The natural affinors on generalized higher order tangent bundles. (English) Zbl 1048.58004

Mikulski, W. M.

Rend. Mat. Appl., VII. Ser. 21, No. 1-4, 339-349 (2001).

The action \(\alpha^{(a)}:\text{GL}(n,\mathbb{R})\times \mathbb{R}\to \mathbb{R},\;\alpha^{(a)}(B,x)=| \det (B)| ^a x\) defines the natural vector bundle \(T^{(0,0),a}M=LM\times _{\alpha^{(a)}}{\mathbb{R}}\) associated to the principal bundle \(LM\) of linear frames on the smooth manifold \(M\). Let \(T^{r*,a}M\) be the vector bundle of all \(r\)-jets of local sections in \(T^{(0,0),a}M\), with target \(0\). Then \(T^{(r),a}M = (T^{r*,a}M)^*\), the dual vector bundle of \(T^{r*,a}M\), is the generalized tangent bundle of order \(r\) of \(M\).
For \(r\geq 1,\;n\geq 2\) and a real number \(a<0\), the author obtains a classification of the affinors of \(T^{(r),a}M\), as well as of the affinors on the extended generalized higher order tangent bundle \(E^{(r),a}\) of \(M\).

Reviewer: Vasile Oproiu (Iaşi)

MSC:

58A32	Natural bundles
58A20	Jets in global analysis
53A55	Differential invariants (local theory), geometric objects

Keywords:

natural bundle; natural transformation; natural affinors