The jet prolongations of $2$-fibred manifolds and the flow operator

Włodzimierz M. Mikulski

Address: Institute of Mathematics, Jagellonian University Reymonta 4, Kraków, Poland


Abstract: Let $r$, $s$, $m$, $n$, $q$ be natural numbers such that $s\ge r$. We prove that any $2$-${\mathcal{F}}\mathbb{M}_{m,n,q}$-natural operator $A\colon T_{\operatorname{2-proj}}\rightsquigarrow TJ^{(s,r)}$ transforming $2$-projectable vector fields $V$ on $(m,n,q)$-dimensional $2$-fibred manifolds $Y\rightarrow X\rightarrow M$ into vector fields $A(V)$ on the $(s,r)$-jet prolongation bundle $J^{(s,r)}Y$ is a constant multiple of the flow operator $\mathcal{J}^{(s,r)}$.

AMSclassification: Primary: 58A20.

Keywords: $(s,r)$-jet, bundle functor, natural operator, flow operator, $2$-fibred manifold, $2$-projectable vector field