Lifting right-invariant vector fields and prolongation of connections

W. M. Mikulski

Annales Polonici Mathematici (2009)

  • Volume: 95, Issue: 3, page 243-252
  • ISSN: 0066-2216

Abstract

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We describe all m ( G ) -gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation W r P = P r M × M J r P of P → M. In other words, we classify all m ( G ) -natural transformations J r L P × M W r P → T W r P = L W r P × M W r P covering the identity of W r P , where J r L P is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all m ( G ) -natural transformations which are similar to the Kumpera-Spencer isomorphism J r L P = L W r P . We formulate axioms which characterize the flow operator of the gauge-bundle W r P → M . We apply the flow operator to prolongations of connections.

How to cite

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W. M. Mikulski. "Lifting right-invariant vector fields and prolongation of connections." Annales Polonici Mathematici 95.3 (2009): 243-252. <http://eudml.org/doc/280786>.

@article{W2009,
abstract = {We describe all $_m(G)$-gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation $W^rP=P^rM×_MJ^rP$ of P → M. In other words, we classify all $_m(G)$-natural transformations $J^rLP×_M W^rP→ TW^rP=LW^rP×_MW^rP$ covering the identity of $W^rP$, where $J^rLP$ is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all $_m(G)$-natural transformations which are similar to the Kumpera-Spencer isomorphism $J^rLP=LW^rP$. We formulate axioms which characterize the flow operator of the gauge-bundle $W^rP → M$. We apply the flow operator to prolongations of connections.},
author = {W. M. Mikulski},
journal = {Annales Polonici Mathematici},
keywords = {higher order principal prolongation; principal connection; Lie algebroid; gauge natural operator},
language = {eng},
number = {3},
pages = {243-252},
title = {Lifting right-invariant vector fields and prolongation of connections},
url = {http://eudml.org/doc/280786},
volume = {95},
year = {2009},
}

TY - JOUR
AU - W. M. Mikulski
TI - Lifting right-invariant vector fields and prolongation of connections
JO - Annales Polonici Mathematici
PY - 2009
VL - 95
IS - 3
SP - 243
EP - 252
AB - We describe all $_m(G)$-gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation $W^rP=P^rM×_MJ^rP$ of P → M. In other words, we classify all $_m(G)$-natural transformations $J^rLP×_M W^rP→ TW^rP=LW^rP×_MW^rP$ covering the identity of $W^rP$, where $J^rLP$ is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all $_m(G)$-natural transformations which are similar to the Kumpera-Spencer isomorphism $J^rLP=LW^rP$. We formulate axioms which characterize the flow operator of the gauge-bundle $W^rP → M$. We apply the flow operator to prolongations of connections.
LA - eng
KW - higher order principal prolongation; principal connection; Lie algebroid; gauge natural operator
UR - http://eudml.org/doc/280786
ER -

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