×

Geometrical objects on subbundles. (English) Zbl 0903.57015

Let \(\xi= (E,\pi,M)\) be a vector bundle with the fibre \(F\cong \mathbb{R}^n\) and let \(L(\xi)= (L(E),p,M)\) be the principal bundle of the frames of \(\xi\). If \(G\subset GL(n, \mathbb{R})\) is a Lie subgroup and \(L(\xi)_G\) is a reduction of the group \(GL(n, \mathbb{R})\) of \(\xi\) to \(G\), then the local trivial bundle \(\xi_G\), associated with \(L(\xi)_G\), is called the \(G\)-reduced bundle of \(\xi\). Especially, the authors study reductions of the group \(G^r_{m,n}\), which is the \(r\)-prolongation of the group \(G^1_{m,n}\), consisting of the matrices of the form \(\left( \begin{smallmatrix} A & 0\\ 0 & B\end{smallmatrix} \right)\) with \(A\in GL(m,\mathbb{R})\) and \(B\in G\). They relate them to Finsler splittings [M. Popescu, On Finsler bundles and Finsler connections, The \(25^{th}\) National Conference of Geometry and Topology, Iassy, 1995 (to appear in Analele Univ. Iassy)] and to fields of geometrical \(G\)-objects on the vector bundle \(\xi\), which are defined by the authors as sections in a certain fibre bundle, associated with the principal bundle \(OG\xi^r\) on the base \(E\) with the group \(G^r_{m,n}\).

MSC:

57R22 Topology of vector bundles and fiber bundles
53C99 Global differential geometry
PDFBibTeX XMLCite
Full Text: EuDML EMIS