In differential geometry by a composite bundle is meant the composition

${\displaystyle \pi :Y\to \Sigma \to X\qquad \qquad (1)}$ 

of fiber bundles

${\displaystyle \pi _{Y\Sigma }:Y\to \Sigma ,\qquad \pi _{\Sigma X}:\Sigma \to X.}$ 

It is provided with bundle coordinates ${\displaystyle (x^{\lambda },\sigma ^{m},y^{i})}$ , where ${\displaystyle (x^{\lambda },\sigma ^{m})}$  are bundle coordinates on a fiber bundle ${\displaystyle \Sigma \to X}$ , i.e., transition functions of coordinates ${\displaystyle \sigma ^{m}}$  are independent of coordinates ${\displaystyle y^{i}}$ .

The following fact provides the above-mentioned physical applications of composite bundles. Given the composite bundle (1), let ${\displaystyle h}$  be a global section of a fiber bundle ${\displaystyle \Sigma \to X}$ , if any. Then the pullback bundle ${\displaystyle Y^{h}=h^{*}Y}$  over ${\displaystyle X}$  is a subbundle of a fiber bundle ${\displaystyle Y\to X}$ .

For instance, let ${\displaystyle P\to X}$  be a principal bundle with a structure Lie group ${\displaystyle G}$  which is reducible to its closed subgroup ${\displaystyle H}$ . There is a composite bundle ${\displaystyle P\to P/H\to X}$  where ${\displaystyle P\to P/H}$  is a principal bundle with a structure group ${\displaystyle H}$  and ${\displaystyle P/H\to X}$  is a fiber bundle associated with ${\displaystyle P\to X}$ . Given a global section ${\displaystyle h}$  of ${\displaystyle P/H\to X}$ , the pullback bundle ${\displaystyle h^{*}P}$  is a reduced principal subbundle of ${\displaystyle P}$  with a structure group ${\displaystyle H}$ . In gauge theory, sections of ${\displaystyle P/H\to X}$  are treated as classical Higgs fields.

Given the composite bundle ${\displaystyle Y\to \Sigma \to X}$  (1), consider the jet manifolds ${\displaystyle J^{1}\Sigma }$ , ${\displaystyle J_{\Sigma }^{1}Y}$ , and ${\displaystyle J^{1}Y}$  of the fiber bundles ${\displaystyle \Sigma \to X}$ , ${\displaystyle Y\to \Sigma }$ , and ${\displaystyle Y\to X}$ , respectively. They are provided with the adapted coordinates ${\displaystyle (x^{\lambda },\sigma ^{m},\sigma _{\lambda }^{m})}$ , ${\displaystyle (x^{\lambda },\sigma ^{m},y^{i},{\widehat {y}}_{\lambda }^{i},y_{m}^{i}),}$ , and ${\displaystyle (x^{\lambda },\sigma ^{m},y^{i},\sigma _{\lambda }^{m},y_{\lambda }^{i}).}$ 

There is the canonical map

${\displaystyle J^{1}\Sigma \times _{\Sigma }J_{\Sigma }^{1}Y\to _{Y}J^{1}Y,\qquad y_{\lambda }^{i}=y_{m}^{i}\sigma _{\lambda }^{m}+{\widehat {y}}_{\lambda }^{i}}$ .

This canonical map defines the relations between connections on fiber bundles ${\displaystyle Y\to X}$ , ${\displaystyle Y\to \Sigma }$  and ${\displaystyle \Sigma \to X}$ . These connections are given by the corresponding tangent-valued connection forms

${\displaystyle \gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\gamma _{\lambda }^{m}\partial _{m}+\gamma _{\lambda }^{i}\partial _{i}),}$ 

${\displaystyle A_{\Sigma }=dx^{\lambda }\otimes (\partial _{\lambda }+A_{\lambda }^{i}\partial _{i})+d\sigma ^{m}\otimes (\partial _{m}+A_{m}^{i}\partial _{i}),}$ 

${\displaystyle \Gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }^{m}\partial _{m}).}$ 

A connection ${\displaystyle A_{\Sigma }}$  on a fiber bundle ${\displaystyle Y\to \Sigma }$  and a connection ${\displaystyle \Gamma }$  on a fiber bundle ${\displaystyle \Sigma \to X}$  define a connection

${\displaystyle \gamma =dx^{\lambda }\otimes (\partial _{\lambda }+\Gamma _{\lambda }^{m}\partial _{m}+(A_{\lambda }^{i}+A_{m}^{i}\Gamma _{\lambda }^{m})\partial _{i})}$ 

on a composite bundle ${\displaystyle Y\to X}$ . It is called the composite connection. This is a unique connection such that the horizontal lift ${\displaystyle \gamma \tau }$  onto ${\displaystyle Y}$  of a vector field ${\displaystyle \tau }$  on ${\displaystyle X}$  by means of the composite connection ${\displaystyle \gamma }$  coincides with the composition ${\displaystyle A_{\Sigma }(\Gamma \tau )}$  of horizontal lifts of ${\displaystyle \tau }$  onto ${\displaystyle \Sigma }$  by means of a connection ${\displaystyle \Gamma }$  and then onto ${\displaystyle Y}$  by means of a connection ${\displaystyle A_{\Sigma }}$ .

Given the composite bundle ${\displaystyle Y}$  (1), there is the following exact sequence of vector bundles over ${\displaystyle Y}$ :

${\displaystyle 0\to V_{\Sigma }Y\to VY\to Y\times _{\Sigma }V\Sigma \to 0,\qquad \qquad (2)}$ 

where ${\displaystyle V_{\Sigma }Y}$  and ${\displaystyle V_{\Sigma }^{*}Y}$  are the vertical tangent bundle and the vertical cotangent bundle of ${\displaystyle Y\to \Sigma }$ . Every connection ${\displaystyle A_{\Sigma }}$  on a fiber bundle ${\displaystyle Y\to \Sigma }$  yields the splitting

${\displaystyle A_{\Sigma }:TY\supset VY\ni {\dot {y}}^{i}\partial _{i}+{\dot {\sigma }}^{m}\partial _{m}\to ({\dot {y}}^{i}-A_{m}^{i}{\dot {\sigma }}^{m})\partial _{i}}$ 

of the exact sequence (2). Using this splitting, one can construct a first order differential operator

${\displaystyle {\widetilde {D}}:J^{1}Y\to T^{*}X\otimes _{Y}V_{\Sigma }Y,\qquad {\widetilde {D}}=dx^{\lambda }\otimes (y_{\lambda }^{i}-A_{\lambda }^{i}-A_{m}^{i}\sigma _{\lambda }^{m})\partial _{i},}$ 

on a composite bundle ${\displaystyle Y\to X}$ . It is called the vertical covariant differential. It possesses the following important property.

Let ${\displaystyle h}$  be a section of a fiber bundle ${\displaystyle \Sigma \to X}$ , and let ${\displaystyle h^{*}Y\subset Y}$  be the pullback bundle over ${\displaystyle X}$ . Every connection ${\displaystyle A_{\Sigma }}$  induces the pullback connection

${\displaystyle A_{h}=dx^{\lambda }\otimes [\partial _{\lambda }+((A_{m}^{i}\circ h)\partial _{\lambda }h^{m}+(A\circ h)_{\lambda }^{i})\partial _{i}]}$ 

on ${\displaystyle h^{*}Y}$ . Then the restriction of a vertical covariant differential ${\displaystyle {\widetilde {D}}}$  to ${\displaystyle J^{1}h^{*}Y\subset J^{1}Y}$  coincides with the familiar covariant differential ${\displaystyle D^{A_{h}}}$  on ${\displaystyle h^{*}Y}$  relative to the pullback connection ${\displaystyle A_{h}}$ .

• 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.

• 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

• Connection (mathematics)
• Connection (fibred manifold)