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 connection on fiber bundles.
Formal definition
Let be a fibered manifold. A (generalized) connection on is a section , where is the jet manifold of .[1]
Connection as a horizontal splitting
Let be a fibered manifold. There is the following canonical short exact sequence of vector bundles over :
where and are the tangent bundles of , respectively, is the vertical tangent bundle of , and is the pullback bundle of onto .
A connection on a fibered manifold is defined as a linear bundle morphism
over which splits the exact sequence (1). A connection always exists.
Sometimes, this connection is called the Ehresmann connection because it yields the horizontal distribution
of and its horizontal decomposition .
At the same time, by an Ehresmann connection also is meant the following construction. Any connection on a fibered manifold yields a horizontal lift of a vector field on onto , but need not defines the similar lift of a path in into . Let and be smooth paths in and , respectively. Then is called the horizontal lift of if , , . A connection is said to be the Ehresmann connection if, for each path in , there exists its horizontal lift through any point . A fibered manifold is a fiber bundle if and only if it admits such an Ehresmann connection.
Connection as a tangent-valued form
Given a fibered manifold , let it be endowed with an atlas of fibered coordinates , and let be a connection on . It yields uniquely the horizontal tangent-valued one-form
on which projects onto the canonical tangent-valued form (tautological one-form or solder form) on , and vice versa. With this form, the horizontal splitting (2) reads
In particular, the connection (3) yields the horizontal lift of any vector field on to a projectable vector field
on .
Connection as a vertical-valued form
The horizontal splitting (2) of the exact sequence (1) defines the corresponding splitting of the dual exact sequence
where and are the cotangent bundles of , respectively, and is the dual bundle to , called the vertical cotangent bundle. This splitting is given by the vertical-valued form
which also represents a connection on a fibered manifold.
Treating a connection as a vertical-valued form, one comes to the following important construction. Given a fibered manifold , let be a morphism and the pullback bundle of by . Then any connection (3) on induces the pullback connection
on .
Connection as a jet bundle section
Let be the jet manifold of sections of a fibered manifold , with coordinates . Due to the canonical imbedding
any connection (3) on a fibered manifold is represented by a global section
of the jet bundle , and vice versa. It is an affine bundle modelled on a vector bundle
There are the following corollaries of this fact.
(i) Connections on a fibered manifold make up an affine space modelled on the vector space of soldering forms
on , i.e., sections of the vector bundle (4).
(ii) Connection coefficients possess the coordinate transformation law
(iii) Every connection on a fibred manifold yields the first order differential operator
on called the covariant differential relative to the connection . If is a section, its covariant differential
and the covariant derivative along a vector field on are defined.
Curvature and torsion
Given the connection (3) on a fibered manifold , its curvature is defined as the Nijenhuis differential
This is a vertical-valued horizontal two-form on .
Given the connection (3) and the soldering form (5), a torsion of with respect to is defined as
Bundle of principal connections
Let be a principal bundle with a structure Lie group . A principal connection on usually is described by a Lie algebra-valued connection one-form on . At the same time, a principal connection on is a global section of the jet bundle which is equivariant with respect to the canonical right action of in . Therefore, it is represented by a global section of the quotient bundle , called the bundle of principal connections. It is an affine bundle modelled on the vector bundle whose typical fiber is the Lie algebra of structure group , and where acts on by the adjoint representation. There is the canonical imbedding of to the quotient bundle which also is called the bundle of principal connections.
Given a basis for a Lie algebra of , the fiber bundle is endowed with bundle coordinates , and its sections are represented by vector-valued one-forms
where are the familiar local connection forms on .
Let us note that the jet bundle of is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition
where
is called the strength form of a principal connection.
See also
Notes
References
- Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural operators in differential geometry (PDF), Springer-Verlag
- Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8
- Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7
- Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory. World Scientific, 2000. ISBN 981-02-2013-8.
External links
- Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013. ISBN 978-3-659-37815-7; arXiv: 0908.1886