On Orbit Dimensions under a Simultaneous Lie Group Action on n Copies of a Manifold

Mireille Boutin

Abstract: 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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