Lifting vector fields to the rth order frame bundle

• Autorzy: J. Kurek , W. M. Mikulski
• Języki publikacji: EN

Abstrakty

EN

We describe all natural operators 𝓐 lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields 𝓐(X) on the rth order frame bundle $L^{r}M = inv J₀^{r}(ℝ^m, M)$ over M. Next, we describe all natural operators 𝓐 lifting vector fields X on m-manifolds M to vector fields on $L^{r}M$. In both cases we deduce that the spaces of all operators 𝓐 in question form free $(m(C^{m+r}_{r}-1) + 1)$-dimensional modules over algebras of all smooth maps $J₀^{r-1}T̃ℝ^m → ℝ$ and $J₀^{r-1}Tℝ^m → ℝ$ respectively, where $Cⁿ_k = n!/(n-k)!k!$. We explicitly construct bases of these modules. In particular, we find that the vector space over ℝ of all natural linear operators lifting vector fields X on m-manifolds M to vector fields on $L^{r}M$ is $(m²C^{m+r-1}_{r-1}(C^{m+r}_r - 1) + 1)$-dimensional.

Kategorie tematyczne

• 58A20: Jets

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2008
• Tom: 111
• Numer: 1
• Strony: 51-58

Daty

wydano

2008

Twórcy

autor

J. Kurek

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

autor

W. M. Mikulski

• Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

Identyfikatory

DOI

10.4064/cm111-1-5