Canonical 1-forms on higher order adapted frame bundles

Jan Kurek and Włodzimierz M. Mikulski

Address:
Institute of Mathematics, Maria Curie-Sklodowska Univesity Pl. M. Curie-Sklodowskiej 1, Lublin, Poland
Institute of Mathematics, Jagiellonian University ul. Reymonta 4, Kraków, Poland

E-mail:
kurek@hektor.umcs.lublin.pl
mikulski@im.uj.edu.pl

Abstract: Let $(M,\mathcal{F})$ be a foliated $m+n$-dimensional manifold $M$ with $n$-dimensional foliation $\mathcal{F}$. Let $V$ be a finite dimensional vector space over $\mathbf{R}$. We describe all canonical (${\mathcal{F}}\mbox {\it ol}_{m,n}$-invariant) $V$-valued $1$-forms $\Theta \colon TP^r(M,{\mathcal{F}}) \rightarrow V$ on the $r$-th order adapted frame bundle $P^r(M,\mathcal{F})$ of $(M,\mathcal{F})$.

AMSclassification: Primary: 58A20; Secondary: 58A32.

Keywords: foliated manifold, infinitesimal automorphism, higher order adapted frame bundle, canonical $1$-form