In
mathematics, and particularly
topology, a fiber bundle (or, in
Commonwealth English
The use of the English language in current and former member countries of the Commonwealth of Nations was largely inherited from British colonisation, with some exceptions. English serves as the medium of inter-Commonwealth relations.
Many ...
: fibre bundle) is a
space that is a
product space, but may have a different
topological structure. Specifically, the similarity between a space
and a product space
is defined using a
continuous surjective map,
that in small regions of
behaves just like a projection from corresponding regions of
to
The map
called the projection or
submersion of the bundle, is regarded as part of the structure of the bundle. The space
is known as the total space of the fiber bundle,
as the base space, and
the fiber.
In the ''trivial'' case,
is just
and the map
is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Aug ...
and
Klein bottle, as well as nontrivial
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 ...
s. Fiber bundles, such as the
tangent bundle of a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
and other more general
vector bundles, play an important role in
differential geometry and
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, as do
principal bundles.
Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as
bundle map
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. The ...
s, and the class of fiber bundles forms a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
*C ...
with respect to such mappings. A bundle map from the base space itself (with the identity mapping as projection) to
is called a
section of
Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transition maps between the local trivial patches lie in a certain
topological group, known as the structure group, acting on the fiber
.
History
In
topology, the terms ''fiber'' (German: ''Faser'') and ''fiber space'' (''gefaserter Raum'') appeared for the first time in a paper by
Herbert Seifert in 1933, 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 ''E'' was not part of the structure, but derived from it as a quotient space of ''E''. The first definition of fiber space was given by
Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.
The theory of fibered spaces, of which
vector bundles,
principal bundles, topological
fibrations and
fibered manifold
In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion
\pi : E \to B\,
that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping
T_y \pi : T_ E ...
s are a special case, is attributed to Seifert,
Heinz Hopf,
Jacques Feldbau
Jacques Feldbau was a French mathematician, born on 22 October 1914 in Strasbourg, of an Alsatian Jewish traditionalist family. He died on 22 April 1945 at the ''Ganacker'' Camp, annex of the concentration camp of Flossenbürg in Germany. As a ...
, Whitney,
Norman Steenrod,
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory.
He was an early member of the Bourbaki group, and is known for his work on the differential ...
,
Jean-Pierre Serre, and others.
Fiber bundles became their own object of study in the period 1935–1940. The first general definition appeared in the works of Whitney.
Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a
sphere bundle In the mathematical field of topology, a sphere bundle is a fiber bundle in which the fibers are spheres S^n of some dimension ''n''. Similarly, in a disk bundle, the fibers are disks D^n. From a topological perspective, there is no difference betw ...
, that is a fiber bundle whose fiber is a sphere of arbitrary dimension.
Formal definition
A fiber bundle is a structure
where
and
are
topological spaces and
is a
continuous surjection satisfying a ''local triviality'' condition outlined below. The space
is called the of the bundle,
the , and
the . The map
is called the (or ). We shall assume in what follows that the base space
is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
.
We require that for every
, there is an open
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of
(which will be called a trivializing neighborhood) such that there is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
(where
is given the
subspace topology, and
is the product space) in such a way that
agrees with the projection onto the first factor. That is, the following diagram should
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
:
Local triviality condition, 230px, center
where
is the natural projection and
is a homeomorphism. The set of all
is called a of the bundle.
Thus for any
, the
preimage
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...
is homeomorphic to
(since this is true of
) and is called the fiber over
Every fiber bundle
is an
open map, since projections of products are open maps. Therefore
carries the
quotient topology determined by the map
A fiber bundle
is often denoted
that, in analogy with a
short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space.
A is a fiber bundle in the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
*C ...
of
smooth manifolds. That is,
and
are required to be smooth manifolds and all the functions above are required to be
smooth map
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s.
Examples
Trivial bundle
Let
and let
be the projection onto the first factor. Then
is a fiber bundle (of
) over
Here
is not just locally a product but ''globally'' one. Any such fiber bundle is called a . Any fiber bundle over a
contractible
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
CW-complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
is trivial.
Nontrivial bundles
Möbius strip
Perhaps the simplest example of a nontrivial bundle
is the
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Aug ...
. It has the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
that runs lengthwise along the center of the strip as a base
and a
line segment for the fiber
, so the Möbius strip is a bundle of the line segment over the circle. A
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of
(where
) is an
arc; in the picture, this is the length of one of the squares. The
preimage
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...
in the picture is a (somewhat twisted) slice of the strip four squares wide and one long (i.e. all the points that project to
).
A homeomorphism (
in ) exists that maps the preimage of
(the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle
would be a
cylinder, but the Möbius strip has an overall "twist". This twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space).
Klein bottle
A similar nontrivial bundle is the
Klein bottle, which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-
torus,
.
Covering map
A
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 fiber bundle such that the bundle projection is a
local homeomorphism. It follows that the fiber is a
discrete space.
Vector and principal bundles
A special class of fiber bundles, called
vector bundles, are those whose fibers are
vector spaces (to qualify as a vector bundle the structure group of the bundle — see below — must be a
linear group). Important examples of vector bundles include the
tangent bundle and
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
of a smooth manifold. From any vector bundle, one can construct the
frame bundle of
bases, which is a principal bundle (see below).
Another special class of fiber bundles, called
principal bundles, are bundles on whose fibers a free and transitive
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
by a group
is given, so that each fiber is a
principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non- ...
. The bundle is often specified along with the group by referring to it as a principal
-bundle. The group
is also the structure group of the bundle. Given a
representation of
on a vector space
, a vector bundle with
as a structure group may be constructed, known as the
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with ...
.
Sphere bundles
A sphere bundle is a fiber bundle whose fiber is an
''n''-sphere. Given a vector bundle
with a
metric (such as the tangent bundle to a
Riemannian manifold) one can construct the associated unit sphere bundle, for which the fiber over a point
is the set of all unit vectors in
. When the vector bundle in question is the tangent bundle
, the unit sphere bundle is known as the
unit tangent bundle.
A sphere bundle is partially characterized by its
Euler class, which is a degree
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
class in the total space of the bundle. In the case
the sphere bundle is called a
circle bundle
In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S^1.
Oriented circle bundles are also known as principal ''U''(1)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circl ...
and the Euler class is equal to the first
Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
, which characterizes the topology of the bundle completely. For any
, given the Euler class of a bundle, one can calculate its cohomology using a
long exact sequence called the
Gysin sequence In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. The Gysin sequence is a useful tool for ...
.
Mapping tori
If
is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
and
is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
then the
mapping torus In mathematics, the mapping torus in topology of a homeomorphism ''f'' of some topological space ''X'' to itself is a particular geometric construction with ''f''. Take the cartesian product of ''X'' with a closed interval ''I'', and glue the bound ...
has a natural structure of a fiber bundle over the
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
with fiber
Mapping tori of homeomorphisms of surfaces are of particular importance in
3-manifold topology.
Quotient spaces
If
is a
topological group and
is a
closed subgroup, then under some circumstances, the
quotient space together with the quotient map
is a fiber bundle, whose fiber is the topological space
. A necessary and sufficient condition for (
) to form a fiber bundle is that the mapping
admits
local cross-sections .
The most general conditions under which the quotient map will admit local cross-sections are not known, although if
is a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
and
a closed subgroup (and thus a Lie subgroup by
Cartan's theorem), then the quotient map is a fiber bundle. One example of this is the
Hopf fibration,
, which is a fiber bundle over the sphere
whose total space is
. From the perspective of Lie groups,
can be identified with the
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
. The abelian subgroup of diagonal matrices is isomorphic to the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
, and the quotient
is
diffeomorphic to the sphere.
More generally, if
is any topological group and
a closed subgroup that also happens to be a Lie group, then
is a fiber bundle.
Sections
A (or cross section) of a fiber bundle
is a continuous map
such that
for all ''x'' in ''B''. Since bundles do not in general have globally defined sections, one of the purposes of the theory is to account for their existence. The
obstruction to the existence of a section can often be measured by a cohomology class, which leads to the theory of
characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
es in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.
The most well-known example is the
hairy ball theorem, where the
Euler class is the obstruction to the
tangent bundle of the 2-sphere having a nowhere vanishing section.
Often one would like to define sections only locally (especially when global sections do not exist). A local section of a fiber bundle is a continuous map
where ''U'' is an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are ...
in ''B'' and
for all ''x'' in ''U''. If
is a local trivialization chart then local sections always exist over ''U''. Such sections are in 1-1 correspondence with continuous maps
. Sections form a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* '' The Sheaf'', a student-run newspaper s ...
.
Structure groups and transition functions
Fiber bundles often come with a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of symmetries that describe the matching conditions between overlapping local trivialization charts. Specifically, let ''G'' be a
topological group that
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
continuously on the fiber space ''F'' on the left. We lose nothing if we require ''G'' to act
faithfully on ''F'' so that it may be thought of as a group of
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
s of ''F''. A ''G''-
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 geographi ...
for the bundle
is a set of local trivialization charts
such that for any
for the overlapping charts
and
the function
is given by
where
is a continuous map called a . Two ''G''-atlases are equivalent if their union is also a ''G''-atlas. A ''G''-bundle is a fiber bundle with an equivalence class of ''G''-atlases. The group ''G'' is called the of the bundle; the analogous term in physics is
gauge group
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
.
In the smooth category, a ''G''-bundle is a smooth fiber bundle where ''G'' is a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
and the corresponding action on ''F'' is smooth and the transition functions are all smooth maps.
The transition functions
satisfy the following conditions
#
#
#
The third condition applies on triple overlaps ''U
i'' ∩ ''U
j'' ∩ ''U
k'' and is called the
cocycle condition (see
Čech cohomology). The importance of this is that the transition functions determine the fiber bundle (if one assumes the Čech cocycle condition).
A
principal ''G''-bundle is a ''G''-bundle where the fiber ''F'' is a
principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non- ...
for the left action of ''G'' itself (equivalently, one can specify that the action of ''G'' on the fiber ''F'' is free and transitive, i.e.
regular). In this case, it is often a matter of convenience to identify ''F'' with ''G'' and so obtain a (right) action of ''G'' on the principal bundle.
Bundle maps
It is useful to have notions of a mapping between two fiber bundles. Suppose that ''M'' and ''N'' are base spaces, and
and
are fiber bundles over ''M'' and ''N'', respectively. A or consists of a pair of continuous functions
such that
That is, the following diagram is
commutative:
For fiber bundles with structure group ''G'' and whose total spaces are (right) ''G''-spaces (such as a principal bundle), bundle morphisms are also required to be ''G''-
equivariant on the fibers. This means that
is also ''G''-morphism from one ''G''-space to another, that is,
for all
and
In case the base spaces ''M'' and ''N'' coincide, then a bundle morphism over ''M'' from the fiber bundle
to
is a map
such that
This means that the bundle map
covers the identity of ''M''. That is,
and the following diagram commutes:
Assume that both
and
are defined over the same base space ''M''. A bundle isomorphism is a bundle map
between
and
such that
and such that
is also a homeomorphism.
[ Or is, at least, invertible in the appropriate category; e.g., a diffeomorphism.]
Differentiable fiber bundles
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 m ...
s, fiber bundles arise naturally as
submersions of one manifold to another. Not every (differentiable) submersion
from a differentiable manifold ''M'' to another differentiable manifold ''N'' gives rise to a differentiable fiber bundle. For one thing, the map must be surjective, and
is called a
fibered manifold
In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion
\pi : E \to B\,
that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping
T_y \pi : T_ E ...
. However, this necessary condition is not quite sufficient, and there are a variety of sufficient conditions in common use.
If ''M'' and ''N'' are
compact and
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
, then any submersion
gives rise to a fiber bundle in the sense that there is a fiber space ''F'' diffeomorphic to each of the fibers such that
is a fiber bundle. (Surjectivity of
follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion
is assumed to be a surjective
proper map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.
Definition
There are several competing defini ...
, meaning that
is compact for every compact subset ''K'' of ''N''. Another sufficient condition, due to , is that if
is a surjective
submersion with ''M'' and ''N''
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 m ...
s such that the preimage
is compact and connected for all
then
admits a compatible fiber bundle structure .
Generalizations
* The notion of a
bundle applies to many more categories in mathematics, at the expense of appropriately modifying the local triviality condition; cf.
principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non- ...
and
torsor (algebraic geometry).
* In topology, a
fibration is a mapping
that has certain
homotopy-theoretic properties in common with fiber bundles. Specifically, under mild technical assumptions a fiber bundle always has the
homotopy lifting property or homotopy covering property (see for details). This is the defining property of a fibration.
* A section of a fiber bundle is a "function whose output range is continuously dependent on the input." This property is formally captured in the notion of
dependent type
In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifier ...
.
See also
*
Affine bundle
*
Algebra bundle
*
Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classe ...
*
Covering map 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 sp ...
*
Equivariant bundle
*
Fibered manifold
In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion
\pi : E \to B\,
that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping
T_y \pi : T_ E ...
*
Fibration
*
Gauge theory
*
Hopf bundle
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hop ...
*
I-bundle
*
Natural bundle In mathematics, a natural bundle is any fiber bundle associated to the ''s''-frame bundle F^s(M) for some s \geq 1. It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold M together with ...
*
Principal bundle
*
Projective bundle
*
Pullback bundle
*
Quasifibration
*
Universal bundle
*
Vector bundle
Notes
References
*
*
*
*
*
*
*
External links
Fiber Bundle PlanetMath
*
*
Sardanashvily, Gennadi, Fibre bundles, jet manifolds and Lagrangian theory. Lectures for theoreticians,
{{DEFAULTSORT:Fiber Bundle