Timeline for When do isometric actions exist?
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| Feb 28, 2011 at 17:54 | comment | added | Dick Palais | No, I'm pretty sure that you can use the fact that locally compact groups are suitable limits of Lie groups to get the result for the general case. My interest in that paper was primarily in proving the important slice theorem, and for that one does need smoothness. BTW, that paper was probably the first place where the concept of a proper group action appears in a published paper. I did NOT invent the term. In fact I worked primarily with a more general concept I called a Cartan G-space. It was Armand Borel who told me he had been investigating proper actions and suggested I use that term. | |
| Feb 28, 2011 at 6:52 | comment | added | Kamran Reihani | Thanks, Dick! But do we have to be in the smooth category to have this fact? As is stated above, using "cut-off" (or generalized Bruhat) functions, one can easily show that the $G$-space $X$ admits an invariant metric inducing the same topology on $X$, where $G$ is a locally compact group, which acts properly on the locally compact metrizable space $X$. | |
| Feb 28, 2011 at 6:32 | history | edited | Dick Palais | CC BY-SA 2.5 |
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| Feb 28, 2011 at 6:22 | history | answered | Dick Palais | CC BY-SA 2.5 |