ON THE NATURAL OPERATORS OF VERTICAL PROLONGATION TYPE ON 1 - CONNECTION BUNDLE

Jan Kurek
1990 Demonstratio Mathematica  
Recently R, Matthes introduced a conoept of an r-connection on fibred manifold as generalization of the classical connections [5] . In this paper we consider the vertioal prolongation of 1-connections on fibred manifold Y -• M onto the vertical bundle VY -M by flows. We prove that the only natural affine operator transforming 1-connections on fibred manifold Y onto 1-connections on the vertical bundle VY is the vertical prolongation by flows. In the paper, we use a general method for finding
more » ... thod for finding all natural operators of certain types elaborated by I. Kolar in The author is grateful to Professor I. Kolar for suggesting the problem, valuable remarks and useful discussions and to Dr J. Slovak for private communication. 1. Let 3T : Y--M be a fibred manifold, dim Y = n + m, dim M = n. Denote by (3 M i J 1 TM -» TM the target jet projection of the first jet prolongation and by py i TY Y the tangent bundle projection. Definition 1 , [5j. 1-connection on fibred manifold JT < Y-is a map (1 ) r i J 1 TM © Y -• TY -161 -
doi:10.1515/dema-1990-0117 fatcat:hokofzoo3bf7tcfrkipdkofni4