This article is cited in
3 papers
On the higher order geometry of Weil bundles over smooth manifolds and over parameter-dependent manifolds
G. N. Bushueva,
V. V. Shurygin Kazan State University
Abstract:
The Weil bundle
$T^{\mathbb A}M_n$ of an
$n$-dimensional smooth manifold
$M_n$ determined by a local algebra
$\mathbb A$ in the sense of A. Weil carries a natural structure of an
$n$-dimensional
$\mathbb A$-smooth manifold. This allows ones to associate with
$T^{\mathbb A}M_n$ the series
$B^r(\mathbb A)T^{\mathbb A}M_n$,
$r=1,\dots,\infty$, of
$\mathbb A$-smooth
$r$-frame bundles. As a set,
$B^r(\mathbb A)T^{\mathbb A}M_n$ consists of
$r$-jets of
$\mathbb A$-smooth germs of diffeomorphisms
$(\mathbb A^n,0)\to T^{\mathbb A}M_n$. We study the structure of
$\mathbb A$-smooth
$r$-frame bundles. In particular, we introduce the structure form of
$B^r(\mathbb A)T^{\mathbb A}M_n$ and study its properties.
Next we consider some categories of
$m$-parameter-dependent manifolds whose objects are trivial bundles
$M_n\times\mathbb R^m\to\mathbb R^m$, 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.
Keywords:
Weil bundle, product preserving bundle functor, higher order connection. Submitted by: B. N. ShapukovReceived: 14.06.2005
Language: English
Fulltext:
PDF file (388 kB)