Pullback of lie algebra and lie group bundles, and their homotopy invariance

We study the pullback Lie algebra (group) bundle of a Lie algebra (group) bundle and show that the Lie algebra bundle of the pullback of a Lie group bundle \mathfrak{G} is isomorphic to the pullback of the Lie algebra bundle of \mathfrak{G}. Then, using the notion of Lie connection on a Lie algebra bundle, we show that the pullbacks of a Lie algebra bundle \xi over a smooth manifold M with respect to two smooth homotopic functions f_۰ , f_۱ : N \rightarrow M are isomorphic to Lie algebra bundles over N.

Keywords:

Lie algebra and Lie group bundles , pullback of bundles , principal bundles , homotopy invariance