In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, in the category of
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s, a fibered manifold is a
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
submersion
that is, a surjective differentiable mapping such that at each point
the tangent mapping
is surjective, or, equivalently, its rank equals
History
In
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, the words fiber (Faser in German) and fiber space (gefaserter Raum) appeared for the first time in a paper by
Herbert Seifert
Herbert Karl Johannes Seifert (; 27 May 1897, Bernstadt – 1 October 1996, Heidelberg) was a German mathematician known for his work in topology.
Biography
Seifert was born in Bernstadt auf dem Eigen, but soon moved to Bautzen, where he attend ...
in
1932
Events January
* January 4 – The British authorities in India arrest and intern Mahatma Gandhi and Vallabhbhai Patel.
* January 9 – Sakuradamon Incident (1932), Sakuradamon Incident: Korean nationalist Lee Bong-chang fails in his effort ...
, but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the base space (topological space) of a fiber (topological) space
was not part of the structure, but derived from it as a quotient space of
The first definition of fiber space is given by
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration t ...
in
1935
Events
January
* January 7 – Italian premier Benito Mussolini and French Foreign Minister Pierre Laval conclude Franco-Italian Agreement of 1935, an agreement, in which each power agrees not to oppose the other's colonial claims.
* ...
under the name sphere space, but in
1940
A calendar from 1940 according to the Gregorian calendar, factoring in the dates of Easter and related holidays, cannot be used again until the year 5280.
Events
Below, the events of World War II have the "WWII" prefix.
January
*January ...
Whitney changed the name to sphere bundle.
The theory of fibered spaces, of which
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s,
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equip ...
s, topological
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
s and fibered manifolds are a special case, is attributed to
Seifert Seifert is a German surname. Notable people with the surname include:
* Alfred Seifert (1850–1901), Czech German painter
* Alfred Seifert (flax miller) (1877–1945), New Zealand flax-miller
* Alwin Seifert (1890–1972), German architect
* Benja ...
,
Hopf,
Feldbau,
Whitney Whitney may refer to:
Film and television
* ''Whitney'' (2015 film), a Whitney Houston biopic starring Yaya DaCosta
* ''Whitney'' (2018 film), a documentary about Whitney Houston
* ''Whitney'' (TV series), an American sitcom that premiered i ...
,
Steenrod,
Ehresmann,
Serre, and others.
Formal definition
A triple
where
and
are differentiable manifolds and
is a surjective submersion, is called a fibered manifold.
is called the total space,
is called the base.
Examples
* Every differentiable
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
is a fibered manifold.
* Every differentiable
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
is a fibered manifold with discrete fiber.
* In general, a fibered manifold needs not to be a fiber bundle: different fibers may have different topologies. An example of this phenomenon may be constructed by taking the trivial bundle
and deleting two points in two different fibers over the base manifold
The result is a new fibered manifold where all the fibers except two are connected.
Properties
* Any surjective submersion
is open: for each open
the set
is open in
* Each fiber
is a closed embedded submanifold of
of dimension
* A fibered manifold admits local sections: For each
there is an open neighborhood
of
in
and a smooth mapping
with
and
* A surjection
is a fibered manifold if and only if there exists a local section
of
(with
) passing through each
Fibered coordinates
Let
(resp.
) be an
-dimensional (resp.
-dimensional) manifold. A fibered manifold
admits fiber charts. We say that a
chart
A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...
on
is a fiber chart, or is adapted to the surjective submersion
if there exists a chart
on
such that
and
where
The above fiber chart condition may be equivalently expressed by
where
is the projection onto the first
coordinates. The chart
is then obviously unique. In view of the above property, the fibered coordinates of a fiber chart
are usually denoted by
where
the coordinates of the corresponding chart
on
are then denoted, with the obvious convention, by
where
Conversely, if a surjection
admits a fibered
atlas
An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth.
Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geographic ...
, then
is a fibered manifold.
Local trivialization and fiber bundles
Let
be a fibered manifold and
any manifold. Then an open covering
of
together with maps
called trivialization maps, such that
is a local trivialization with respect to
A fibered manifold together with a manifold
is a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
with typical fiber (or just fiber)
if it admits a local trivialization with respect to
The atlas
is then called a bundle atlas.
See also
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Notes
References
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Historical
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External links
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{{Manifolds
Differential geometry
Fiber bundles
Manifolds