A construction of a connection on $GY\to Y$ from a connection on $Y\to M$ by means of classical linear connections on $M$ and $Y$

Mikulski, W. M.

Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 759-770.

Résumé

Let $G$ be a bundle functor of order $(r,s,q)$, $s\geq r\leq q$, on the category $\Cal F\Cal M_{m,n}$ of $(m,n)$-dimensional fibered manifolds and local fibered diffeomorphisms. Given a general connection $\Gamma$ on an $\Cal F\Cal M_{m,n}$-object $Y\to M$ we construct a general connection $\Cal G(\Gamma,\lambda,\Lambda)$ on $GY\to Y$ be means of an auxiliary $q$-th order linear connection $\lambda$ on $M$ and an $s$-th order linear connection $\Lambda$ on $Y$. Then we construct a general connection $\Cal G (\Gamma,\nabla_1,\nabla_2)$ on $GY\to Y$ by means of auxiliary classical linear connections $\nabla_1$ on $M$ and $\nabla_2$ on $Y$. In the case $G=J^1$ we determine all general connections $\Cal D(\Gamma,\nabla)$ on $J^1Y\to Y$ from general connections $\Gamma$ on $Y\to M$ by means of torsion free projectable classical linear connections $\nabla$ on $Y$.

Détail

Classification : 58A05, 58A20, 58A32
Keywords: general connection; classical linear connection; bundle functor; natural operator