On prolongation of connections

Włodzimierz M. Mikulski¹

¹ Institute of Mathematics Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 97 (2010) no. 2, pp. 101-121

Résumé

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 $Y\to M$.

Détail

DOI : 10.4064/ap97-2-1

Mots-clés : fibred manifold m dimensional base n dimensional fibres positive integers present construction rth order holonomic connections mit gamma nabla general connections mit gamma means torsion classical linear connections nabla prove construction rth order holonomic connections mit gamma nabla to general connections mit gamma means torsion classical linear connections nabla equal applying bundle functor mathcal mathcal mathcal mathcal fibred manifolds present construction cal rth order holonomic connections cal mit theta nabla qth order holonomic connections mit theta to means torsion classical linear connections nabla have well known classical construction cal mit gamma nabla applying construct so called mit gamma nabla lift wider class geometric objects appendix present direct proof recent result saying geq geq there construction rth order holonomic connections mit gamma to general connections mit gamma