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Section (fiber bundle)
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In the mathematical field of topology, a section (or cross section)[1] of a fiber bundle is a continuous right inverse of the projection function
. In other words, if
is a fiber bundle over a base space,
:
This article relies largely or entirely on a single source. (July 2022) |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Bundle_section.svg/640px-Bundle_section.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Vector_field.svg/320px-Vector_field.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Vector_bundle_with_section.png/640px-Vector_bundle_with_section.png)
then a section of that fiber bundle is a continuous map,
such that
for all
.
A section is an abstract characterization of what it means to be a graph. The graph of a function can be identified with a function taking its values in the Cartesian product
, of
and
:
Let be the projection onto the first factor:
. Then a graph is any function
for which
.
The language of fibre bundles allows this notion of a section to be generalized to the case when is not necessarily a Cartesian product. If
is a fibre bundle, then a section is a choice of point
in each of the fibres. The condition
simply means that the section at a point
must lie over
. (See image.)
For example, when is a vector bundle a section of
is an element of the vector space
lying over each point
. In particular, a vector field on a smooth manifold
is a choice of tangent vector at each point of
: this is a section of the tangent bundle of
. Likewise, a 1-form on
is a section of the cotangent bundle.
Sections, particularly of principal bundles and vector bundles, are also very important tools in differential geometry. In this setting, the base space is a smooth manifold
, and
is assumed to be a smooth fiber bundle over
(i.e.,
is a smooth manifold and
is a smooth map). In this case, one considers the space of smooth sections of
over an open set
, denoted
. It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g.,
sections, or sections with regularity in the sense of Hölder conditions or Sobolev spaces).