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Tangent bundles and gauge groups. (English) Zbl 1351.58005

Let \(T^kM\) be the \(k\)th order iterated tangent bundle of an \(n\)-dimensional manifold \(M\). The authors show that the differentials \(T^ka\) of a diffeomorphism \(a\) of a smooth manifold \(M\) induce in the fibers of \(T^kM\) linear transformations, which embody the action of the \(k\)th order gauge group \(G_k\approx T^k(\mathrm{GL}(n,\mathbb R))\subset\mathrm{GL}(2^kn,\mathbb R)\). This action extends in a natural way to the osculating subbundles \(\mathrm{Osc}^{k-1}M\subset T^kM\).

MSC:

58A20 Jets in global analysis
58D17 Manifolds of metrics (especially Riemannian)
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