Lifting vector fields to the $r$th order frame bundle

J. Kurek¹ ; W. M. Mikulski²

¹ Institute of Mathematics Maria Curie-Sk/lodowska University Pl. Marii Curie-Sk/lodowskiej 1 20-031 Lublin, Poland
² Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Colloquium Mathematicum, Tome 111 (2008) no. 1, pp. 51-58.

Résumé

We describe all natural operators $\mathcal{A}$ lifting nowhere vanishing vector fields $X$ on $m$-dimensional manifolds $M$ to vector fields $\mathcal{A}(X)$ on the $r$th order frame bundle $L^rM=\mathop{\rm inv} J^r_0(\mathbb{R}^m, M)$ over~$M$. Next, we describe all natural operators $\mathcal{A}$ lifting vector fields $X$ on $m$-manifolds $M$ to vector fields on $L^rM$. In both cases we deduce that the spaces of all operators $\mathcal{A}$ in question form free $(m(C^{m+r}_r-1)+1)$-dimensional modules over algebras of all smooth maps $J^{r-1}_0\widetilde T\mathbb{R}^m\to\mathbb{R}$ and $J^{r-1}_0T\mathbb{R}^m\to\mathbb{R}$ respectively, where $C^n_k={n!/(n-k)!k!}$. We explicitly construct bases of these modules. In particular, we find that the vector space over $\mathbb{R}$ of all natural linear operators lifting vector fields $X$ on $m$-manifolds $M$ to vector fields on $L^rM$ is $(m^2C^{m+r-1}_{r-1}(C^{m+r}_r-1)+1)$-dimensional.

Détail

DOI : 10.4064/cm111-1-5

Keywords: describe natural operators mathcal lifting nowhere vanishing vector fields m dimensional manifolds vector fields mathcal rth order frame bundle mathop inv mathbb describe natural operators mathcal lifting vector fields m manifolds vector fields cases deduce spaces operators mathcal question form r dimensional modules algebras smooth maps r widetilde mathbb mathbb r mathbb mathbb respectively where n k explicitly construct bases these modules particular vector space mathbb natural linear operators lifting vector fields m manifolds vector fields r r r dimensional