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Journal of Lie Theory 18 (2008), No. 3, 517--521
Copyright Heldermann Verlag 2008

Closedness of the Tangent Spaces to the Orbits of Proper Actions

Madeleine Jotz
Dép. de Mathématiques, Ecole Polytechnique Fédérale, 1015 Lausanne, Switzerland

Karl-Hermann Neeb
Fachbereich Mathematik, Technische Universität, Schlossgartenstrasse 7, 64289 Darmstadt, Germany


We show that for any proper action of a Banach-Lie group $G$ on a Banach manifold $M$, the corresponding tangent maps ${\frak g} \to T_x(M)$ have closed range for each $x \in M$, i.e., the tangent spaces of the orbits are closed. As a consequence, for each free proper action on a Hilbert manifold, the quotient $M/G$ carries a natural manifold structure.

Keywords: Banach Lie group, Banach manifold, proper action.

MSC: 22E65, 58B25, 57E20

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