Prolongations of isometric actions to vector bundles

Abstract

In this paper, we define an isometry on a total space of a vector bundle E by using a given isometry on thebase manifold M. For this definition, we assume that the total space of the bundle is equipped with a special metricwhich has been introduced in one of our previous papers. We prove that the set of these derived isometries on E forman imbedded Lie subgroup Ge of the isometry group I(E). Using this new subgroup, we construct two different principalbundle structures based one on E and the other on the orbit space E/Ge.