
Journal of Lie Theory 12 (2002), No. 1, 191203 Copyright Heldermann Verlag 2002 On Orbit Dimensions under a Simultaneous Lie Group Action on n Copies of a Manifold Mireille Boutin 127 Vincent Hall, 206 Church Street S.E., Minneapolis, MN 55455, U.S.A. We show that the maximal orbit dimension of a simultaneous Lie group action on n copies of a manifold does not pseudostabilize when n increases. We also show that if a Lie group action is (locally) effective on subsets of a manifold, then the induced Cartesian action is locally free on an open and dense subset of a sufficiently big (but finite) number of copies of the manifold. The latter is the analogue for the Cartesian action to OlverOvsiannikov's theorem on jet bundles and is an important fact relative to the moving frame method and the computation of joint invariants. Some interesting corollaries are presented. [ Fulltextpdf (178 KB)] 