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.


58A07 Real-analytic and Nash manifolds
58D19 Group actions and symmetry properties
Full Text: DOI EuDML


[1] [G] Grauert, H.:On Levi’s problem and the imbedding of real-analytic manifolds. Ann. of Math,68, 2, 460–472 (1958) · Zbl 0108.07804
[2] [Gr] Grove, K.:Center of mass and G-local triviality ofG-bundles, Proc. AMS54, 352–354 (1976) · Zbl 0293.57018
[3] [Gr, Ka] Grove, K., Karcher, H.:How to conjugate C 1-close group actions. Math. Z.132, 11–20 (19 · Zbl 0245.57016
[4] [H, H, K] Heinzner P., Huckleberry A. T., Kutzschebauch F.:Abels’ Theorem in the real analytic case and applications to complexifications. To appear in Complex Analysis and Geometry, Lecture Notes in Pure and Applied Mathematics, Marcel Decker 1994
[5] [Hi] Hirsch, M.:Differential topology. Berlin Heidelberg New York: Springer (1976) · Zbl 0356.57001
[6] [I] Illman, S.:Every proper smooth action of a Lie group is equivalent to a real analytic action: A contribution to Hilberts fifth problem. Preprint MPI/93-3 Bonn (1993)
[7] [Ka] Karcher, H.:Riemannian Center of Mass and Mollifier Smoothing. Comm. on Pure and Appl. Math.30 509–541 (1977) · Zbl 0354.57005
[8] [K, N] Kobayashi, S., Nomizu, K.:Foundations of Differential Geometry, Vol. II. Interscience Publishers, New York-London-Sydney, (1969) · Zbl 0175.48504
[9] [Ku] Kutzschebauch, F.:Eigentliche Wirkungen von Liegruppen auf reell-analytischen Mannigfaltigkeiten, Schriftenreihe des Graduiertenkollegs Geometrie und Mathematische Physik, Ruhr-Universität-Bochum,5, 1–53 (1994) · Zbl 0840.22017
[10] [M, S] Matumoto, T., Shiota, M.:Unique triangulation of the orbit space of a differentiable transformation group and its applications, Adv. Stud. Pure Math.9, 41–55 (1986) · Zbl 0657.57016
[11] [M, Z] Montgomery, D., Zippin, L.:Topological Transformation Groups, Interscience Publishers, New York and London, (1955) · Zbl 0068.01904
[12] [Pa 1] Palais, R.S.:On the existence of slices for actions of non-compact Lie groups. Ann. of Math.73, 2, 295–323 (1961) · Zbl 0103.01802
[13] [Pa 2] Palais, R.S.:Equivalence of nearby differentiable actions of a compact group. Bull. Amer. math. Soc.67, 362–364 (1961) · Zbl 0102.38101
[14] [W] Whitney, H.:Differentiable manifolds, Ann. of Math.37, 645–680 (1936) · JFM 62.1454.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.