Geodesic
On Orbit Dimensions under a Simultaneous Lie Group Action on n Copies of a Manifold
Journal of Lie theory, Tome 12 (2002) no. 1, pp. 191-203 Cet article a éte moissonné depuis la source Heldermann Verlag

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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. 
@article{JLT_2002_12_1_JLT_2002_12_1_a8,
     author = {M. Boutin },
     title = {On {Orbit} {Dimensions} under a {Simultaneous} {Lie} {Group} {Action} on n {Copies} of a {Manifold}},
     journal = {Journal of Lie theory},
     pages = {191--203},
     year = {2002},
     volume = {12},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JLT_2002_12_1_JLT_2002_12_1_a8/}
}
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M. Boutin . On Orbit Dimensions under a Simultaneous Lie Group Action on n Copies of a Manifold. Journal of Lie theory, Tome 12 (2002) no. 1, pp. 191-203. http://geodesic.mathdoc.fr/item/JLT_2002_12_1_JLT_2002_12_1_a8/