On prolongations of projectable connections

Jan Kurek¹ ; Włodzimierz M. Mikulski²

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

Annales Polonici Mathematici, Tome 101 (2011) no. 3, pp. 237-250.

Résumé

We extend the concept of $r$-order connections on fibred manifolds to the one of $(r,s,q)$-order projectable connections on fibred-fibred manifolds, where $r,s,q$ are arbitrary non-negative integers with $s\geq r\leq q$. Similarly to the fibred manifold case, given a bundle functor $F$ of order $r$ on $(m_1,m_2,n_1,n_2)$-dimensional fibred-fibred manifolds $Y\to M$, we construct a general connection $\mathcal{F}({\mit\Gamma},{\mit\Lambda}):FY\to J^1FY$ on $FY\to M$ from a projectable general (i.e. $(1,1,1)$-order) connection ${\mit\Gamma}:Y\to J^{1,1,1}Y$ on $Y\to M$ by means of an $(r,r,r)$-order projectable linear connection ${\mit\Lambda}:TM\to J^{r,r,r}TM$ on $M$. In particular, for $F=J^{1,1,1}$ we construct a general connection $\mathcal{J}^{1,1,1}({\mit\Gamma},\nabla): J^{1,1,1}Y\to J^1J^{1,1,1}Y$ on $J^{1,1,1}Y\to M$ from a projectable general connection ${\mit\Gamma}$ on $Y\to M$ by means of a torsion-free projectable classical linear connection $\nabla$ on $M$. Next, we observe that the curvature of ${\mit\Gamma}$ can be considered as $\mathcal{R}_{\mit\Gamma}:J^{1,1,1}Y\to T^*M\otimes VJ^{1,1,1}Y$. The main result is that if $m_1\geq 2$ and $n_2\geq 1$, then all general connections $D({\mit\Gamma},\nabla):J^{1,1,1}Y\to J^1J^{1,1,1}Y$ on $J^{1,1,1}Y\to M$ canonically depending on ${\mit\Gamma}$ and $\nabla$ form the one-parameter family $\mathcal{J}^{1,1,1}({\mit\Gamma},\nabla)+t\mathcal{R}_{\mit\Gamma}$, $t\in\mathbb{R}$. A similar classification of all general connections $D({\mit\Gamma},\nabla):J^1Y\to J^1J^1Y$ on $J^1Y\to M$ from $({\mit\Gamma}, \nabla)$ is presented.

Détail

DOI : 10.4064/ap101-3-4

Mots-clés : extend concept r order connections fibred manifolds order projectable connections fibred fibred manifolds where arbitrary non negative integers geq leq similarly fibred manifold given bundle functor order dimensional fibred fibred manifolds construct general connection mathcal mit gamma mit lambda to projectable general order connection mit gamma means order projectable linear connection mit lambda particular construct general connection mathcal mit gamma nabla projectable general connection mit gamma means torsion free projectable classical linear connection nabla observe curvature mit gamma considered mathcal mit gamma *m otimes main result geq geq general connections mit gamma nabla canonically depending mit gamma nabla form one parameter family mathcal mit gamma nabla mathcal mit gamma mathbb similar classification general connections mit gamma nabla mit gamma nabla presented