On the higher order geometry of Weil bundles over smooth manifolds and over parameter-dependent manifolds
(Lobachevskii Journal of Mathematics, Volume XVIII)
The Weil bundle T AMn of an n-dimensional smooth manifold Mn 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 AMn the series B r(A)T AMn, r=1,...,∞, of A-smooth r-frame bundles. As a set, B r(A)T AMn consists of r-jets of A-smooth germs of diffeomorphisms (An,0)→ T AMn. We study the structure of A-smooth r-frame bundles. In particular, we introduce the structure form of B r(A)T AMn and study its properties. Next we consider some categories of m-parameter-dependent manifolds whose objects are trivial bundles Mn × Rm → Rm, 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.