On the higher order geometry of Weil bundles over smooth manifolds and over parameter-dependent manifolds

(Lobachevskii Journal of Mathematics, Vol.18, pp.53-105 )

The Weil bundle T^A M_n of an n-dimensional smooth manifold M_n determined by a local algebra A in the sense of A. Weil carries a natural structure of an n-dimensional A-smooth manifold. This allows ones to associate with T^A M_n the series B^r(A)T^A M_n , r=1,∞, of A-smooth r-frame bundles. As a set, B^r(A)T^A M_n consists of r-jets of A-smooth germs of diffeomorphisms (Aⁿ,0) → T^A M_n. We study the structure of A-smooth r-frame bundles. In particular, we introduce the structure form of B^r(A)T^A M_n and study its properties.
Next we consider some categories of m-parameter-dependent manifolds whose objects are trivial bundles Mⁿ× R^m→ R^m, define (generalized) Weil bundles and higher order frame bundles of m-parameter-dependent manifolds and study the structure of these bundles. We also show that product preserving bundle functors on the introduced categories of m-parameter-dependent manifolds are equivalent to generalized Weil functors.

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