffeomorphism between smooth manifolds and is a smooth function. Then the return transport is defined by respect by

The return is thus a smooth function.

If one limits the function on an open subset of a, we obtain as a smooth function. The return is thus a morphism between the sheaves of smooth functions of and.

## The return of a vector bundle

Let and be topological spaces, a vector bundle over and a continuous map. Then the retracted vector bundle is defined by

Together with the projection. Usually you listed this vector bundle by calling it even pullback bundle of respect.

Is a section in the vector bundles, then the retracted section by

Is given for all.

The retracted vector bundle is a special case of a fiber product. In the field of differential geometry are usually smooth manifolds instead of arbitrary topological spaces and considered. Then, it is also required in addition that the image and the vector bundles are differentiable.

## Dual operator

Let and be two vector bundles and a continuous map, so that the corresponding return transport is. The dual operator of the return transport is the push forward of.

## Repatriation of certain objects

In this section and are smooth manifolds and is a smooth map.

### Smooth functions

The set of smooth functions can be interpreted in a natural way with the vector space of smooth sections of the vector bundle. According to the repatriation of a smooth function can also be interpreted as a return of an element of the vector bundle.

### Differential forms

Since the amount of differential forms form a vector bundle, one can examine the return transport of a differential form.

If a differentiable map and a k- form, then the differential form for revoked, which in the case of 1 -forms by

Given for tangent vectors at the point.

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