## On the uniqueness of the analyticity of a proper $$G$$-action.(English)Zbl 0858.58001

The author proves the following theorem: Let $$G$$ be a Lie group with only finitely many components. Furthermore, let $$G$$ act real-analytically and properly on the real-analytical manifolds $$X$$ and $$Y$$. If there is a smooth $$G$$-equivariant diffeomorphism from $$X$$ to $$Y$$ then there is a real-analytic $$G$$-equivariant diffeomorphism from $$X$$ to $$Y$$. The proof involves the existence of a global slice for the action of $$G$$ and the use of a nonlinear averaging process called the center of mass construction.

### MSC:

 58A07 Real-analytic and Nash manifolds 58D19 Group actions and symmetry properties
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### References:

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