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

Abstract

The Weil bundle Tdouble-struck A signMn of an n-dimensional smooth manifold Mn determined by a local algebra double-struck A sign in the sense of A. Weil carries a natural structure of an n-dimensional A-smooth manifold. This allows ones to associate with T double-struck A signMn the series Br(double- struck A sign)Tdouble-struck A signMn, r = 1,...,∞, of double-struck A sign-smooth r-frame bundles. As a set, Br(double- struck A sign)Tdouble-struck A signMn consists of r-jets of double-struck A sign-smooth germs of diffeomorphisms (double-struck A signn, 0) → Tdouble-struck A signMn. We study the structure of double-struck A sign-smooth r-frame bundles. In particular, we introduce the structure form of Br(double-struck A sign)Tdouble-struck A signMn 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.