Tangent bundles and gauge groups

The differentials $T^ka$ $(k\ge1)$ of a diffeomorphism $a$ of a smooth manifold $M$ induce in the fibers of the fiber bundles $T^kM,$ i.e., in the corresponding tangent spaces, linear transformations, which embody the action of the gauge group $\mathcal G_k$. This action extends in a natural way to the osculating subbundles $\mathrm{Osc}^{k-1}M\subset T^kM$.