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## On prolongation of connections

### Tom 97 / 2010

Annales Polonici Mathematici 97 (2010), 101-121 MSC: 58A05, 58A20, 58A32. DOI: 10.4064/ap97-2-1

#### Streszczenie

Let $Y\to M$ 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 $r$th order holonomic connections $B^r({\mit\Gamma},\nabla):Y\to J^rY$ on $Y\to M$ from general connections ${\mit\Gamma}:Y\to J^1Y$ on $Y\to M$ by means of torsion free classical linear connections $\nabla$ on $M$. Then we prove that any construction $B$ of $r$th order holonomic connections $B({\mit\Gamma},\nabla):Y\to J^rY$ on $Y\to M$ from general connections ${\mit\Gamma}:Y\to J^1Y$ on $Y\to M$ by means of torsion free classical linear connections $\nabla$ on $M$ is equal to $B^r$. Applying $B^r$, for any bundle functor $F:\mathcal F\mathcal M_{m,n}\to\mathcal F\mathcal M$ on fibred $(m,n)$-manifolds we present a construction $\cal F^r_q$ of $r$th order holonomic connections $\cal F^r_q({\mit\Theta},\nabla):FY\to J^r(FY)$ on $FY\to M$ from $q$th order holonomic connections ${\mit\Theta}:Y\to J^qY$ on $Y\to M$ by means of torsion free classical linear connections $\nabla$ on $M$ (for $q=r=1$ we have a well-known classical construction $\cal F({\mit\Gamma},\nabla):FY\to J^1(FY)$). Applying $B^r$ we also construct a so-called $({\mit\Gamma},\nabla)$-lift of a wider class of geometric objects. In Appendix, we present a direct proof of a (recent) result saying that for $r\geq 3$ and $m\geq 2$ there is no construction $A$ of $r$th order holonomic connections $A({\mit\Gamma}):Y\to J^rY$ on $Y\to M$ from general connections ${\mit\Gamma}:Y\to J^1Y$ on $Y\to M$.

#### Autorzy

• Włodzimierz M. MikulskiInstitute of Mathematics
Jagiellonian University
Łojasiewicza 6
30-348 Kraków, Poland
e-mail

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