Proper smooth \(G\)-manifolds have complete \(G\)-invariant Riemannian metrics
The main result of the paper states that if \(G\) is a (not necessarily compact) Lie group and \(M\) is a proper smooth \(G\)-manifold then there exists a smooth \(G\)-equivariant embedding of \(M\) into some Hilbert \(G\)-space as a closed submanifold. Such a result in the topological case is classical; it was proved by R. S. Palais in early 60s. In the proof regular representations \(L^2(G)\) and \(L^2(K)\), where \(K\) is a compact subgroup of \(G\), are considered, and results about extending some representations and embedding homogeneous spaces smoothly into Hilbert \(G\)-spaces are shown. As a consequence of the main theorem it is proved that any proper smooth \(G\)-manifold admits a complete \(G\)-invariant smooth Riemannian metric.
scientific article; zbMATH DE number 5012980