On lifting of connections to Weil bundles

Jan Kurek¹ ; Włodzimierz M. Mikulski²

¹Institute of Mathematics Maria Curie-Skłodowska University Pl. Marii Curie-Skłodowskiej 1 20-031 Lublin, Poland
²Chair of Geometry Faculty of Mathematics and Computer Science Jagiellonian University Łojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 103 (2012) no. 3, pp. 319-324

Résumé

We prove that the problem of finding all $\mathcal M f_m$-natural operators $B:Q\rightsquigarrow QT^A$ lifting classical linear connections $\nabla$ on $m$-manifolds $M$ to classical linear connections $B_M(\nabla)$ on the Weil bundle $T^AM$ corresponding to a $p$-dimensional (over $\mathbb R$) Weil algebra $A$ is equivalent to the one of finding all $\mathcal M f_m$-natural operators $C:Q\rightsquigarrow (T^1_{p-1},T^*\otimes T^*\otimes T)$ transforming classical linear connections $\nabla$ on $m$-manifolds $M$ into base-preserving fibred maps $C_M(\nabla):T^1_{p-1}M=\bigoplus^{p-1}_MTM\to T^*M\otimes T^*M\otimes TM$.

Détail

DOI : 10.4064/ap103-3-7

Keywords: prove problem finding mathcal m natural operators rightsquigarrow lifting classical linear connections nabla m manifolds classical linear connections nabla weil bundle corresponding p dimensional mathbb weil algebra equivalent finding mathcal m natural operators rightsquigarrow p * otimes * otimes transforming classical linear connections nabla m manifolds base preserving fibred maps nabla p bigoplus p mtm *m otimes *m otimes