Let be an
affine bundle
In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.. (page 60)
Formal definition
Let \overline\pi:\overline Y\to X be a vector bundle with a typical fiber a ...
modelled over a vector bundle . A
connection on is called the affine connection if it as a section of the
jet bundle
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. ...
of is an affine bundle morphism over . In particular, this is 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 ...
on the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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 ma ...
. (That is, the connection on an affine bundle is an example of an affine connection; it is not, however, a general definition of an affine connection. These are related but distinct concepts both unfortunately making use of the adjective "affine".)
With respect to affine bundle coordinates on , an affine connection on is given by the
tangent-valued connection form
:
An affine bundle is a fiber bundle with a
general affine structure group of affine transformations of its typical fiber of dimension . Therefore, an affine connection is associated to a
principal connection
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-conne ...
. It always exists.
For any affine connection , the corresponding
linear derivative of an affine morphism defines a unique
linear connection on a vector bundle . With respect to linear bundle coordinates on , this connection reads
:
Since every vector bundle is an affine bundle, any linear connection on
a vector bundle also is an affine connection.
If is a vector bundle, both an affine connection and an associated linear connection are
connections on the same vector bundle , and their difference is a basic soldering form on
:
Thus, every affine connection on a vector bundle is a sum of a linear connection and a basic soldering form on .
Due to the canonical vertical splitting , this soldering form is brought into a
vector-valued form
:
where is a fiber basis for .
Given an affine connection on a vector bundle , let and be the
curvatures of a connection and the associated linear connection , respectively. It is readily observed that , where
:
is the
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
of with respect to the basic soldering form .
In particular, consider the tangent bundle of a manifold coordinated by . There is the canonical soldering form
:
on which coincides with the
tautological one-form
In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus p ...
:
on due to the canonical vertical splitting . Given an arbitrary linear connection on , the corresponding affine connection
:
on is 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 ...
. The torsion of the Cartan connection with respect to the soldering form coincides with the
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
of a linear connection , and its curvature is a sum of the curvature and the torsion of .
See also
*
Connection (fibred manifold)
In differential geometry, a fibered manifold is surjective submersion of smooth manifolds . Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connect ...
*
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 ...
*
Connection (vector bundle)
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The mo ...
*
Connection (mathematics) In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are var ...
*
Affine gauge theory
References
*
*
Differential geometry
Connection (mathematics)
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