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), 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. ISBN0-521-36948-7.
Mangiarotti, L., Sardanashvily, G., Connections in Classical and Quantum Field Theory. World Scientific, 2000. ISBN981-02-2013-8.
External links[]
Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory, Lambert Academic Publishing, 2013. ISBN978-3-659-37815-7; arXiv:0908.1886