In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and especially
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
and
gauge theory
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 under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
, a connection is a device that defines a notion of
parallel transport
In differential geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on ...
on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-connection on a
principal G-bundle over a
smooth 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 may ...
''
'' is a particular type of connection that is compatible with the
action
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ...
of the group ''
''.
A principal connection can be viewed as a special case of the notion of an
Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any
fiber bundle
In mathematics, and particularly topology, a fiber bundle ( ''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 pr ...
associated to ''
'' via the
associated bundle Associated may refer to:
*Associated, former name of Avon, Contra Costa County, California
*Associated Hebrew Schools of Toronto, a school in Canada
*Associated Newspapers, former name of DMG Media, a British publishing company
See also
*Associatio ...
construction. In particular, on any
associated vector bundle the principal connection induces a
covariant derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
* Covariance matrix, a matrix of covariances between a number of variables
* Covariance or cross-covariance between ...
, an operator that can differentiate
sections
Section, Sectioning, or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
of that bundle along
tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a
linear connection on the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on ...
of a
smooth 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 may ...
.
Formal definition

Let
be a smooth
principal ''G''-bundle over a
smooth 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 may ...
. Then a principal
-connection on
is a differential 1-form on
with values in the Lie algebra of
which is
-equivariant and reproduces the Lie algebra generators of the fundamental vector fields on
.
In other words, it is an element ''ω'' of
such that
#
where
denotes right multiplication by
, and
is the
adjoint representation on
(explicitly,
);
# if
and
is
the vector field on ''P'' associated to ''ξ'' by differentiating the ''G'' action on ''P'', then
(identically on
).
Sometimes the term ''principal
-connection'' refers to the pair
and
itself is called the
connection form
In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Carta ...
or connection 1-form of the principal connection.
Computational remarks
Most known non-trivial computations of principal ''
''-connections are done with
homogeneous space
In mathematics, a homogeneous space is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the action of a group. Homogeneous spaces occur in the theories of Lie groups, algebraic groups and ...
s because of the triviality of the (co)tangent bundle. (For example, let
, be a principal ''
''-bundle over
.) This means that 1-forms on the total space are canonically isomorphic to
, where
is the dual lie algebra, hence ''
''-connections are in bijection with
.
Relation to Ehresmann connections
A principal ''
''-connection
on
determines an
Ehresmann connection on
in the following way. First note that the fundamental vector fields generating the
action on
provide a bundle isomorphism (covering the identity of
) from the
bundle to
, where
is the kernel of the
tangent mapping which is called the
vertical bundle
In mathematics, the vertical bundle and the horizontal bundle are Vector bundle, vector bundles associated to a Fiber bundle#Differentiable fiber bundles, smooth fiber bundle. More precisely, given a smooth fiber bundle \pi\colon E\to B, the verti ...
of
. It follows that
determines uniquely a bundle map
which is the identity on
. Such a projection
is uniquely determined by its kernel, which is a smooth subbundle
of
(called the
horizontal bundle) such that
. This is an Ehresmann connection.
Conversely, an Ehresmann connection
(or
) on
defines a principal
-connection
if and only if it is
-equivariant in the sense that
.
Pull back via trivializing section
A trivializing section of a principal bundle ''
'' is given by a section ''s'' of ''
'' over an open subset ''
'' of ''
''. Then the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
''s''
*''ω'' of a principal connection is a 1-form on ''
'' with values in
.
If the section ''s'' is replaced by a new section ''sg'', defined by (''sg'')(''x'') = ''s''(''x'')''g''(''x''), where ''g'':''M''→''G'' is a smooth map, then
. The principal connection is uniquely determined by this family of
-valued 1-forms, and these 1-forms are also called connection forms or connection 1-forms, particularly in older or more physics-oriented literature.
Bundle of principal connections
The group ''
'' acts on the
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
''
'' by right translation. The
quotient space ''TP''/''G'' is also a manifold, and inherits the structure of a
fibre bundle
In mathematics, and particularly topology, a fiber bundle ( ''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 ...
over ''TM'' which shall be denoted ''dπ'':''TP''/''G''→''TM''. Let ρ:''TP''/''G''→''M'' be the projection onto ''M''. The fibres of the bundle ''TP''/''G'' under the projection ρ carry an additive structure.
The bundle ''TP''/''G'' is called the bundle of principal connections . A
section
Section, Sectioning, or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
Γ of dπ:''TP''/''G''→''TM'' such that Γ : ''TM'' → ''TP''/''G'' is a linear morphism of vector bundles over ''M'', can be identified with a principal connection in ''P''. Conversely, a principal connection as defined above gives rise to such a section Γ of ''TP''/''G''.
Finally, let Γ be a principal connection in this sense. Let ''q'':''TP''→''TP''/''G'' be the quotient map. The horizontal distribution of the connection is the bundle
:
We see again the link to the horizontal bundle and thus Ehresmann connection.
Affine property
If ''ω'' and ''ω''′ are principal connections on a principal bundle ''P'', then the difference is a
-valued 1-form on ''P'' that is not only ''G''-equivariant, but horizontal in the sense that it vanishes on any section of the vertical bundle ''V'' of ''P''. Hence it is basic and so is determined by a 1-form on ''M'' with values in the
adjoint bundle
:
Conversely, any such one form defines (via pullback) a ''G''-equivariant horizontal 1-form on ''P'', and the space of principal ''G''-connections is an
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
for this space of 1-forms.
Examples
Maurer-Cartan connection
For the trivial principal
-bundle
where
, there is a canonical connection
pg 49called the Maurer-Cartan connection. It is defined at a point
by
for
which is a composition
defining the 1-form. Note that
is the
Maurer-Cartan form on the Lie group
and
.
Trivial bundle
For a trivial principal
-bundle
, the identity section
given by
defines a 1-1 correspondence
between connections on
and
-valued 1-forms on
pg 53. For a
-valued 1-form
on
, there is a unique 1-form
on
such that
#
for
a vertical vector
#
for any
Then given this 1-form, a connection on
can be constructed by taking the sum
giving an actual connection on
. This unique 1-form can be constructed by first looking at it restricted to
for
. Then,
is determined by
because
and we can get
by taking
Similarly, the form
defines a 1-form giving the properties 1 and 2 listed above.
Extending this to non-trivial bundles
This statement can be refined
pg 55 even further for non-trivial bundles
by considering an open covering
of
with
trivializations and transition functions
. Then, there is a 1-1 correspondence between connections on
and collections of 1-forms
which satisfy
on the intersections
for
the
Maurer-Cartan form on
,
in matrix form.
Global reformulation of space of connections
For a principal
bundle
the set of connections in
is an affine space
pg 57 for the vector space
where
is the associated adjoint vector bundle. This implies for any two connections
there exists a form
such that
We denote the set of connections as
, or just
if the context is clear.
Connection on the complex Hopf-bundle
We
pg 94 can construct
as a principal
-bundle
where
and
is the projection map
Note the Lie algebra of
is just the complex plane. The 1-form
defined as
forms a connection, which can be checked by verifying the definition. For any fixed
we have
and since
, we have
-invariance. This is because the adjoint action is trivial since the Lie algebra is Abelian. For constructing the splitting, note for any
we have a short exact sequence
where
is defined as
so it acts as scaling in the fiber (which restricts to the corresponding
-action). Taking
we get
where the second equality follows because we are considering
a vertical tangent vector, and
. The notation is somewhat confusing, but if we expand out each term
it becomes more clear (where
).
Induced covariant and exterior derivatives
For any
linear representation
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
''W'' of ''G'' there is an
associated vector bundle over ''M'', and a principal connection induces a
covariant derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
* Covariance matrix, a matrix of covariances between a number of variables
* Covariance or cross-covariance between ...
on any such vector bundle. This covariant derivative can be defined using the fact that the space of sections of
over ''M'' is isomorphic to the space of ''G''-equivariant ''W''-valued functions on ''P''. More generally, the space of ''k''-forms
with values in is identified with the space of ''G''-equivariant and horizontal ''W''-valued ''k''-forms on ''P''. If ''α'' is such a ''k''-form, then its
exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
d''α'', although ''G''-equivariant, is no longer horizontal. However, the combination d''α''+''ω''Λ''α'' is. This defines an
exterior covariant derivative
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of ...
d
''ω'' from
-valued ''k''-forms on ''M'' to
-valued (''k''+1)-forms on ''M''. In particular, when ''k''=0, we obtain a covariant derivative on
.
Curvature form
The
curvature form of a principal ''G''-connection ''ω'' is the
-valued 2-form Ω defined by
:
It is ''G''-equivariant and horizontal, hence corresponds to a 2-form on ''M'' with values in
. The identification of the curvature with this quantity is sometimes called the ''(Cartan's) second structure equation''.
Historically, the emergence of the structure equations are found in the development of the
Cartan connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...
. When transposed into the context of
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s, the structure equations are known as the
Maurer–Cartan equations: they are the same equations, but in a different setting and notation.
Flat connections and characterization of bundles with flat connections
We say that a connection
is flat if its curvature form
. There is a useful characterization of principal bundles with flat connections; that is, a principal
-bundle
has a flat connection
pg 68 if and only if there exists an open covering
with trivializations
such that all transition functions
are constant. This is useful because it gives a recipe for constructing flat principal
-bundles over smooth manifolds; namely taking an open cover and defining trivializations with constant transition functions.
Connections on frame bundles and torsion
If the principal bundle ''P'' is the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on ...
, or (more generally) if it has a
solder form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuiti ...
, then the connection is an example of an
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
, and the curvature is not the only invariant, since the additional structure of the solder form ''θ'', which is an equivariant R
''n''-valued 1-form on ''P'', should be taken into account. In particular, the
torsion form on ''P'', is an R
''n''-valued 2-form Θ defined by
:
Θ is ''G''-equivariant and horizontal, and so it descends to a tangent-valued 2-form on ''M'', called the ''torsion''. This equation is sometimes called the ''(Cartan's) first structure equation''.
Definition in algebraic geometry
If ''X'' is a scheme (or more generally, stack, derived stack, or even prestack), we can associate to it its so-called ''de Rham stack'', denoted ''X
dR''. This has the property that a principal ''G'' bundle over ''X
dR'' is the same thing as a ''G'' bundle with *flat* connection over ''X''.
References
*
*
*
{{Manifolds
Connection (mathematics)
Differential geometry
Fiber bundles
Maps of manifolds
Smooth functions