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Composite bundles play a prominent role in gauge theory with symmetry breaking, e.g., gauge gravitation theory, non-autonomous mechanics where is the time axis, e.g., mechanics with time-dependent parameters, and so on. There are the important relations between connections on fiber bundles , and .
Composite bundle
In differential geometry by a composite bundle is meant the composition
of fiber bundles
It is provided with bundle coordinates , where are bundle coordinates on a fiber bundle , i.e., transition functions of coordinates are independent of coordinates .
The following fact provides the above mentioned physical applications of composite bundles. Given the composite bundle (1), let be a global section of a fiber bundle , if any. Then the pullback bundle over is a subbundle of a fiber bundle .
Composite principal bundle
For instance, let be a principal bundle with a structure Lie group which is reducible to its closed subgroup . There is a composite bundle where is a principal bundle with a structure group and is a fiber bundle associated with . Given a global section of , the pullback bundle is a reduced principal subbundle of with a structure group . In gauge theory, sections of are treated as classical Higgs fields.
Jet manifolds of a composite bundle
Given the composite bundle (1), let us consider the jet manifolds , , and of the fiber bundles , , and , respectively. They are provided with the adapted coordinates , , and
There is the canonical map
Composite connection
This canonical map defines the relations between connections on fiber bundles , and . These connections are given by the corresponding tangent-valued connection forms
A connection on a fiber bundle and a connection on a fiber bundle define a connection
on a composite bundle . It is called the composite connection. This is a unique connection such that the horizontal lift onto of a vector field on by means of the composite connection coincides with the composition of horizontal lifts of onto by means of a connection and then onto by means of a connection .
Vertical covariant differential
Given the composite bundle (1), there is the following exact sequence of vector bundles over :
where and are the vertical tangent bundle and the vertical cotangent bundle of . Every connection on a fiber bundle yields the splitting
of the exact sequence (2). Using this splitting, one can construct a first order differential operator
on a composite bundle . It is called the vertical covariant differential. It possesses the following important property.
Let be a section of a fiber bundle , and let be the pullback bundle over . Every connection induces the pullback connection
on . Then the restriction of a vertical covariant differential to coincides with the familiar covariant differential on relative to the pullback connection .
References
- Saunders, D., The geometry of jet bundles. Cambridge University Press, 1989. 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