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 $Y$
Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 759-770
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library
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 $Y$.
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 $Y$.
Classification :
58A05, 58A20, 58A32
Keywords: general connection; classical linear connection; bundle functor; natural operator
Keywords: general connection; classical linear connection; bundle functor; natural operator
@article{CMUC_2005_46_4_a14,
author = {Mikulski, W. M.},
title = {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 $Y$},
journal = {Commentationes Mathematicae Universitatis Carolinae},
pages = {759--770},
year = {2005},
volume = {46},
number = {4},
mrnumber = {2259506},
zbl = {1121.58001},
language = {en},
url = {http://geodesic.mathdoc.fr/item/CMUC_2005_46_4_a14/}
}
TY - JOUR AU - Mikulski, W. M. TI - 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 $Y$ JO - Commentationes Mathematicae Universitatis Carolinae PY - 2005 SP - 759 EP - 770 VL - 46 IS - 4 UR - http://geodesic.mathdoc.fr/item/CMUC_2005_46_4_a14/ LA - en ID - CMUC_2005_46_4_a14 ER -
%0 Journal Article %A Mikulski, W. M. %T 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 $Y$ %J Commentationes Mathematicae Universitatis Carolinae %D 2005 %P 759-770 %V 46 %N 4 %U http://geodesic.mathdoc.fr/item/CMUC_2005_46_4_a14/ %G en %F CMUC_2005_46_4_a14
Mikulski, W. M. 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 $Y$. Commentationes Mathematicae Universitatis Carolinae, Tome 46 (2005) no. 4, pp. 759-770. http://geodesic.mathdoc.fr/item/CMUC_2005_46_4_a14/