## Galina N. Bushueva and Vadim V. Shurygin

# 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 (*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.

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