Proper smooth \(G\)-manifolds have complete \(G\)-invariant Riemannian metrics
type: Item
instance of: scholarly article
title: Proper smooth \(G\)-manifolds have complete \(G\)-invariant Riemannian metrics
zbMATH Open document ID: 1090.57022
author: Marja Kankaanrinta
published in: Topology and its Applications
publication date: 2006-03-16T00:00:00Z
review text: 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.
reviewed by: Tomasz Rybicki
Mathematics Subject Classification ID: 57S20
Mathematics Subject Classification ID: 57R91
zbMATH DE Number: 5012980
zbMATH Keywords: proper
zbMATH Keywords: Lie group
zbMATH Keywords: Hilbert space
zbMATH Keywords: Riemannian metric
zbMATH Keywords: regular representation
zbMATH Keywords: smooth vector
Wikidata QID: Q115340853
MaRDI profile type: Publication
full work available at URL: j.topol.2005.01.034
OpenAlex ID: W2082450604
cites work: Q5658847
cites work: Q5565773
cites work: Q5723435
cites work: Q5615626
cites work: Equivariant embeddings in euclidean space
cites work: Q3257175
cites work: On the existence of slices for actions of non-compact Lie groups
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DOI: 10.1016/J.TOPOL.2005.01.034
description: scientific article; zbMATH DE number 5012980