On prolongation of connections

• Autorzy: Włodzimierz M. Mikulski
• Języki publikacji: EN

Abstrakty

EN

Let Y → M be a fibred manifold with m-dimensional base and n-dimensional fibres. Let r, m,n be positive integers. We present a construction $B^r$ of rth order holonomic connections $B^r(Γ,∇):Y → J^rY$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M. Then we prove that any construction B of rth order holonomic connections $B(Γ,∇):Y → J^rY$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M is equal to $B^r$. Applying $B^r$, for any bundle functor $F:ℱℳ_{m,n} →ℱℳ$ on fibred (m,n)-manifolds we present a construction $ℱ^r_q$ of rth order holonomic connections $ℱ^r_q(Θ,∇):FY → J^r(FY)$ on FY → M from qth order holonomic connections $Θ:Y → J^qY$ on Y → M by means of torsion free classical linear connections ∇ on M (for q=r=1 we have a well-known classical construction ℱ(Γ,∇):FY → J¹(FY)). Applying $B^r$ we also construct a so-called (Γ,∇)-lift of a wider class of geometric objects. In Appendix, we present a direct proof of a (recent) result saying that for r ≥ 3 and m ≥ 2 there is no construction A of rth order holonomic connections $A(Γ):Y → J^rY$ on Y → M from general connections Γ:Y → J¹Y on Y → M.

Kategorie tematyczne

• 58A05: Differentiable manifolds, foundations
• 58A32: Natural bundles
• 58A20: Jets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2010
• Tom: 97
• Numer: 2
• Strony: 101-121

Daty

wydano

2010

Twórcy

autor

Włodzimierz M. Mikulski

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

Identyfikatory

DOI

10.4064/ap97-2-1