# 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 *^{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 *(A*^{n},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 *M*_{n} × 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.