Lifting to the $r$-frame bundle by means of connections

J. Kurek¹ ; W. M. Mikulski²

¹ Institute of Mathematics Maria Skłodowska-Curie University pl. Marii Skłodowskiej-Curie 1 20-031 Lublin, Poland
² Institute of Mathematics Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 97 (2010) no. 1, pp. 63-71.

Résumé

Let $m$ and $r$ be natural numbers and let $P^r:\mathcal M f_m\to\mathcal F\mathcal M$ be the $r$th order frame bundle functor. Let $F:\mathcal M f_m\to\mathcal F\mathcal M$ and $G:\mathcal M f_k\to\mathcal F\mathcal M$ be natural bundles, where $k=\dim (P^r\mathbb R^m)$. We describe all $\mathcal M f_m$-natural operators $A$ transforming sections $\sigma$ of $FM\to M$ and classical linear connections $\nabla$ on $M$ into sections $A(\sigma,\nabla)$ of $G(P^rM)\to P^rM$. We apply this general classification result to many important natural bundles $F$ and $G$ and obtain many particular classifications.

Détail

DOI : 10.4064/ap97-1-5

Keywords: natural numbers mathcal mathcal mathcal rth order frame bundle functor mathcal mathcal mathcal mathcal mathcal mathcal natural bundles where dim mathbb describe mathcal m natural operators transforming sections sigma classical linear connections nabla sections sigma nabla apply general classification result many important natural bundles and obtain many particular classifications