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 the base manifold M . For this definition, we assume that the total space of the bundle is equipped with a special metric which has been introduced in one of our previous papers. We prove that the set of these derived isometrics on E form an imbedded Lie subgroup (G) over tilde of the isometry group I(E) . Using this new subgroup, we construct two different principal bundle structures based one on E and the other on the orbit space E/(G) over tilde.