On prolongations of projectable connections

• Autorzy: Jan Kurek , Włodzimierz M. Mikulski
• Języki publikacji: EN

Abstrakty

EN

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 $𝒥^{1,1,1}(Γ,∇): J^{1,1,1}Y → J¹J^{1,1,1}Y$ 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 $𝓡_Γ:J^{1,1,1}Y → T*M ⊗ VJ^{1,1,1}Y$. 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 $𝒥^{1,1,1}(Γ,∇) + t𝓡_Γ$, t ∈ ℝ. A similar classification of all general connections D(Γ,∇):J¹Y → J¹J¹Y on J¹Y → M from (Γ,∇) is presented.

Kategorie tematyczne

• 58A32: Natural bundles
• 58A20: Jets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2011
• Tom: 101
• Numer: 3
• Strony: 237-250

Daty

wydano

2011

Twórcy

autor

Jan Kurek

• Institute of Mathematics, Maria Curie-Skłodowska University, Pl. M. Curie-Skłodowskiej 1, 20-031 Lublin, Poland

autor

Włodzimierz M. Mikulski

• Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

Identyfikatory

DOI

10.4064/ap101-3-4