Connection (composite bundle) について

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Connection (composite bundle) ：ウィキペディア英語版

Connection (composite bundle)
Composite bundles

Y\to \Sigma \to X

X=\mathbb R

is the time axis, e.g., mechanics with time-dependent parameters, and so on. There are the important relations between connections on fiber bundles

Y\to X

Y\to \Sigma

and

\Sigma\to X

.
==Composite bundle==

In differential geometry by a composite bundle is meant the composition
:

\pi: Y\to \Sigma\to X  \qquad\qquad (1)

of fiber bundles
:

\pi_: Y\to\Sigma, \qquad  \pi_: \Sigma\to X.

It is provided with bundle coordinates

(x^\lambda,\sigma^m,y^i)

, where

(x^\lambda,\sigma^m)

are bundle coordinates on a fiber bundle

\Sigma\to X

, i.e., transition functions of coordinates

\sigma^m

are independent of coordinates

y^i

.
The following fact provides the above mentioned physical applications of composite bundles. Given the composite bundle (1), let

h

be a global section
of a fiber bundle

\Sigma\to X

, if any. Then the pullback bundle

Y^h=h^*Y

over

X

is a subbundle of a fiber bundle

Y\to X

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