Three basic results for real analytic proper \(G\)-manifolds
The authors call a Lie group \(G\) ``good'' if it can be embedded as a closed subgroup into a Lie group with finitely many connected components. This class of Lie groups contains in particular all linear groups. They prove a number of results on real-analytic actions of ``good'' Lie groups. \textit{F. Kutzschebauch} [Eigentliche Wirkungen von Liegruppen auf reell-analytischen Mannigfaltigkeiten, Schriftenreihe des Graduiertenkollegs Geometrie u. Mathem. Phys. 5 (1994; Zbl 0840.22017)] proved that given a Lie group \(G\) with finitely many connected components, and a real-analytic proper \(G\)-manifold \(M\), there exists a \(G\)-invariant real-analytic Riemannian metric. In the article under review this is generalized to the case where \(G\) is a ``good'' Lie group with possibly infinitely many connected components. \textit{F. Kutzschebauch} [loc. cit.] also proved that under the same conditions \(M\) admits a \(G\)-equivariant real analytic linearization if the number of isotropy types is finite. This result is also generalized, from the case of ``finitely many connected components'' to the case where \(G\) is an arbitrary linear Lie group. By a result of \textit{T. Matumoto} and \textit{M. Shiota} [Adv. Stud. Pure Math. 9, 41-55 (1987; Zbl 0657.57016)] it has been known that for a compact Lie group \(G\) acting on real analytic manifolds \(M\) and \(N\) analytically, the space of real-analytic equivariant maps is dense in the space of \(C^r\)-equivariant maps with respect to the strong (Whitney) topology. This cannot be directly generalized to arbitrary proper actions by non-compact groups, because for a non-compact Lie group acting properly on manifolds \(M\) and \(N\) the space of equivariant maps is discrete in the strong (Whitney) topology. For this reason the authors developed a new topology, called ``strong-weak'' topology, see [Math. Ann. 316, No.~1, 139-168 (2000; Zbl 0940.57031]. The authors show that using this topology the aforementioned approximation result generalizes, i.e., given a ``good'' Lie group acting properly and analytically on real-analytic manifolds \(M\) and \(N\) every \(G\)-equivariant \(C^r\)-map can be approximated (with respect to the ``strong-weak''-topology) by a real-analytic equivariant map. For actions of compact Lie groups the ``strong-weak'' topology reduces to the strong topology, thus this is indeed a generalization.
scientific article; zbMATH DE number 1406575