Journal of Lie Theory 12 (2002), No. 1, 191--203
Copyright Heldermann Verlag 2002
On Orbit Dimensions under a Simultaneous Lie Group Action on n Copies of a Manifold
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 pseudo-stabilize 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 Olver-Ovsiannikov'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.
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