On prolongations of projectable connections

Jan Kurek, and Włodzimierz M. Mikulski. "On prolongations of projectable connections." Annales Polonici Mathematici 101.3 (2011): 237-250. <http://eudml.org/doc/280202>.

@article{JanKurek2011,
abstract = {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 ≥ r ≤ q. Similarly to the fibred manifold case, given a bundle functor F of order r on (m₁,m₂,n₁,n₂)-dimensional fibred-fibred manifolds Y → M, we construct a general connection ℱ(Γ,Λ):FY → J¹FY on FY → M from a projectable general (i.e. (1,1,1)-order) connection $Γ:Y → J^\{1,1,1\}Y$ on Y → M by means of an (r,r,r)-order projectable linear connection $Λ:TM → J^\{r,r,r\}TM$ on M. In particular, for $F = J^\{1,1,1\}$ we construct a general connection $^\{1,1,1\}(Γ,∇): J^\{1,1,1\}Y → J¹J^\{1,1,1\}Y$ on $J^\{1,1,1\}Y → M$ from a projectable general connection Γ on Y → M by means of a torsion-free projectable classical linear connection ∇ on M. Next, we observe that the curvature of Γ can be considered as $_Γ:J^\{1,1,1\}Y → T*M ⊗ VJ^\{1,1,1\}Y$. The main result is that if m₁ ≥ 2 and n₂ ≥ 1, then all general connections $D(Γ,∇):J^\{1,1,1\}Y → J¹J^\{1,1,1\}Y$ on $J^\{1,1,1\}Y → M$ canonically depending on Γ and ∇ form the one-parameter family $^\{1,1,1\}(Γ,∇) + t_Γ$, t ∈ ℝ. A similar classification of all general connections D(Γ,∇):J¹Y → J¹J¹Y on J¹Y → M from (Γ,∇) is presented.},
author = {Jan Kurek, Włodzimierz M. Mikulski},
journal = {Annales Polonici Mathematici},
keywords = {fibered-fibered manifolds; the -prolongation; -order projectable connection; projectable general connection; natural operator},
language = {eng},
number = {3},
pages = {237-250},
title = {On prolongations of projectable connections},
url = {http://eudml.org/doc/280202},
volume = {101},
year = {2011},
}

TY - JOUR
AU - Jan Kurek
AU - Włodzimierz M. Mikulski
TI - On prolongations of projectable connections
JO - Annales Polonici Mathematici
PY - 2011
VL - 101
IS - 3
SP - 237
EP - 250
AB - 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 ≥ r ≤ q. Similarly to the fibred manifold case, given a bundle functor F of order r on (m₁,m₂,n₁,n₂)-dimensional fibred-fibred manifolds Y → M, we construct a general connection ℱ(Γ,Λ):FY → J¹FY on FY → M from a projectable general (i.e. (1,1,1)-order) connection $Γ:Y → J^{1,1,1}Y$ on Y → M by means of an (r,r,r)-order projectable linear connection $Λ:TM → J^{r,r,r}TM$ on M. In particular, for $F = J^{1,1,1}$ we construct a general connection $^{1,1,1}(Γ,∇): J^{1,1,1}Y → J¹J^{1,1,1}Y$ on $J^{1,1,1}Y → M$ from a projectable general connection Γ on Y → M by means of a torsion-free projectable classical linear connection ∇ on M. Next, we observe that the curvature of Γ can be considered as $_Γ:J^{1,1,1}Y → T*M ⊗ VJ^{1,1,1}Y$. The main result is that if m₁ ≥ 2 and n₂ ≥ 1, then all general connections $D(Γ,∇):J^{1,1,1}Y → J¹J^{1,1,1}Y$ on $J^{1,1,1}Y → M$ canonically depending on Γ and ∇ form the one-parameter family $^{1,1,1}(Γ,∇) + t_Γ$, t ∈ ℝ. A similar classification of all general connections D(Γ,∇):J¹Y → J¹J¹Y on J¹Y → M from (Γ,∇) is presented.
LA - eng
KW - fibered-fibered manifolds; the -prolongation; -order projectable connection; projectable general connection; natural operator
UR - http://eudml.org/doc/280202
ER -